arXiv:1407.0445v3 [math.DG] 22 May 2017

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC
SPACES
DAVID FISHER AND KEVIN WHYTE
Abstract. This paper opens the study of quasi-isometric embeddings
of symmetric spaces. The main focus is on the case of equal and higher
rank. In this context some expected rigidity survives, but some surprising examples also exist. In particular there exist quasi-isometric
embeddings between spaces X and Y where there is no isometric embedding of X in Y . A key ingredient in our proofs of rigidity results
is a direct generalization of the Mostow-Morse Lemma in higher rank.
Typically this lemma is replaced by the quasi-flat theorem which says
that maximal quasi-flat is within bounded distance of a finite union of
flats. We improve this by showing that the quasi-flat is in fact flat off
of a subset of codimension 2.

1. Introduction
The rigidity theorems of Mostow and Margulis are among the most celebrated results about the intersection of discrete groups and geometry. With
the rise of Gromov’s program for the geometric study of discrete groups,
coarse analogues of these results were among the most desired results [G].
There are many possible translations of these theorems to a coarse setting,
and so results and questions in this direction (see [F] for a good survey).
We first recall two basic definitions:
Definition 1.1. Let (X, dX ) and (Y, dY ) be metric spaces. Given real numbers K≥1 and C≥0,a map f : X→Y is called a (K, C)-quasi-isometry if
(1)

1
K dX (x1 , x2 ) − C≤dY
x2 in X, and,

(f (x1 ), f (x2 ))≤KdX (x1 , x2 ) + C for all x1 and

(2) the C neighborhood of f (X) is all of Y .
If f satisfies (1) but not (2), then f is called a (K, C)-quasi-isometric embedding.
Remark: Throughout this paper, all symmetric spaces will have no compact or Euclidean factors.
First author partially supported by NSF grant DMS-1308291. Second author partially
supported by NSF Grant DMS-1007236. The authors would also like to thank the FIM
at ETHZ for hospitality and support at several points during the development of these
ideas.
1

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DAVID FISHER AND KEVIN WHYTE

To quickly summarize the current state of knowledge, good analogues of
Mostow rigidity are now known in the coarse setting. In particular, from
the work of many people, one can deduce:
Theorem 1.2. Two symmetric spaces are quasi-isometric if and only if they
are isometric (after a possible rescaling). Further, any finitely generated
group quasi-isometric to a symmetric space is virtually a cocompact lattice
in its isometry group.
See [F] for a detailed discussion and attribution. This theorem is not quite
as strong as one would like to parallel Mostow rigidity - in particular, self
quasi-isometries of symmetric spaces or quasi-isometries between cocompact
lattices in the same semi-simple Lie group can be quite wild, while Mostow’s
result says group isomorphisms are induced by isometries. That is simply
the truth for Hn and CHn . For the other irreducible symmetric spaces, by
results of Kleiner-Leeb and Pansu one has more [KL1, P]:
Theorem 1.3. Let X be an irreducible symmetric space of higher rank, or
quaternionic or Cayley hyperbolic spaces. Every quasi-isometry of X is at
bounded distance from an isometry.
This is a very satisfying analogue of Mostow’s rigidity results can be
used to give quick proofs of Mostow’s results for cocompact lattices (coarse
analogues are also known for lattices which are not cocompact, but those are
generally more difficult, see [E]). Margulis’ super-rigidity results allow for
distinct domain and range and homomorphisms rather than isomorphisms,
and finding analogues of those results is an important problem in geometric
group theory. The most obvious geometric question along these lines is to
ask whether quasi-isometric embeddings of one symmetric space in another
must be at bounded distance of the inclusion of a totally geodesic symmetric
subspace, a question raised in e.g. in [F].
In this paper we make the first significant progress on this question. Perhaps the most surprising piece of the puzzle is that exotic embeddings exist
even assuming the domain and range of have equal rank and rank at least
2.
Theorem 1.4. For any r > 1 there are quasi-isometric embeddings of
SLr+1 (R)/O(r + 1) into Sp2r (R)/U (2r).
The theorem gives quasi-isometric embeddings between spaces of rank r
when there are no isometric embeddings in these cases, so the embeddings
are definitely exotic. See section 2 for a more detailed discussion of the behavior of these embeddings. Our construction is more general, and produces
quasi-isometric embeddings in most of the cases where our rigidity results
fail. The heart of the construction is
Theorem 1.5. Let G1 and G2 be semi-simple Lie groups of equal real rank
with Iwasawa decompositions Gi = Ki Ai Ni . Every injective homomorphim

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

3

of the solvable Lie group A1 N1 as a subgroup of A2 N2 is a quasi-isometric
embedding.
This is relevant to the above as every symmetric space K\G is isometric to the solvable Lie group AN coming from the Iwasawa decomposition
G = KAN . We refer to quasi-isometries constructed using Iwasawa decompositions and homomorhisms as in Theorem 1.5 as AN -maps. Isometric
embeddings, arising from homomorphisms from G1 into G2 are trivially also
AN -maps. As another application of Theorem 1.5, we have:
Theorem 1.6. There is a quasi-isometric embedding of H2 ×H2 in SL(3, R)/O(3).
More generally there is a quasi-isometric embedding of the product of n
copies of H2 into SL(n + 1, R)/O(n + 1).
It is not hard to adapt the proof of Theorem 1.6 to produce quasi-isometric
Q
P
embeddings of ki=1 SL(ni + 1, R) into SL(n + 1, R) whenever ki=1 ni = n.
We make no attempt to be complete in our discussion of quasi-isometric
embeddings in the reducible case, focusing instead on a particular construction that shows a somewhat more dramatic failure of rigidity. By combining
Theorems 1.6 and 1.4 with a simple twist described in Section 2 we prove
the following theorem:
Theorem 1.7. There exists a quasi-isometric embeddings of the product of
H2 × H2 into SP (4, R)/U (2) that is not at bounded distance from an AN map.
The behavior of these maps on flats is particularly surprising and we describe
it in detail in Section 2. It is probably true that the same type of argument
yields quasi-isometries that are not AN maps from (H2 )r to Sp(4r, R)/U (2r)
but proving this by our method would be involve some quite complicated
combinatorics. While this paper was being revised, Nguyen produced examples of quasi-isometric embeddings of (H2 )n into SL(n + 1, C) where every
flat maps over more than a single flat, and so the map is transparently not
at bounded distance from an AN map [N].
In the other direction, despite the constructions in Theorems 1.4, 1.5,
1.6 and 1.7, there is a substantial amount of rigidity for quasi-isometric
embeddings in equal and higher rank. We prove that quite often one has
only isometric embeddings. The key issue seems to be linear embeddings
of the patterns of hyperplanes in the restricted root system. We refer to
this pattern of hyperplanes as the Weyl pattern of the symmetric space.
The reason to believe that rigidity should occur in equal and higher rank is
the quasi-flat theorem of Eskin-Farb and Kleiner-Leeb which shows that the
image of a flat in this setting is within a bounded distance of a finite union of
flats. To prove our rigidity results, we require a substantial improvement to
this statement that we describe immediately after stating our main rigidity
result.

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DAVID FISHER AND KEVIN WHYTE

Theorem 1.8. Let X1 and X2 be irreducible symmetric spaces or Euclidean
buildings, both of rank r > 1. Let C1 and C2 be their Weyl patterns on Rr .
Then:
(1) If there are no elements of GLr (R) embedding C1 into C2 then there
are no quasi-isometric embeddings of X1 into X2 .
(2) Fix K > 1. If all embeddings of C1 into C2 which are K-quasiconformal are conformal then all (K, C)-quasi-isometric embeddings
of X1 into X2 are at bounded distance from a totally geodesic embedding. In particular, if all pattern embeddings are conformal then
all quasi-isometric embeddings are at bounded distance from totally
geodesic.
Remark: While part (1) of the conclusion of Theorem 1.8 follows easily
for reducible symmetric spaces and buildings, the assumption of part (2)
never holds in the reducible case, not even in the context of Corollary 1.11.
Results in the reducible case have recently been obtained by Nguyen [N].
To explain the key technical mechanism in the proof, our version of the
Mostow-Morse Lemma, we pass to asymptotic cones and bilipschitz maps.
We do so simply to have cleaner statements of results. We are now consider
a building Y of rank k and in this context Kleiner and Leeb showed that
any bilipschitz map f Rk → Y has image contained in a finite union of flats
[KL1]. One can also deduce this by taking asymptotic cones of a theorem
of Eskin and Farb [EsF]. Given such a map f , we call a point x ∈ Rk flat if
there exists r > 0 such that B(x, r) ⊂ Rk maps to a single flat in Y . Given
f , we will call the set of non-flat points for f the exceptional set for f .
Lemma 1.9 (Higher rank Mostow-Morse). Let Y be an affine building of
real rank r and f : Rk → Y a bilipschitz map. Then the exceptional set
Z for f has codimension 2. In fact Z is contained in a finite collection of
bilipschitz images linear spaces of dimension d − 2.
Since codimension 2 sets in one dimensional flats are empty, this statement
is a direct generalization of the standard Mostow-Morse lemma in hyperbolic
spaces. This Lemma is proven in subsection 3.1 and is a necessary component of our proofs. While we do not investigate this here, it seems that
Lemma 1.9 should be true in a much broader context, namely any setting
where there is a) a quasi-flat theorem and b) some combinatorial control on
intersections of flats, including perhaps settings like [BHS, H]. We prove the
lemma from the quasi-flats theorem, it seems quite difficult to prove directly
without first proving that theorem. There is a coarse analogue of Lemma
1.9 for quasi-flats in symmetric spaces and buildings, but the coarse version
of the statement becomes quite cumbersome.
We use Lemma 1.9 in proving both parts of Theorem 1.8, though a proof
of the first part is possible, though much more cumbersome, without it. To
prove both parts of the theorem we pass to asymptotic cones and study the
map restricted to flats. To prove the first part of the theorem, we look at a

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

5

flat point x of a flat F where we have a bilischitz map of domains in Rr . By
Rademacher-Stepanov this map is differentiable almost everywhere and the
goal is to see that the derivative is a linear embedding of patterns. To check
this, one uses that the pattern arises as the intersection of F with transverse
flats to control the derivative. To prove this one uses that these intersections
are codimension one in both flats and so are not contained in the exceptional
sets. The proof of the second conclusion is more difficult. The key step is to
show that a flat maps to a single flat. The hypothesis allows us to see that
the derivative along a flat is everywhere conformal and so the map on the
flat is smooth off the exceptional set by a result of Gehring. The hypotheses
imply that the isometric part of the derivative is an element of the Weyl
group at every point and the fact that the exceptional set is codimension 2
allows us to conclude this Weyl group element is constant and then check
that the rescaling must be the same at every point. This shows the map on
the flat is globally isometric.
The use of Lemma 1.9 leads to a short elegant proof in our context and
also allows for significant simplification of the proofs of [E, D]. To conclude the proof of Theorem 1.8 in the setting of symmetric spaces requires
an additional input, namely the main result of [KL2] which we use to conclude that the image of the quasi-isometric embedding, which is a union of
isometrically embedded flats, is actually a sub-symmetric space.
We now state some corollaries of Theorem 1.8. In section 4 we classify all
pattern embeddings (note that Bn , Cn , and BCn all have the same pattern,
which we will call BCn ). The conclusion is that these are all conformal
except for some exceptional embeddings of An into BCn or F4 and some
exceptional embeddings of A2 and BC2 into G2 .
Corollary 1.10. Among equal rank, irreducible buildings or symmetric spaces
all quasi-isometric embeddings are at bound distance from totally geodesic
unless the domain is of type An and the range of type Bn , Cn , BCn or F4
or unless the source is type D4 and the range is type A4 or unless the range
is of type G2 .
As simple example of this is, for example, that all quasi-isometric embeddings of SLn (R) into SLn (C) (for n > 2) are near isometric embeddings. As
discussed in sections 2 and 4 our rigidity results and construction are close to
complementary. See section 5 for a discussion of the few cases where neither
applies. We also have an additional corollary concerning quasi-isometric
embeddings with “small constants”.
Corollary 1.11. Let X and Y be irreducible buildings or symmetric spaces
of equal rank r > 1. Then there exists a constant K, depending only on
the Weyl patterns of X and Y such that for any K ′ < K and any C, any
(K, C)-quasi-isometric embedding of X into Y is at bounded distance from
an isometry.

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DAVID FISHER AND KEVIN WHYTE

Remark: One might expect the gap in the Corollary to be a result of
a limiting argument, taking (Ki , C)-quasi-isometric embeddings for some
sequence Ki → 1 and then analysing the case of (1, C) embeddings. That
(1, C) embeddings are necessarily isometric can be proven using a somewhat
simplified version of the proof of Theorem 1.8. This limiting argument
would yield only some non-explicit K slightly larger than 1. If one proves
the Corollary directly from Theorem 1.8 instead, one obtains an explicitly
computable K that depends only on the Weyl patterns of X and Y .
The ideas used in this paper find further application in work of the first
author and Nguyen on quasi-isometric embeddings of non-uniform higher
rank lattices [FN].
We end the introduction with some remarks on the case of embeddings
between rank one spaces and embeddings where the rank is allowed to increase. Without the assumptions of equal rank and higher rank the picture
becomes quite opaque and it is not clear that any rigidity remains. In rank
one, a theorem of Bonk and Schramm shows, in particular, that every rank
one symmetric space quasi-isometrically embeds in Hn for n sufficiently large
(and hence into CHn and HHn ) [BS]. Thus quasi-isometric embeddings exist
in many setting where there are no isometric embeddings. One can iterate
these constructions and combine them with isometric embeddings to build
various exotic quasi-isometric embeddings between say HHn and HHm for
some m >> n. Further, even when isometric embeddings do exist, quasiisometric embeddings can be quite far from isometric, even equivariantly,
for example quasi-fuchsian groups in H3 and more general bendings of cocompact lattices in Hn inside of Isom(Hn+1 ) are quasi-isometric as along as
they remain convex cocompact.
When the rank of the target is higher than that of the domain things are
also not well controlled, even if both are higher rank (since rank increases
under quasi-isometric embeddings, there are none where the rank of the
domain exceeds that of the target). For example, any X can be quasiisometrically embedded in X × R as the graph of any Lipschitz function
X → R. In particular, there are many strange embeddings of SL(n, R) into
SL(n + 1, R) of this type as SL(n, R)× R is a totally geodesic (and full rank)
subspace of SL(n + 1, R).
In both of these cases it still seems to be a difficult question to determine
precisely when quasi-isometric embeddings exist, see Section 5 for some open
problems along these lines.

2. AN -maps
In this section we describe a construction of some “exotic” quasi-isometric
embeddings. These examples are built from injective homomorphisms of
solvable Lie groups. Let G be a semi-simple Lie group, with Iwasawa decomposition G = KAN . The solvable group AN acts simply transitively on

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

7

X = K\G by isometries, and so is quasi-isometric to X. The key observation
in our construction is:
Proposition 2.1. 1.5 Let A = Rk for some k ≥ 1, and let N1 and N2 be
two nilpotent Lie groups with A actions so that Ni is uniformly exponentially distorted in ANi . If f : N1 → N2 is an A-equivariant embedding of
Lie groups then the induced embedding AN1 → AN2 is a quasi-isometric
embedding.
Remark: We call maps satisfying the hyphotheses of the proposition AN
maps.
Proof. We use the notation kgkG throughout for the distance in G from g
to the identity. Here uniformly exponentially distorted means that there is
a C > 0 so that for all n ∈ N we have ||n||AN ≤ C log(1 + ||n||N ). This
holds for the solvable groups AN coming from semi-simple Lie groups as N
is spanned by non-trivial root spaces. It holds in many other, but not all,
solvable Lie groups.
Let λ be such that f is λ-Lipschitz (such a constant exists as the map
AN1 to AN2 is a homomorphism). The content of the proposition is that
there is a linear lower bound on distances in AN2 in terms of the distances
in AN1 . Since f is a homomorphism, it suffices to find such a bound for
distance from the identity.
The following holds because the exponential map for nilpotent groups is
polynomial.
Claim 2.2. Let N and N ′ be simply connected nilpotent Lie groups. For
any embedding N as a closed subgroup of N ′ there is a polynomial P so that
for all n ∈ N we have ||n||N ≤ P (||n||N ′ ).
We now return to the proof of the proposition. Let g = an be an element
of AN as above. The triangle inequality gives ||g|| ≤ ||n|| + ||a||. Note that
all norms appearing in this argument are norms in the group AN , not in
any subgroup. Since the projection to A is distance decreasing, ||g|| ≥ ||a||.
We also have:
||g|| = d(an, Id) ≥ d(an, a) − d(a, Id) = ||n|| − ||a||
Thus, if ||a|| ≤ 12 ||n||, we have ||g|| ≥ 12 ||n||. Otherwise, ||g|| ≥ ||a||. So
we have:
1
1
max(||a||, ||n||) ≥ (||a|| + ||n||)
2
4
Thus the distance to the identity is bilipschitz equivalent to the distance
in the product A × N where A and N are given the norms induced by their
inclusions in AN . Since f is a linear isomorphism along A is it bilipschitz
there. For f embedding N1 into N2 we have that the norm in N2 is bounded
by a polynomial in the norm in N1 by the claim, and so, by uniform exponential distortion, the norm of an n in AN2 is bounded by the logarithm of
||a|| + ||n|| ≥ ||g|| ≥

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DAVID FISHER AND KEVIN WHYTE

a polynomial of an exponential of the norm in AN1 . This yields the desired
linear lower bound.

For G = KAN we have that N splits as a sum of positive root spaces
N = ⊕λ Eλ where λ is a linear function on A describing the action of A on
the part of the Lie algebra of N tangent to Eλ . An A-equivariant embedding
of N1 into N2 must send these root spaces into root spaces.
If λ1 and λ2 are positive roots in G1 then either λ1 + λ2 = λ for a root
λ, in which case [Eλ1 , Eλ2 ] = Eλ or λ1 + λ2 is not a root, in which case Eλ1
and Eλ2 commute. Since a homomorphism respects brackets, it follows that
if Eλi maps into the Eηi and λ1 + λ2 = λ then η1 + η2 = η for a root η and
Eλ maps into the η-root space of N2 .
When λ1 + λ2 is not a root in G1 , we must have that the images in the
η1 and η2 root spaces in G2 that commute. For simplicity, assume that G2
is R-split or complex. Then this implies that η1 + η2 is not a root in G2 .
Under this assumption, we have an additive map λ 7→ η sending positive
roots for G1 to positive roots for G2 such that two roots in the domain sum
to a root iff they do in the range. This linear map is nothing but the map
induced on A. Conversely, given such a map, one clearly has an embedding
of solvable groups. In summary:
Proposition 2.3. Let G1 and G2 be semisimple real Lie groups with restricted root systems R1 and R2 of equal rank. There is an embedding of
AN1 into AN2 if there is a linear automorphism T of A carrying positive
roots into positive root and such that λ1 and λ2 sum to a root iff T (λ1 )
and T (λ2 ) do. If the Gi are both R-split, both complex, or G1 split and G2
complex then the converse holds .
Proposition 2.4. There is a quasi-isometric embedding Sln+1 (R) into Sp2n (R).
Proof. By the previous proposition, we must find a linear map Rn → Rn
which sends the positive roots of An to positive roots of Cn respecting when
roots sum to roots. Represent the root systems as:
An with positive roots xi − xj for i > j on the space
Rn = {(x0 , · · · , xn ) : x0 + · · · xn = 0}
and
Cn with positive roots 2yi , yi + yj , and yi − yj with i > j on
Rn = {(y1 , · · · , yn )}
then the map:
T (x0 , · · · xn ) =

1
(2x1 −(x0 +xn ), 2x2 −(x0 +xn ), · · · , 2xn−1 −(x0 +xn ), xn −x0 )
2

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

9

satisfies the requirements. The corresponding root for xi − xj for i > j > 0
is yi − yj , for xi − x0 for 0 < i < n is yi + yn and for xn − x0 is 2yn .
Some additional work is required to show that this matching of roots
spaces actually gives a homomorphism of nilpotent groups. Essentially given
two roots α and β which sum to a root α + β, one has that corresponding
root spaces satisfy [Xα , Xβ ] = cα,β Xα+β and one needs to check that it is
possible to chose the cα,β consistently over all choices of α and β. For this
particular case, one can could construct an explicit matrix embedding to
check this condition. For a more theoretical discussion see subsection 4.2
and particularly Corollary 4.13.


Remark: This same map gives a quasi-isometric embedding Sln+1 (C) into
Sp2n (C), or any other An to Cn situation where the domain is R-split (or
complex if the target is as well). The exact same construction works over padic fields. It is interesting to compare this to the linear pattern embeddings
classified in section 4.
Remark: There is some earlier work using AN maps to build quasi-isometric
embeddings, but only in cases where A is one dimensional. In particular, Brady and Farb use a similar construction to show that H3 quasiisometrically embeds in H2 × H2 [BF]. In rank one, this construction is
quite flexible and has other applications [BF, Fo, L], but it remains quite
surprising that such a construction is possible in the more combinatorially
complex higher rank case.
It is interesting to look at the geometry of this map in more detail. Restricted to a flat in Sln+1 corresponding to a coset of A in AN the map f
is precisely the linear map T above. The hyperplanes in Sp2n that do not
come from those in Sln+1 come from the roots 2yi for i < n and yi + yj for
i < j < n. These correspond to the functions 2xi = x0 + xn (for 0 < i < n)
and xi + xj = x0 + xn (with 0 < i < j < n). The chambers of the flat in the
domain are given by the ordering of the coordinates, and how these missing
hyperplanes subdivide such a chamber depends on where x0 and xn sit in
this ordering.
A flat in An has (n + 1)! chambers and one in Cn has 2n n!. Thus
2n
chambers. The minimally subthe“average” chamber must map across n+1
divided chambers are those where x0 and xn are either the two smallest or
two largest in the ordering (these come in four families from the ordering
between x0 and xn and whether they are minimal or maximal) and map to
single chambers in Cn . Those maximally subdivided are those with x0 and
xn the minimum and maximum (in either order).
For concreteness we look at n = 2. There are six chambers in our flat,
corresponding to the permutations of {0, 1, 2}. There are two chambers
in which x0 and x2 are the minimum and maximum. The ordering x0 <
x1 < x2 is the unique chamber fixed by N (call this chamber C0 ) and the
ordering x2 < x1 < x0 is the opposite chamber. Both of these are sent to

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DAVID FISHER AND KEVIN WHYTE

two chambers in Sp4 /U (2). The four orderings in which x1 is either minimal
or maximal are all sent to single chambers.
This gives a picture of what happens for an arbitrary flat. Divide the flat
into its six chamber, and label each according to whether it is the chamber
fixed by N (there is just one such), adjacent to this chamber, opposite to
this chamber (the generic case), or adjacent to an opposite chamber. What
happens to every chamber is then determined as above. It is not hard to
see that the flats in Sl3 are of the following types:
• A flat passing through C0 . These are described above, and have
two chambers (opposite to each other in the flat) that map to two
chambers, and the four others which map to single chambers. Thus
the flat maps to a total of eight chambers which is necessarily a
single flat. The map on flats if given by the linear map T above.
• There are two chambers of F (next to each other in F ) which are
opposite to C0 ,their neighbors in F which are adjacent to opposites
to C0 , and two chambers adjacent to C0 . This maps across eight total
chambers, and hence has image at bounded distance from a single
flat. Unlike the previous case, however, the map is not globally linear
as a map R2 → R2 .
• There are two chambers of F (next to each other in F ) which are both
adjacent to C0 or both adjacent to opposite to C0 . The remaining
four chambers are all opposite to C0 . Such a flat maps to an image
quasi-flat with ten chambers, and hence is not near a single flat.
• (The generic case) All chambers in F are opposite to C0 . In this
case all six chambers map across two chambers in the range. The
image quasi-flat has twelve chambers and so in not near a single flat.
It might be interesting to work out precisely what these quasi-flats look
like. In particular, in analogy with Proposition 3.1, what does the locus of
points where the image essentially bends (is not near a single flat) look like?
The existence of these coherent families of quasi-flats not close to flats is
genuinely surprising.
We now prove Theorem 1.6, which we can do by providing an explicit
map onQmatrices. The Iwasawa decomposition of (H2 )n can be realized as
AN = Ai Ni where each Ai is a two by two diagonal matrix


ai
0
0 ai −1



and each Ni is a two by two diagonal matrix


1 ui
0 1



where each ai is a non-zero real number and each ui is a real number.
The desired AN embedding is then explicitly:

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES



α1 u1 u2 . . .
un
 0 α2 0 . . .
0

 ..
..
..
..
..
 .
.
.
.
.
0 0 . . . 0 α1 −1 . . . αn −1

11







where the αi are easily computed explicitly. For instance in the case n = 2,
we have α1 = a21 a22 and α2 = a22 .
We can proceed to the proof of Theorem 1.7. First note that in an AN map there is always one chamber at infinity, the one fixed by AN such that
any every flat passing through it maps lineary to a flat in the range. We
show a map is not an AN map by showing this does not occur and in fact
that no flat maps linearly from domain to range. We begin by describing the image of flats in H2 × H2 into SL(3, R)/O(3) and then combining
this with the description given above for flats SL(3, R)/O(3) mapping into
SP (4, R)/U (2).
For this description, we will think of H2 as being metrics on R2 up to
scalars, and SL3 (R)/SO3 (R) as the same on R3 . The embedding is then:
choose to planes P1 and P2 in R3 , and identify each with R2 . A point of
H2 × H2 can then be thought of a a metric (up to scalar) on each of P1
and P2 . Rescale so they agree on their line of intersection (let’s call that
l), and extend that to an inner-product on R3 by having P1 and P2 meet
orthogonally.
Note that this tells us the image of the embedding, namely all metrics
on R3 for which P1 and P2 meet orthogonally. Also note that there is a
Lipschitz inverse, mapping SL3 (R)/SO3 (R) to H2 × H2 just by restricting
the metric to P1 and P2 .
The geodesics in H2 correspond to the set of metrics keeping a pair of lines
orthogonal, and similarly a maximal flat in SL3 (R)/SO3 (R) corresponds to
the set of metrics keeping a triple of lines orthogonal. The construction
includes a distinguished line l = P1 ∩ P2 . Choosing another line l1 ⊂ P1 and
l2 ⊂ P2 , we get a product of two geodesics in H2 × H2 which naturally maps
to the flat in SL3 (R)/SO3 (R) in which l, l1 , l2 are orthogonal.
This flat in H2 ×H2 is subdivided into four quadrants - if we choose vectors
v ∈ l, u1 ∈ l1 , u2 ∈ l2 then we can think of the quadrants as determined by
which of v and vi is longer in the metrics on Pi . The flat in SL3 (R)/SO3 (R)
is similarly divided into six chambers according to the ordering of sizes of
v, v1 , and v2 . From this point of view we can see how some chambers in
the domain map to two chambers in the range and others to only one. For
example, if we know that v is shorter than v1 in P1 and v is longer than
v2 in P2 then, once we rescale to get the metrics to agree on v, we know
that the order is ||v2 || < ||v|| < ||v1 || and we have a quadrant that maps
to a single chamber. On the other hand, if v is shorter than v1 in P1 and
shorter than v2 in P2 , after rescaling we might have ||v|| < ||v1 || < ||v2 || or
||v|| < ||v2 || < ||v1 ||, so this quadrant gets mapped across two chambers.

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DAVID FISHER AND KEVIN WHYTE

Viewing the chambers in H2 × H2 as a points in ∂H2 × ∂H2 , what we see is
that a chamber in which neither point is l is mapped to two chambers, as the
one chamber with points points l while those in which one is l and one is not
are mapped to a single chamber. Thinking of chambers in SL3 (R)/SO3 (R)
as flags V1 ⊂ V2 ⊂ R3 and the points at infinity in H2 as lines in R2 what
we have is this:
• A chamber in H2 × H2 with points at infinity l1 and l2 , neither equal
to l maps to the two chambers l1 ⊂ V and l2 ⊂ V where V is the
span of l1 and l2 .
• A chamber in H2 × H2 with points at infinity l1 and l, l1 6= l maps
to the chamber l1 ⊂ P1 . Likewise, a chamber with endpoints l and
l2 maps to the one corresponding to the flag l2 ⊂ P2 .
• The chamber with both points at infinity equal to l maps to the two
chambers l ⊂ P1 and l ⊂ P2 .
From this it is easy to determine what happens to flats : if none of the
endpoints is l the image is a quasi-flat made of eight chambers, if both
geodesics have l as an endpoint the image is a single flat (and the map is
linear here). Finally if one of the geodesics has l as an endpoint and the
other does not the image is again a single flat ( the metrics for which the
three endpoints that are not l are orthogonal ) but the map is not linear one half of the flat maps to four of the chambers and the other half to two.
The embedding of SL3 (R)/SO3 (R) into Sp4 (R)/U (2) can be described
similarly (by extending a metric on R3 to one on R4 compatible with the
standard symplectic form) but is more complicated. The summary for chambers follows as described above. In this terminology, there is a distinguished
flag V1 ⊂ V2 in R3 , and this flag as well as those opposite to it (meaning
those U1 ⊂ U2 for which U1 6⊂ V2 and V1 6⊂ U2 ) map to two chambers. Flags
neither equal to nor opposite to V1 ⊂ V2 map to a single chamber.
We can compose these constructions to get an exotic embedding of H2 ×
2
H into Sp4 (R)/U (2). One can do this fixing one Iwasawa decomposition
of SL(3, R), but one can also do this allowing the map ι1 : H2 × H2 →
SL(3, R)/O(3) and the map ι2 : SL(3, R)/O(3) → SP (4, R)/U (2) to be
constructed using different choices of Iwasawa decompositions for SL(3, R).
The choice of AN in an Iwasawa decomposition depends on a choice of a
Weyl chamber at infinity or equivalently of a full flag on R3 .
The nature of this map will depend on how P1 , P2 , and l are situated
relative to the flag V1 ⊂ V2 . Consider the “generic” case, where l 6⊂ V2 and
V1 6⊂ P1 ∪ P2 . This corresponds to a “generic” choice of two Weyl chambers
in SL(3, R)/O(3), i.e. a choice of two opposite chambers. There are then
two special lines, namely l1 = P1 ∩ V2 and l2 = P2 ∩ V2 .
• The chamber in H2 × H2 with endpoints (l1 , l2 ) maps to the two
chambers corresponding to the flags l1 ⊂ V2 and l2 ⊂ V2 . These two
are both adjacent to V1 ⊂ V2 and so each map to a single chamber.
The image under the composition thus is two chambers.

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

13

• The chamber (l1 , l) maps to the single flag l1 ⊂ P1 . This is an
adjacent chamber and so the image under the composition is a single
chamber. The same holds for the symmetric case (l, l2 ).
• The chamber (l, l) maps to the two flags l ⊂ P1 and l ⊂ P2 . Both of
these are opposite, and so the image under the composition is four
chambers.
/ {l, l2 } maps to the two flags l1 ⊂ U
• A chamber (l1 , l2′ ) where l2′ ∈
′
and l2 ⊂ U , for U the span of l1 and l2′ . The first is adjacent and the
second opposite, so the image has three chambers. The same holds
for the symmetric cases (l1′ , l2 ).
• A chamber (l, l2′ ) (with l2′ as above) maps to the single flag l2′ ⊂ P2 .
This is opposite and so maps to two chambers.
• There is a bijection between the lines in P1 \ {l1 , l} and lines in P2 \
{l2 , l} defined by saying two lines correspond iff their span contains
V1 . A chamber with endpoints (l1′ , l2′ ) maps to the two flags l1′ ⊂ U
and l2′ ⊂ U where U is the span of l1′ and l2′ . These are both adjacent
if l1′ and l2′ correspond under the above bijection, and otherwise are
both opposite. In the former case the image under the composition
is two chambers, and in the latter is four.
Remark: the bijection in the last item extends to the full boundaries by
sending l1 → l2 and l → l. The graph of this bijection is the boundary of
an isometrically embedded H2 in H2 × H2 , and the composite embedding
”bends” interestingly along it.
Which flats in H2 × H2 map nicely to Sp4 (R)/U (2)? If the image is to
be a single flat and the map is to be linear, then every chamber must map
to two. A quick enumeration of the possibilities shows this never happens.
The only cases where the image is a single flat (eight total chambers) are:
• The flat ll1 × ll2 : here the quadrant (l, l) maps to half the flat, the
quadrants (l1 , l) and (l, l2 ) map to single chambers, and the quadrant
(l1 , l2 ) maps to two chambers.
• The flats of the form l1 l1′ × ll2′ where l1′ and l2′ correspond ( or the
symmetric case ll1′ × l2 l2′ ): The quadrant (l1 , l2′ ) maps to three chambers, (l1 , l) maps to a single chamber, and the other two map to two
chambers each.
Thus no flat maps linearly to one flat and the map cannot be an AN map.
It is easy to check that a generic flat maps to 16 chambers and one can
enumerate all possible behaviors by continuing the analysis above. It would
be interesting to have a less computational approach to showing that a map
is not AN . In [N], Nguyen gives some examples quite similiar to those we
just discussed, where one can check that a map is not AN by checking that
no flat maps to a single flat, but the example above shows that this is not
the only obstruction.

14

DAVID FISHER AND KEVIN WHYTE

3. Rigidity
If X1 quasi-isometrically embeds in X2 of equal rank, then every maximal
flat in X1 gives a quasi-flat in X2 . By results of [KL1] and [EsF], such a
quasi-flat is at bounded distance from a finite union of flats. As we want to
control the intersection of two quasi-flats we need more information on precisely what a quasi-flat can look like. For simplicity we describe the results
for bilipschitz embeddings of Rn into Euclidean buildings, since the results
for quasi-flats in Eucldiean buildings or symmetric spaces follow formally
by passage to asymptotic cones. For background on buildings, symmetric
spaces, asymptotic cones and their relations to one another, we refer the
reader to [KL1].
3.1. Structure of Quasi-Flats in Buildings. In this subsection, X will
also be a Euclidean building of rank d. By a flat we mean a top dimensional,
isometrically embedded Rd in X. Such a flat comes with a distinguished family of codimension one affine subspaces, which we call Weyl hyperplanes. An
affine subspace of a flat which is the intersection of the hyperplanes containing it is called a subflat. Each subflat comes with an induced pattern of
hyperplanes by intersection, and we continue to call these Weyl hyperplanes
(of the subflat). By a Weyl box in a (sub)flat we mean a convex set which is
a finite intersection of closed Weyl halfspaces. We now state a more precise
form of Lemma 1.9 from the introduction. The description of the exceptional set here takes place in the range rather than the domain, but the two
are, of course, related by a bilipschitz map.
Lemma 3.1. For any K, there is an n so that for any K-bilipschitz embedding f of Rd into X there is a finite set {S1 , S2 , · · · , Sn } of codimension 2
subflats of X so that every point x in the image of f not contained in any
of the Si has a neighborhood contained a flat.
Proof. We begin by discussing the local structure of a union of two flats. For
this we need some vocabulary. Given a finite collection of disjoint rays, we
form a star by identifying their initial points. A fan is a product F = S × Rk
where S is a star. If S consists of exactly 4 rays, then F is a union of two
flats intersecting transversely.
Lemma 3.2. There is a k (depending only on X) so that for any flats F
and F ′ of X, off of at most k codimension two subflats, each point of F ∪ F ′
has a neighborhood which is either contained entirely in a flat or which is
contained in a fan.
Proof. The intersection F ∩ F ′ is a Weyl box in a (sub)flat of each. If this
subflat is codimension two (or more) then we are done (with k = 1). If it
is codimension one then the intersection is a box in a hyperplane H. The
interior points of the box have neighborhoods which are the contained in the
union of two flats which forms a fan. Thus the points at which the conclusion
of lemma fails are the boundary points of the intersection. The number of

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

15

faces of such a box is bounded by the combinatorics of the Coxeter system
of X, and each face is contained in a codimension two subflat of F and F ′ .
Finally, if the intersection is of codimension zero, then the conclusion of
the lemma holds in the interior and exterior of the intersection (these points
are locally contained in a single flat) and at points in the interior of faces of
the box of intersection (these are locally in a pair of transverse flats forming
a fan). Thus the points at which it fails are contained in the codimension
two and higher facets of the boundary of the box. As before, the number of
these facets is bounded by the Coxeter system, and each facet is contained
in a codimension two subflat.

Corollary 3.3. For any j there is an m so that for any collection F1 , F2 , · · · , Fj
of flats, off of at most m codimension two subflats (called exceptional),
each point of ∪Fi has a neighborhood which is either contained entirely in
a flat or which is contained in the union of finite flats meeting pairwise
transversely along a common hyperplane H.
Proof. Let x be a point of the union ∪Fi off of the exceptional sets produced by the previous lemma for the pairwise intersections. If the various
hyperplanes through x along which the flats branch are the same in a neighborhood of x then we are done. If two of the flats intersect in something
codimension two or more then we add that subflat to the
 collection of exceptional subflats (the number of such is bounded by 2j ). Thus we need
only address points at which three or more flats intersect.
If three flats intersect pairwise in a subflat of codimension two more then
we add this intersection to the set of exceptional subflats.
Finally, if a three flats intersect near x in a box in a hyperplane H then
the lemma holds near x unless unless x is on the boundary of the box, and
is thus in one of the codimension two subspaces containing the faces of the
box. The number of these is similarly bounded by j and X.

We can now finish the proof of Lemma 3.1. By [KL1][Corollary 7.2.4],
there is an s depending only on K so that image of f is a subset of the union
of at most s flats. Let Y be the union of these flats. Applying the corollary
gives a family of exceptional subflats off of which every point of Y is has a
neighborhood which is either flat (in which case there is nothing to prove) or
finite collection of Weyl halfspaces meeting along a common hyperplane H.
In the latter case, we locally have an embedding of Rd into Rd−1 × S where
S is finite union of rays meeting at a single point. As such an embedding
must visit only two of the rays in S, we have the image contained in a union
of two half spaces meeting along a hyperplane. Such a union is itself a flat
in X.

3.2. Flats go to flats. In this subsection, we prove that under hypotheses
analogous to those of Theorem 1.8, bilipschitz embeddings of Euclidean
buildings take flats to flats.

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DAVID FISHER AND KEVIN WHYTE

Proposition 3.4. Let X and Y be Euclidean buildings of equal rank, such
that every K-quasi-conformal linear embedding of patterns is conformal. The
every K-bilipschitz embedding of X in Y sends flats to flats and is a similarity along every flat.
Lemma 3.5. Let φ be an embedding as above and F a flat in X. For any
hyperplane H of F , almost every point x ∈ H has a neighborhood mapped
to a hyperplane of Y .
Proof. By Proposition 3.1, we know φ(F ) is locally contained in a flat off of
a finite set of codimension 2 subflats. Let Σ ⊂ F be the pre-image of these
exceptional subflats. For any x ∈ F \ Σ, φ is locally a map into a single flat.
By Rademacher’s theorem, φ is differentiable at almost every point in such
a neighborhood, with derivative K-quasi-conformal linear map. Since these
neighborhoods cover F \ Σ, this holds at almost every point of F \ Σ.
Let x be such a point of differentiability and H a hyperplane of F through
x, let y = φ(x) and G be a flat in Y which locally contains φ(F ) near y.
Choose F ′ a flat in X which intersects F in H. The image of F ′ is also
a bilipschitz image of a flat, and so has the structure given by Proposition
3.1. We have φ(H) = φ(F ) ∩ φ(F ′ ). Off of the exceptional points of φ(F ′ )
this is locally the intersection of two flats which are transverse as F and
F ′ are. Thus φ(H) is (locally) contained in the closure of a finite union of
hyperplanes in G. By the differentiability of φ at x, φ(H) much locally be
equal to a single such hyperplane.

Proof of Proposition 3.4. We now finish the proof of Proposition 3.4. As in
the proof of the lemma, let x be a point of differentiability of φ in F \ Σ.
Lemma 3.5 implies the derivative is a linear embedding of patterns at x, and
so is 1-quasiconformal. As this holds almost everywhere, work of Gehring
implies φ is smooth on F \ Σ. By the assumed rigidity of patterns, the
derivative at each point is a scalar multiple of one of a finite number of
isometric pattern embeddings, where the scalar is bounded between K1 and
K. Since Σ cannot disconnect as it is codimension at least two and the map
is C 1 off of Σ, we know that off of Σ we, in fact, have a scalar multiple of a
single isometry. Comparing the scaling along different lines in the pattern,
we see that, possibly after re-scaling the metric on Y , that φ is isometric
on F \ Σ. By continuity, φ is therefore an isometry on all of F . This forces
the image to be a single flat. A similar use of Gehring’s theorem occurs in
[D]. As all of our maps are not just quasi-conformal but bilipschitz, we are
really using an easy special case of this theorem.

3.3. Proof of rigidity results. We know prove our main rigidity reslts
from the results of the last section. First, we conclude the analogue of our
rigidity results for bilipschitz embeddings of buildings:
Theorem 3.6. Let X and Y be buildings of equal rank. If there is a bilipschitz embedding of X in Y then there is a linear embedding of their Weyl

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

17

chamber patterns. If all such pattern embeddings which K-quasi-conformal
are conformal, then all K-bilipschitz embeddings of X in Y are, up to rescaling, isometric embedding of subbuildings.
Proof. The first conclusion is an immediate consequence of Lemma 3.5. For
the second, by Proposition 3.4, each flat in X maps by a similarity to a flat
in Y . Since the dilation must be equal on two flats intersecting in anything
of positive dimension, and all flats can be connected by such a chain of flats,
we can rescale the metric on Y so that flats map isometrically to flats. Since
every geodesic is contained in a flat, the map is globally isometric and totally
geodesic.

The proof of our main rigidity result requires one really new ingredient,
the main technical result of [KL2].
Proof of Theorem 1.8. Given a quasi-isometric embedding of X1 into X2
satisfying the condition of the first bullet point in Theorem 1.8, by passing
to the asymptotic cones, we obtain a bilipschitz embedding of buildings. By
Theorem 3.6 this is impossible as there are no linear embeddings of their
Coxeter systems. If X1 and X2 satisfy the conditions of the second bullet
point in Theorem 1.8, then the map on asymptotic cones is an isometric
embedding by Theorem 3.6 and so in particular, every flat is mapped to
a single flat. By [KL1, Lemma 7.1.1.], the fact that flats go to individual
flats in all asymptotic cones implies that every flat maps to within bounded
distance ( depending only on the quasi-isometry constants ) of a single flat.
Since intersections of flats encode the Weyl chamber patterns, we know
that the Weyl hyperplanes map to within uniformly bounded distance of
Weyl hyperplanes. These affine foliations are quite rigid - in particular, by
[MSW, Lemma 7.1.12], the quasi-isometry is at bounded distance from an
affine map preserving patterns. Thus every apartment in the Tits boundary
of the domain maps to a well defined apartment in the Tits boundary of the
domain. Further, this maps respects the decomposition into chambers. It is
now easy to check that the image is a sub-building: the isometric embedding
of patterns induces an inclusion of Weyl groups W ′ < W , and charts from
the domain building structure push forward to charts in the range which are
W ′ compatible and therefore W compatible. By [KL2, Theorem 3.1] it is
the boundary of a sub symmetric space (or building) Y ′ ⊂ Y , and the image
of the quasi-isometric embedding is at bounded distance from Y ′ . So the
embedding is a quasi-isometry X to Y ′ , and is therefore at bounded distance
from an isometry by the main results of either [EsF] or [KL1]. (We quote
these last results only for simplicity of exposition. At this point, we know
much more, e.g. that the quasi-isometry from X to Y ′ is bounded distance
from an isometry along each flat, so one can use only a small part of the
arguments from those papers.)


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DAVID FISHER AND KEVIN WHYTE

4. Pattern embeddings
In this section we do the necessary algebraic analysis to deduce Corollary
1.10 from Theorem 1.8. Since it does not require too much more work, we
also analyze essentially all settings in which AN maps are possible, at least
when the range is classical.
The geometric information needed in our rigidity results (and the algebraic information needed for our constructions of exotic embeddings via
AN −maps) is closely related to the restricted root systems of semi-simple
groups over R. Let X be a symmetric space or Euclidean building of rank r.
The maximal flats are r-dimensional Euclidean vector spaces (canonically
identified with A). Each such flat has a finite collection affine foliations,
along which the flat meets other maximal flats transversely. These are precisely the level sets of the roots.
One results about QI-embeddings say that if X1 quasi-isometrically embeds in X2 ( of equal rank ) then there is a linear isomorphism A1 → A2
which preserves these foliations, and that if we know all such linear maps are
conformal then the QI-embedding is at bounded distance from isometric.
Using the metric on the flats we can used duality to work in terms of the
root vectors rather than the hyperplanes (which are their perpendiculars).
In the following, we study pattern preserving maps of flats using this duality.
4.1. Pattern Embeddings.
Definition 4.1. By a pattern embedding of a root system on R1 in a root
system on R2 we mean a linear embedding of the underlying vector spaces
which sends the root vectors of R1 to scalar multiples of the roots of R2 . We
say the pattern embedding is equal rank if the root systems are equal rank
(and so the embedding is induced by an isomorphism of vector spaces), and
that the embedding is conformal if the linear map is a conformal embedding.
We want to discuss in more detail when a non-conformal, equal rank
pattern embeddings exists of one root system in another. We analyze the
reducible case in terms of irreducible components, so we begin by studying
general pattern embeddings in the irreducible case. Because we are using
the dual notion of pattern, our results on pattern embeddings in unequal
rank do not say anything about linear embeddings sending affine foliations
to affine foliations.
In rank two, which is somewhat special, we are simply discussing patterns
of lines in the plane. The irreducible patterns are A2 , BC2 , and G2 - which
are, respectively: three, four, and six lines arranged symmetrically, and the
reducible pattern A1 × A1 = D2 which is two perpendicular lines. It is
straight forward to check that there are pattern embeddings for any pair
where the range has at least as many lines as the domain. These rank two
embeddings will be the building blocks for our study of the higher rank cases.
To this end we note that there are three basic non-conformal embeddings
: A2 into BC2 , D2 into A2 , and D2 into BC2 (for this last case there are

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

19

two distinct embeddings, one conformal and one not). We have excluded
the case of target G2 as it does not occur inside any higher rank irreducible
root system.
In rank n > 2 there are three families of irreducible patterns: An , BCn ,
Dn (this only for n > 3 as D3 = A3 and D2 is reducible ), plus the exceptional E6 , E7 , E8 , and F4 (see, for example, [Kn], for a detailed description
of these exceptional root systems). For reference, the classical root systems
are:
(1) An for n ≥ 2 with V = {(x0 , · · · , xn ) ∈ Rn+1 : x0 + · · · + xn = 0}
and roots xi − xj .
(2) Bn for n ≥ 2 with V = Rn and roots {xi ± xj } ∪ {±xi }.
(3) Cn for n ≥ 2 with V = Rn and roots {xi ± xj } ∪ {±2xi }.
(4) BCn for n ≥ 2 with V = Rn and roots {xi ± xj } ∪ {±xi } ∪ {±2xi }.
(5) Dn for n ≥ 4 with V = Rn and roots {±xi ± xj }.
While the root systems Bn , Cn , and BCn are all distinct, they all have
the same root patterns. We will refer to this pattern, somewhat sloppily, as
BCn .
Given a root system on V , a two dimensional subspace U ⊂ V is special
if its intersection with the roots is irreducible (so A2 or BC2 , as G2 does
not occur inside any irreducible root systems other than itself). Two roots
whose span is special we will call related. A root system all of whose
special subspaces are of type A2 is called simply laced (these are the A,
D, and E families). As above, simple counting the roots in various special
planes shows that a pattern that embeds in a simply laced system is itself
simply laced.
We need some facts about special planes in root systems. We will call a
special plane full if it is of type A2 in a simply-laced root system or of type
BC2 in doubly-laced root system. (All root systems except G2 which are
not simply laced are doubly laced, G2 is triply laced, but we will not need
this, or the usual definitions of these concepts, here.)
Lemma 4.2. If R is an irreducible root system then
• Any two full planes are equivalent under the action of the Weyl
group.
• For any two roots r and r ′ there is a sequence of roots r = r0 , r1 , · · · , rn =
r ′ such that the span of ri and ri+1 is a full special plane for all i.
Proof. Let U ⊂ V be a special plane. Since R is finite, we can choose a
linear functional on V which vanishes on U but not on any root outside U .
By perturbing this functional we can find a functional which is non-zero on
all roots and whose values on roots in U is much smaller than its values on
any other roots. Using this functional to define the positive roots, we see
that the positive roots in U are generated by the positive simple roots in
U . Thus U comes from an edge of the corresponding Dynkin diagram. It
is standard that the Weyl group is transitive both on positive systems and

20

DAVID FISHER AND KEVIN WHYTE

on simple edges of the Dynkin diagram spanned by roots of a fixed length
[Kn]. Since any diagram contains at most one doubled edge, the first part
of the lemma follows.
For the second part, we first consider the weaker equivalence generated
by all special planes. Since two roots which are not orthogonal necessarily
span a special plane, the equivalence classes give a decomposition of V into
orthogonal factors, with every root contained in a factor. Thus irreducibility
implies there is only one factor.
This completes the second clam for simply laced root systems. For doubly
laced systems we need to see that two roots in a common special plane of
type A2 are connected by a chain of full special planes. The first paragraph
above shows that there are at most two Weyl orbits of such planes (one
spanned by short roots and one by long roots), and it is trivial to check the
claim for one such plane of each type in both BCn and F4 .

Lemma 4.3. A pattern embedding of irreducible root patterns that maps all
special planes to special planes of the same type is conformal.
Proof. The starting point is the observation made above that such a map
is conformal on all special planes. Thus the metric is scaled by the same
factor along any two related root lines in the domain. By irreducibility this
means the map scales all root lines by the same factor. The metric on V
is determined by its restrictions to the root lines (this is a linear algebra
exercise for An and Dn , and all the other root systems are supersets of
these).

Corollary 4.4. Every pattern automorphism of an irreducible pattern is
conformal.
This can also be shown directly : after composing with an element of the
Weyl group, the automorphism preserves the positive cone, and therefor the
simple roots. Since it must also preserve the types of the special plane, this
means it corresponds to an automorphism of the Dynkin diagram ( ignoring
the direction of multiple edges). These are isometric automorphisms of the
root systems if the directions are preserved ( automatic except for BC2 , G2 ,
and F4 which have a conformal pattern automorphism switching the long
and short roots ).
Lemma 4.5. Let P and P ′ be irreducible root systems of equal rank (excluding G2 ). If P and P ′ are both simply laced, or neither is, then every
embedding of P in P ′ is conformal.
Proof. If both are simply laced, this is an immediate corollary of the previous
lemma. The argument when neither are is the same as there once one notes
that for non simply laces root systems, irreducibility implies than any two
roots are connected by a chain of special planes of type BC2 .


QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

21

Conformal embeddings corresponding to inclusions of sub root systems.
Up to symmetries, these are all the obvious inclusions of one root system as
a subset of another in the coordinates above, which we call standard ([Kn]).
Each root system includes canonically into the higher rank root systems in
the same family, and An−1 ⊂ Dn ⊂ BCn . Here the isomorphism of A3 and
D3 causes some confusion, as the two inclusions into BCn for n > 3 are
distinct.
By the above lemmas, the only possibilities for non-conformal pattern
embedding above rank two are one of A, D or E into one of BC or F . We
will put off a discussion of the exceptional root systems until later, and first
address the classical examples:
Lemma 4.6. There are, up to the actions of the Weyl groups (and rescaling), two non-conformal pattern embeddings of An into BCm ( for m ≥ n)
:
• The roots xi − xj for i > j > 0 map to yi − yj , and the roots xi − x0
map to yi .
• The roots xi − xj for i > j > 0 map to yi − yj , and the roots xi − x0
map to yi + y1 (meaning the line to 2y1 when i = 1).
Proof. Since the embedding is non-conformal, the prior lemmas imply that
there is some special plane in An which maps to one of type BC2 in BCn .
The Weyl groups are transitive on special planes in the domain and on type
BC2 planes in the range. Thus we may assume that the plane spanned by
x1 − x0 and x2 − x1 maps to the plane spanned by y1 and y2 in the range.
Up to automorphism there are two such maps, depending on whether the
missing root line is one of the yi or y1 ± y2 . Thus we may arrange for the
map to behave as indicated for the roots xi − xj with 2 ≥ i > j.
Assume, for induction, that we can apply an automorphism to put the
map in one of the two standard forms for all the roots xi −xj with k ≥ i > j.
The root xk+1 − xk is related to all the roots xk − xi for k > i, so its
image must be related to yk − yi for all k > i ≥ 1 and to either yk or
yk + y1 respectively. It must also be outside the span of those roots because
xk+1 − xk is in the domain. The only such roots are ±yk ± ys for s > k.
By applying an automorphism of BCn fixing the first k-coordinates, we may
assume s = k + 1 and that yk+1 occurs with positive sign. So xk+1 − xk
maps to λ(yk+1 ± yk ) with λ > 0.
Since (xk+1 − xk ) + (xk − xi ) = xk+1 − xi is a root in the domain for all
0 < i < k, the image must be a scalar multiple of a root. By induction and
linearity, the image is λyk+1 + (1 ± λ)yk − yi . Since no roots in the range
have three non-zero coefficients and λ > 0, we must have xk+1 − xk mapping
to yk+1 − yk as claimed.

Lemma 4.7. Every pattern embedding of Dn (for n ≥ 4) into BCm is
conformal (and hence standard)

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DAVID FISHER AND KEVIN WHYTE

Proof. As before, a non-conformal embedding must send some special plane
to one of type BC2 . The Weyl group of Dn are transitive on special planes,
so we may assume one of the planes mapping non-conformally occurs in the
canonical An−1 inside Dn . By the remark above, this means the pattern
embedding can be taken to be one of the two above maps sending An−1 into
BCn−1 ⊂ BCn . Explicitly, the map does one of the following:
• The roots xi − xj for i > j > 1 map to yi − yj , and the roots xi − x1
map to yi (for i > 1).
• The roots xi − xj for i > j > 1 map to yi − yj , and the roots xi − x1
map to yi + y2 for i > 1 (meaning the line to 2y2 when i = 2).
Consider the root x1 + x2 . It is not in the span of An−1 so its image
must have nontrivial y1 component. It must also be related to all yi − y2
the images of xi − x2 ( for all i > 2). The only such roots are ±y1 ± y2 .
However, none of these are related to the images of all the xi − x1 for i > 2,
so no such pattern embeddings exist.

For the exceptional root systems the conformal examples are all standard. Since B4 and C4 are both isometric sub root systems of F4 , there are
non-conformal embeddings (and even AN -maps) from A4 into F4 . Likewise,
there is a unique non-conformal embedding of D4 into F4 , via a straightforward calculation as above starting with A3 ⊂ D4 into B3 or C3 inside
F4 . For completeness we give it here on a basis (the rest is determined by
linearity):
•
•
•
•

x2 − x1
x3 − x2
x4 − x3
x1 + x2

7 y1
→
7→ y2
7→ 12 (y4 + y3 − y2 − y1 )
7→ 21 (y4 + y3 + y2 − y1 )

This gives a complete picture of pattern embeddings between irreducible
root systems. We next turn to the cases where the domain is reducible. For
simplicity, we will only discuss the classical root systems here as the other
work out similarly. Likewise we will only consider the case of the domain as
a product of two factors as those with more can all be built by compositions
of such.
Suppose R = R1 ⊕ R2 with each Ri irreducible. To embed R into R′
of equal rank one needs embeddings of both Ri into R′ so that the image
vector spaces Vi decompose V ′ as an internal direct sum. We will call the
embedding standard if it is conformal on both of the components and their
images are orthogonal.
The earlier lemmas tell us what the image subspaces can look like:
• An includes in Dn+1 naturally as the root vectors with coordinate
sum zero. Composing with signed permutations of the coordinates
of Dn+1 one can get an embedding of An with image the hyperplane
Σεi yi = 0 for any εi = ±1.

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

23

• The previous embedding can be composed with the inclusions of
Dn+1 into Dm or BCm for m > n. The image subspace is a hyperplane as above in the subspace spanned by n + 1 of the coordinates.
These are all the conformal embeddings of An ( for n > 3 ) into
these spaces.
• An includes into BCn non-conformally (in two ways) which is equal
rank. This can be composed with the inclusions of BCn into BCm
for m > n. The image subspaces are the span of n coordinates.
These are all of the non-conformal embeddings of An into BCm .
• Dn includes into BCn conformally and equal rank. This can be
composed with any inclusion BCn into BCm . The image is the span
of n coordinates. These are all the pattern embeddings Dn into BCm
(note that for n = 3 this is also an embedding of A3 into BCm not
covered by the earlier cases ).
• For n ≤ m, An includes into Am , Dn into Dm , and BCn into BCm .
These are conformal with image the span of n-coordinates.
The upshot is that the image subspace when the domain is rank n is
always either the span of n-coordinates or a hyperplane of the specified type
in the span of n + 1-coordinates, and the latter happens only if the domain
pattern is An .
Consider an embedding R1 ⊕R2 into R′ of equal rank. If both factors have
images of the first type ( the span of a subset of coordinates of size equal to
the rank of Ri inside Dm or BCm ) then these subsets of coordinates must
be complementary to get V ′ = V1 ⊕ V2 . This exhausts the possibilities when
no factor is of type A, and in these cases all the embeddings are conformal.
It also includes all the cases where An only embeds non-conformally, and all
such embeddings are simply products of embeddings of irreducible composed
with one of the conformal embeddings of products of D and BC factors into
D or BC.
When the target is type A, the factors must be so as well. This case is
non-conformal :
Lemma 4.8. There are equal rank, non-standard pattern embeddings of
Am ⊕ An into Am+n .
Proof. Divide the m + n + 1 coordinates for Am+n into two sets, one of size
m + 1 and on of size n + 1, which overlap in a single element. The natural
inclusions of An and Am as the spans of these are conformal embeddings
of An and Am . Their images are disjoint ( as the intersection consists of
vectors with coordinate sum zero supported in a single coordinate ) and
together span.

Similarly,
Lemma 4.9. There is a non-standard pattern embedding An ⊕ Am into
Dm+n .

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DAVID FISHER AND KEVIN WHYTE

Proof. The embeddings of the factors are given by Ak includes in Dk+1 as
the vectors of coordinate sum zero, composed with an isometric embedding
of Dk+1 into Dm+n . The image of the underlying vector space is therefor
a hyperplane in the span of k + 1 coordinates defined by Σ ± yi = 0 for
some collection of signs. If the two sets of coordinates overlap in fewer
than two coordinates then the image isn’t full rank. If the overlap in more
than two then they are not disjoint. Thus the only embedding comes from
overlapping in exactly two coordinates ( say y1 and y2 ) and where the sign
choices agree for one of these two and differ for the other. Explicitly, after
signed permutations, it must be:
(x0 , x1 , · · · , xn ) × (z0 , z1 , · · · , zm ) 7→ (x0 − z0 , x1 + z1 , x2 , · · · , xn , y2 , · · · , ym )

In the same way we get non-standard pattern embeddings An × Dm into
Dn+m and An × BCm into BCn+m with all factors conformal. Up to compositions, we now have a complete list. To be more precise, we say that a
pattern embedding is maximal if it only factors as a composition of two
embeddings trivially (meaning that one embedding is actually an isomorphism). The above constructions together with the list of possible image
subspaces for the irreducible embeddings give a complete list.
Corollary 4.10. The maximal, equal rank, non-standard pattern embeddings among products of the classical root systems ( with irreducible range)
are:
• D2 = A1 × A1 into A2
• An into BCn (in two ways for n > 2)
• An × Am into An+m
• An × Am into Dn+m
• An × Dm into Dn+m
• An × BCm into BCn+m
In all of the reducible cases the factors embed conformally but the images
are not orthogonal.
Proof. All listed embeddings have been constructed above. It remains to
show the list is complete. This is a direct consequence of the classification
of irreducible embeddings above and the discussion there of what the image
vector spaces can be. First the case when the image is type A:
The factors are all type A and embedded conformally. Let the target be
Ar , which we view in the standard way as a subset of vectors in Rr+1 with
coordinate sum zero. Each factor has image the vectors lying in the span of
a subset of the coordinates. Two such sets cannot overlap in more than one
coordinate or their intersection would contain a root vector. If two overlap
in exactly one factor then we have two factors embedding as in the lemma.
Thus our embedding is a composition of two nontrivial embeddings unless

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

25

these are the only two factors. The last possibility is that all factors land in
non-overlapping sets of coordinates. In this case the span of the images is a
subspace of Rr+1 of co-dimension equal to the number of factors. Since we
are assuming equal rank, this implies there is only one factor and our map
is therefore an isomorphism (and, in particular, conformal).
When the target is Dn or BCn we argue similarly, but things are simpler
as the roots span the full Rn in the stand coordinates for these root spaces.
Again, the image each irreducible factor is either the span of a subset of the
coordinates, or a hyperplane in such. Any two images of the first type must
map to disjoint sets of coordinates (as they must be disjoint subspaces).
The map thus factors through the map that combines these factors first,
hence is not maximal unless there are only the two factors. Since the map is
non-conformal while these image spaces are orthogonal, at least one of the
factors must be Ak embedding non-conformally into BCk , but then again
the map is non-maximal.
Thus there is at most one factor of the first type. Similarly, the argument
used above for An targets gives that maximality implies there is only one of
the second type as well. These are precisely the cases of one An factor, the
last three on the list.
Finally, suppose there are no factors of the first type and two of the second.
This is then an embedding of An × Am into either Dn+m or BCn+m . The
former is the third listed embedding. Since the images are assumed of the
hyperplane type, the classification of embeddings of A into BC says the
image lands in D ⊂ BC, which contradicts maximality.

4.2. Relations with AN -maps. Recall that if A ⊂ G is a maximal R−split
torus then diagonalizing the action of A on the Lie algebra of G gives the
collection of eigenvalue functions λi : A → R, which are the restricted roots.
If we choose a hyperplane in A to divide the roots into positive and negative,
the N in the KAN decomposition is the span of the root spaces Eλ ⊂ G
with λ positive. Hence, by equivariance, an AN -map from A1 N1 into A2 N2
gives an isomorphism A1 to A2 which sends positive roots to positive roots.
In particular, it gives a pattern embedding.
A basic Lie theory fact is that if r, r ′ , and r + r ′ are roots the letting
E∗ be the corresponding root spaces, one has [Er , Er′ ] = Er+r′ . The nontrivial part here is that the bracket is surjective. In the R-split case (or
the complex case) this says that any non-zero vectors in Er and Er′ have
non-zero bracket. For an AN -map, it follows that the image of the positive
roots is closed, meaning that if r, r ′ are roots in the domain then r + r ′ is
a root in the range iff it is in the domain. This puts strong limitations on
the kinds of non-conformality that can occur.
To make this precise, and extend to the non-split case, we focus on the
two kinds of basic non-conformality : a D2 mapping into an A2 or nonconformally into a BC2 , and an A2 into a BC2 .

26

DAVID FISHER AND KEVIN WHYTE

Lemma 4.11. Let f be an AN -map from A1 N1 into A2 N2 . If r and r ′
are positive roots in the domain which generate a D2 and which map to
non-orthogonal roots s and s′ then s + s′ is not a root.
Lemma 4.12. Let f be an AN -map from A1 N1 into A2 N2 . Suppose P is a
special plane for the domain of type A2 containing positive roots r, r ′ , and
r +r ′ and which map to roots s, s′ and s+s′ in a plane of type BC2 . Neither
s + 2s′ nor s′ + 2s is a root.
In both cases the conclusion is a restriction which roots in the image plane
are positive, constraining them to lie in a single quadrant. Note that this is
the same as saying the image is closed without any assumptions about the
sizes of the root spaces (in particular, nothing is assumed R-split).
We need a basic fact from the theory of semi-simple Lie algebras (which
is also responsible for the fact quoted above in the split case) : given a root
space Er and 0 6= v ∈ Er there is a vector u ∈ E−r so that [v, u] = Hr where
Hr is the element of the maximal torus so that for any root space Et and
any w ∈ Et , [Hr , w] = (r · t)w.
Proof. For the first lemma, let v and v ′ be non-zero vectors in the root space
Es and Es′ which are in the image of f and let u be as above relative to v.
Since r + r ′ is not a root, the images of Er and Er′ commute, in particular
[v, v ′ ] = 0. Then, by the Jacobi identity:
0 = [[v, v ′ ], u] = [[v, u], v ′ ]+[v, [v ′ , u]] = [Hs , v ′ ]+[v, [v ′ , u]] = (s·s′ )v ′ +[v, [v ′ , u]]
By assumption, s · s′ 6= 0, so [v, [v ′ , u]] 6= 0, which implies that [v ′ , u] 6= 0
so s′ − s must be a root. Thus if s + s′ is a root, there is a string of roots
(s′ − s), (s′ − s) + s, (s′ − s) + 2s. This happens only in BC2 and only if s
and s′ are orthogonal.
The second lemma is similar.
Let z and z ′ be elements of the root spaces Er and Er′ with z ′′ = [z, z ′ ] ∈
Er+r′ non-zero ( possible as [Er , Er′ ] = Er+r′ ). Let v ′ , v ′ , and v ′′ = [v, v ′ ]
be their images in the root spaces Es , Es′ , and Es+s′ respectively.
Suppose 2s + s′ is a root in the range (the other case is symmetric). This
gives a root string of length three in BC2 : s′ , s + s′ , 2s + s′ . As this is the
maximal length of a root string, neither s′ − s nor s′ + 3s is a root in the
range ( and s and s + s′ must be orthogonal). Since r ′ + 2r is not a root
in the domain ( the root system A2 has only length two root strings), we
have [v ′′ , v] = 0. Choose a u ∈ E−s in the range as before. We have, by the
Jacobi identity:
0 = [0, u] = [[v ′′ , v], u] = [[v ′′ , u], v]+[v ′′ , [v, u]] = [[v ′′ , u], v]−((s+s′ )·s)v = [[v ′′ , u], v]
The Jacobi identity says 0 = [v ′′ , u] = [[v, v ′ ], u] = [[v, u].v ′ ] + [v, [v ′ , u]].
The second term on the right is zero as s′ − s is not a root, and the first is
(sṡ′ )v ′ . Thus s · s′ = 0 which is a contradiction as s · (s + s′ ) = 0.

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

27


We now turn to discussing the prior pattern embeddings in this context.
Since an AN -map must send roots to roots, the two pattern embeddings of
An into BCn therefore could come from AN -maps into Bn or Cn respectively
(or either into BCn ).
For the first map, the A2 -planes that get sent to BC2 planes are the planes
Pij spanned by xi − x0 , xj − x0 , and xi − xj , which map to yi , yj , and yi − yi
inside the plane spanned by yi and yj . The ”missing” roots in the image
are yi + yj and −yi − yj . Thus the lemma above says that yi and yj must
have opposite signs. For this to be true for all i and j implies that n = 2.
Thus this map can arise from an AN -map only for A2 into BC2 (in which
case this map is the same as the other pattern embedding).
For the second map, the A2 planes mapping to BC2 planes are the planes
Pi spanned by xi − x0 , xi − x1 , and x1 − x0 , which map to yi + y1 , yi − y1 ,
and 2y1 inside the plane spanned by yi and y1 . The lemma then says yi − y1
and yi + y1 must be the same sign for all i. This happens, for example, if
yn > · · · > y1 > 0. This map does indeed arise an an AN -map (with this
order), as described previously.
In the case of the embedding of D4 into F4 , it does send roots to roots.
However the image is not closed under addition for any ordering, so there is
no AN -map.
For the reducible case, an AN -map for An ×Am → An+m was constructed
explicitly already, and the An × Dm → Dn+m , An × BCm → BCn+m , etc.
behave similarly. In all of these cases one can take the Dynkin diagram
for the target system and remove one edge so that the two components are
the diagrams for the two factors of the domain. If one orders the roots on
the factors so that one is positive and one is negative, additivity is always
satisfied.
The one case that does not fit this pattern is An × Am → Dn+m . If we
call the roots in the An by xi − xj for n ≥ i > j ≥ 0, for Am by zi − zj for
m ≥ i > j ≥ 0, and for Dn+m by yi ± yj for n + m1 > i > j ≥ 0 then the
mapped described sends:
• xi − xj to yi − yj
• zi − zj to yi+n−1 − yj+n−1 for i > j > 1
• zi − z1 to yi+n−1 − y1 for i > 1
• zi − z0 to yi+n−1 + y0 for i > 1
• z1 − z0 to y1 + y0
Since the individual factors map conformally, the only conditions from
the lemmas are the positive roots from one factor must have non-negative
inner-product with those from the other. From this we conclude:
• xi − x1 and zj − z1 have the same sign for all i and j at least 2
• xi − x0 and zj − z0 have opposite signs for all i and j at least 2
• x1 − x0 has opposite signs to both zi − z0 and zi − z1 for i > 1
• z1 − z0 has opposite signs to both xi − x0 and xi − x1 for i > 1

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DAVID FISHER AND KEVIN WHYTE

This is not possible as x2 − x1 has the same sign as z2 − z1 , which is
opposite to x1 − x0 and therefore the same as z2 − z0 hence opposite to
x2 − x0 . So x2 − x1 and x2 − x0 have opposite signs, which means it is
impossible for both to be opposite to z1 − z0 .
Finally we observe that the closed condition is in general sufficient for
the existence of an AN -map when the domain is split, so it is exactly these
pattern embeddings that arise from AN -maps. Since the subset if closed,
it is immediate that the span of the image root spaces gives a subgroup
of N2 which is graded exactly as N1 is. Thus the content of the claim is
that this implies the existence of an AN -map. To see this, let AN and
AN ′ correspond to the R-split groups for the given root systems. Let ∆ be
the set of simple roots in the domain. For each s ∈ ∆ choose an arbitrary
isomorphism Es → Es′ between the root spaces in domain and range. The
rest is determined inductively. Suppose we already have an isomorpshism
Er → Er′ defined, and s ∈ ∆ with r +s a root (note that the closed condition
says this happens in the domain iff it does in the range, so there is no
′
ambiguity). Then [Er , Es ] = Er+s and [Er′ , Es′ ] = Er+s
, so the map on
Er+s is determined. We need to see it is well-defined. If r is a positive root
then we need to see that all ways of writing r = s1 + s2 + · · · sn with all
partial sums roots give the same map Er → Er′ . To avoid too much Lie
theory, we prove this only for domain An , which is sufficient for all the maps
constructed here.
Lemma 4.13. Let s and t be simple roots whose sum is not a root. Suppose
r is a positive root such that r + s, r + t, and r + s + t are all roots. Then
the maps (Er ⊗ Es ) ⊗ Et → Er+s+t and (Er ⊗ Et ) ⊗ Es → Er+s+t are equal.
Proof. This is essentially just the Jacobi identity. Let u∗ be vectors in the
relevant root spaces. Then:
[ur , us ], ut ] = [[ur , ut ], us ] + [ur , [us , ut ]]
The second term on the right vanishes as Es and Et are assumed to
commute.

Thus we can freely permute adjacent terms provided the simple roots
commute. If we express xi − xj as above as a sum of simple roots in An
(namely xs+1 − xs for varying s) then we have the partial sums xik − xjk
where at each step either ik decreases by one or jk increases by one. The
simple roots of those two types always commute, so the lemma allows us to
move any one sum to any other.
5. Further Questions
We collect here some further questions concerning quasi-isometric embeddings of symmetric spaces left open by our work and which we believe would
be a useful guide to further research.

QUASI-ISOMETRIC EMBEDDINGS OF SYMMETRIC SPACES

29

• Can one classify the quasi-isometric embeddings of Sl3 (R) into Sp4 (R)?
It seems unlikely that the AN -maps are the only ones, but we do
not know how to construct others. Similarly, can one rule out quasiisometric embeddings for spaces of An to Cn type where no AN maps exist? Can one rule out embeddings that realize the other
linear pattern embedding from lemma 4.6 in section 4? Might it be
that the existence of a quasi-isometric embedding forces (maybe via
a limiting argument) the existence of an AN -map?
• When do quasi-isometric embeddings exist when rank increases? As
a start, if X quasi-isometrically embeds in Y × Rd for some d, does
that imply X quasi-isometrically embeds in Y ? More ambitiously,
is there any sense in which one can describe all the quasi-isometric
embeddings when rank increases? Again, as a start, are all quasiisometric embeddings of X in X × R graphs of Lipschitz functions?
• What can one say about uniformly proper embeddings? These give
perhaps a more natural geometric analogue of Margulis’ superrigidity. As a warning, recall that the horospheres give a uniformly proper
embedding of Rn in Hn+1 , so rank is not well behaved.

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David Fisher
Department of Mathematics
Indiana University
Bloomington, IN 47401
E-mail: fisherdm@indiana.edu
Kevin Whyte
Department of Mathematics
University of Illinois at Chicago
Chicago, Il 60607
E-mail: kwhyte@math.uic.edu

