arXiv:1702.00954v2 [math.DG] 22 Dec 2017

The growth of the first non-Euclidean filling function of
the quaternionic Heisenberg group
Moritz Gruber

Abstract. The filling volume functions of the n-th quaternionic Heisenberg group grow, up to
dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling
volume function in dimension n+1, which is strictly faster than the growth rate of the (n + 1)dimensional filling volume function of the Euclidean space.
Keywords: isoperimetric inequalities, filling functions, nilpotent Lie groups

1 Introduction

In general, isoperimetric inequalities delimit the maximal volume needed to fill a boundary of a given
volume. Different types of isoperimetric inequalities arise by specifying restrictions to the boundaries and
fillings. An interesting class of such isoperimetric inequalities is formed by the filling volume functions.
These describe the difficulty to fill Lipschitz cycles by Lipschitz chains. In [3] we proved an Euclidean
behaviour for the filling volume function in dimension m + 1 of the nth quaternionic Heisenberg group
m+1
n
for m ∈ {1, ..., n − 1}, i.e. it grows like l m . The known results for the filling volume functions of
HH
n+2
the complex Heisenberg group HCn (see [7],[8]) suggest a growth like l n for the filling volume function of
n
HH
in dimension n + 1. Our technique in [3] could only confirm the super-Euclidean behaviour without
telling the exact growth. Now we are able to state it:
n
Theorem 1. Let HH
be the nth quaternionic Heisenberg group, equipped with a left-invariant Riemannian
metric. Then:
n+2
n+1
FH
.
n (l) ∼ l n
H

This theorem was part of the author’s dissertation [4] at the Karlsruhe Institute of Technology.

2 Filling volume functions & Heisenberg groups

2.1 Filling volume functions

Filling volume functions describe the difficulty to fill a Lipschitz cycle of a given mass by a Lipschitz
chain. In the following let X be a metric space and m ∈ N. Further we denote by volm the m-dimensional
Hausdorff-measure of X and by ∆m we denote the m-simplex equipped with an Euclidean metric.

2

2 Filling volume functions & Heisenberg groups

P
A Lipschitz m-chain a in X is a (finite) formal sum a = j zj αj of Lipschitz maps αj : ∆m → X
P
with coefficients zj ∈ Z. The boundary of a Lipschitz m-chain a = j zj αj is defined as the Lipschitz
(m − 1)-chain
m
X
X

(−1)i αj|∆m
zj
∂a =
i
i=0

j

th
where ∆m
face of ∆m . A Lipschitz m-chain a with zero-boundary, i.e. ∂a = 0, is called a
i denotes the i
Lipschitz m-cycle. A filling of a Lipschitz m-cycle a is a Lipschitz (m + 1)-chain b with boundary ∂b = a.
We define the mass of a Lipschitz m-chain a as the total volume of its summands:
X
zj volm (αj ) .
mass(a) :=
j

In the case that X is a Riemannian manifold, the volume of such a summand is given by volm (αj ) =
R
∆m Jαj dλ, where dλ denotes the m-dimensional Lebesgue-measure and Jαj is the jacobian of αj . This
is well defined, as Lipschitz maps are, by Rademacher’s Theorem, almost everywhere differentiable.
Given an m-cycle, one is interested in the filling with the smallest mass. In the next step one can vary
the cycle and examine how large the ratio between the mass of the optimal filling and the mass of the
cycle can get. This leads to a family of invariants of the space X, the filling volume functions:
Definition. Let n ∈ N and let X be an n-connected metric space. For m ≤ n the (m + 1)-dimensional
filling volume function of X is given by
m+1
FX
(l) = sup inf mass(b)
a

b

∀l ∈ R+ ,

where the infimum is taken over all (m + 1)-chains b with ∂b = a and the supremum is taken over all
m-cycles a with mass(a) ≤ l.
As we are mostly interested in the large scale geometry of the space X, the exact description of the
filling volume functions is of less importance to us. Indeed we only look at the asymptotic behaviour of
the functions. We do this by the following equivalence relation, which makes the growth rate of the filling
volume functions a quasi-isometry invariant.
Definition. Let f, g : R+ → R+ be functions. Then we write f 4 g if there is a constant C > 0 with
f (l) ≤ Cg(Cl) + Cl + C

∀l ∈ R+ .

If f 4 g and g 4 f we write f ∼ g. This defines an equivalence relation.
We read the notation f 4 g as “f is bounded from above by g” respectively “g is bounded from below
by f ” according whether we are more interested in f or g.
Proposition 2.1 (see for example [6, Lemma 1]). Let X and Y be n-connected Riemannian manifolds.
Then:
m+1
X quasi-isometric to Y ⇒ FX
∼ FYm+1 ∀m ≤ n.
Any two left-invariant Riemannian metrics on a Lie group are Lipschitz-equivalent. So, by the above
proposition, the behaviour of the filling volume functions of a Lie group equipped with a left-invariant
Riemannian metric does not depend on the choice of this metric.

3

2 Filling volume functions & Heisenberg groups

Let’s look at the example of the filling volume functions of the n-dimensional Euclidean space. They
were first computed by Federer and Fleming in [2].
Example. The filling volume functions of the Euclidean space E n are
FEm+1
(l) ∼ l
n

m+1
m

for m ≤ n − 1.

This enables us to use the terms Euclidean, sub-Euclidean and super-Euclidean filling volume function
for filling volume functions with the same, strictly slower respectively strictly faster growth rate than
m+1
l m .
The following theorem generalises the Euclidean case to spaces with non-positive curvature. For a
proof see [5].
Theorem 2.2. The filling volume functions of an n-dimensional Hadamard space X are
m+1
FX
(l) 4 l

m+1
m

for m ≤ n − 1.

We are interested in this, as the fact that a Riemannian manifold with non-positive curvature has
Euclidean or sub-Euclidean filling volume functions in all dimensions yields a sufficient criterion for the
existence positive curvature: Let M be a Riemannian manifold with a super-Euclidean filling volume
function in some dimension, then it can’t be of non-positive curvature.

2.2 Heisenberg groups

The complex Heisenberg group HCn is a higher
enberg group

 1 x
Heis = 0 1

0 0

dimensional analogue of the classical 3-dimensional Heis

z

y | x, y, z ∈ R ≤ GL3 (R).

1

Equipped with a left invariant Riemannian metric it appears as a horosphere in the complex-hyperbolic
space SU(n, 1)/ S(U(n) × U(1)). As an abstract Lie group it can be defined as follows:
Definition. The nth complex Heisenberg group HCn is the unique simply connected Lie group with Lie
algebra hnC generated by
BC := {h1 , ..., hn , k1 , ..., kn , K}
and with
[km , hm ] = K
for all m ∈ {1, ..., n} and with all other brackets of generators equal to zero. This is a 2-step nilpotent
Lie group of dimension 2n + 1 with Lie algebra hnC = V1 ⊕ V2 , where V1 = span{h1 , ..., hn , k1 , ..., kn } ∼
= Cn
and V2 = span{K} ∼
= R.
= Im C ∼
n
The group we examine in this paper is the quaternionic Heisenberg group HH
. It appears similar to the
complex Heisenberg group as a horosphere, now in the quaternionic-hyperbolic space Sp(n, 1)/(Sp(n) ×
Sp(1)).

2 Filling volume functions & Heisenberg groups

4

n
Definition. The nth quaternionic Heisenberg group HH
is the unique simply connected Lie group with
n
Lie algebra hH generated by

BH := {h1 , ..., hn , i1 , ..., in , j1 , ..., jn , k1 , ..., kn , I, J, K}
and with
[km , jm ] = I = [im , hm ], [im , km ] = J = [jm , hm ], [jm , im ] = K = [km , hm ]
for all m ∈ {1, ..., n} and with all other brackets of generators equal to zero. This is a 2-step nilpotent Lie
group of dimension 4n+3 with Lie algebra hnH = V1 ⊕V2 where V1 = span{h1 , ..., hn , i1 , ..., in , j1 , ..., jn , k1 , ..., kn } ∼
=
Hn and V2 = span{I, J, K} ∼
= Im H, where H denotes the Hamilton quaternions.
These nilpotent Lie groups are representatives of a large class of nicely behaving Lie groups:
Definition. A simply connected, d-step nilpotent Lie group G with Lie algebra g is stratified, if there is
a decomposition g = V1 ⊕ ... ⊕ Vd of the Lie algebra with [V1 , Vj ] = Vj+1 .
Let G be a stratified nilpotent Lie group. Then for every t > 0 there is an automorphism sbt : g → g
which maps v ∈ Vj to tj vj . We denote the corresponding automorphism exp ◦b
st of the Lie group by st .
2.3 A technical criterion for lower bounds

For our proof of the lower bound on the (n + 1)-dimensional filling volume function of the quaternionic
n
Heisenberg group HH
, we will apply a technique based on Stokes theorem.
Proposition 2.3 (see [1, Proposition 1.2]). Let G be a stratified nilpotent Lie group equipped with a
left-invariant Riemannian metric and let m ∈ N. If there exists a Lipschitz (m + 1)-chain b and a closed
G-invariant (m + 1)-form η in G and constants C, r, s > 0 such that
1) mass(st (∂b)) ≤ Ctr ,
2)

R

b

η>0,

3) s∗t η = ts η ,
s

then holds FGm+1 (l) < l r .
Burillo used this criterion to compute lower bounds on the filling volume functions of the complex
n
Heisenberg group HCn . We will use this criterion as well as the property that HH
contains HCn .
The change-over from the complex to the quaternionic Heisenberg group complicates the construction of
the form η, as there are more non-trivial brackets and therefore G-invariant (n+1)-forms with s∗t η = tn+2 η
get very rare. In [4] we have even seen, that in the case of the octonionic Heisenberg group there are no
such forms.

5

3 The proof of Theorem 1
3 The proof of Theorem 1

In this section we compute the (n + 1)-dimensional filling volume function of the quaternionic Heisenberg
n
n
group HH
. As HH
fulfils the conditions of [3, Theorem 1], we already have the super-Euclidean upper
bound:
n+2
n+1
FH
.
n (l) 4 l n
H

Thus it remains to prove the corresponding lower bound.
n
Proposition 3.1. Let HH
be the nth quaternionic Heisenberg group. Then:
n+1
FH
n (l) < l

n+2
n

H

.

Proof. We are going to use Proposition 2.3. To this end, we have to construct a Lipschitz (n + 1)-chain b
n
n
n
in HH
and a closed HH
-invariant (n + 1)-form η on HH
with the correct scaling behaviour. We start with
the constructions Burillo did in the proof of [1, Theorem 2.1] to obtain the lower bound for the filling
n+1
volume function FH
of the complex Heisenberg group. We denote the there constructed (n + 1)-chain
n
C
′
by b and the corresponding (n + 1)-form by γ.
Now let BC be the usual basis of the Lie algebra hnC of the complex Heisenberg group HCn and BH the
n
basis of the Lie algebra hnH of the quaternionic Heisenberg group HH
(compare Section 2.2). Then the
n
n
complex Heisenberg group HC embeds as a Lie subgroup into the quaternionic Heisenberg group HH
.
We do this on the Lie algebra level via the map
Φ : hnC → hnH

defined by hi 7→ hi , ki 7→ ki , K 7→ K .

Hence we can consider the (n + 1)-chain b′ constructed in [1] as an (n + 1)-chain in the quaternionic
n
Heisenberg group HH
. We set b := b′ . The above embedding respects the grading of the Lie algebras,
i.e. vectors of the first layer are mapped to vectors of the first layer and vectors of the second layer are
mapped to vectors of the second layer. Therefore the boundary ∂b of the chain b has the same scaling
n
behaviour in the quaternionic Heisenberg group HH
as it had in the complex Heisenberg group HCn , i.e.
n
mass(st (∂b)) ≤ mass(∂b) · t .
n
n
, such that η restricts
Consequently, it remains to construct a closed, HH
-invariant (n+1)-form η on HH
n
on the embedded HC to the (n + 1)-form γ (this implies condition 2) in Proposition 2.3) and such that
η satisfies
s∗t η = tn+2 η .
n
) by
The form γ is given (with respect to the above notation for HCn ⊂ HH

γ = (−1)n · K ∗ ∧ h∗1 ∧ ... ∧ h∗n
where for v ∈ hnH the symbol v ∗ denotes the dual form of v.
We start the construction of η by defining some special n-forms. For this let v0,m := h∗m and v1,m := i∗m
for m ∈ {1, ..., n}. Further let
Aeven := {a ∈ {0, 1}n |

n
X

m=1

am is even}

6

3 The proof of Theorem 1

Aodd := {a ∈ {0, 1}n |

n
X

am is odd}

m=1

Then we define the n-forms

for a ∈ {0, 1}n

v(a) := va1 ,1 ∧ va2 ,2 ∧ ... ∧ van ,n
and assign them signs by
(

sign(a) :=
We set
η :=

X

1

if

−1 if

Pn

m=1

am ≡ 0 or 3 (mod 4)

m=1

am ≡ 1 or 2 (mod 4)

Pn

sign(a)v(a) ∧ K ∗ −

a∈Aeven

X

sign(a)v(a) ∧ J ∗

a∈Aodd

The only form v(a), which is not zero when restricted to HCn , is
v(0, ..., 0) = h∗1 ∧ h∗2 ∧ ... ∧ h∗n
As the sign of this form is “+”, the form η coincides with γ on HCn .
Further holds
s∗t η = tn+2 η
as each summand in η consists of n dual forms of vectors of the first layer of the grading of the Lie algebra
hnH which scale linearly and one dual form of a vector of the second layer of the grading which scales
quadratically. It remains to show that η is closed. For this purpose we will use the bijections
τr : Aodd → Aeven , (a1 , ..., ar , ..., an ) 7→ (a1 , ..., 1 − ar , ..., an ).
Then:
(−1)n · dη =

X

sign(a)v(a) ∧ dK ∗ −

a∈Aeven

=

X

sign(a)v(a) ∧
−

X

a∈Aeven
r=1,...,n

sign(a)v(a) ∧ dJ ∗

a∈Aodd

a∈Aeven
r=1,...,n

=

X

X

(jr∗

∧

i∗r

+ kr∗ ∧ h∗r )

sign(a)v(a) ∧ (i∗r ∧ kr∗ + jr∗ ∧ h∗r )

a∈Aodd
r=1,...,n


sign(a)v(a) ∧ (jr∗ ∧ i∗r + kr∗ ∧ h∗r )


− sign(τr (a))v(τr (a)) ∧ (i∗r ∧ kr∗ + jr∗ ∧ h∗r )

Each of term of the sum is zero as the following computations show:

P
We treat the case ar = 1, the case ar = 0 works analogue. As a ∈ Aeven we have
m am ≡
P
0 or 2 (mod 4) and therefore m τr (a)m ≡ 3 or 1 (mod 4) as the value at position r changes from 1 to
0. By the definition of the sign we have sign(a) = sign(τr (a)). As ar = 1 the form v(a) contains i∗r (and

7

References
not h∗r ) and the form v(τr (a)) contains h∗r (and not i∗r ). Therefore such a summand looks like
sign(a) · v(a) ∧ kr∗ ∧ h∗r − v(τr (a)) ∧ i∗r ∧ kr∗



The forms v(a) and v(τr (a)) only differ at position r. To see that v(a) ∧ kr∗ ∧ h∗r and v(τr (a)) ∧ i∗r ∧ kr∗
coincide, we have to bring all index r terms at coinciding positions and compare the generated signs.
We do this by bringing i∗r to the last position, hr ∗ to the second-to-last position and kr∗ the third-to-last
position. In the case of v(a) ∧ kr∗ ∧ h∗r this changes the sign by (−1)n−r+2 · 1 · 1 = (−1)n−r+2 . In the case
of v(τr (a)) ∧ i∗r ∧ kr∗ this changes the sign by (−1) · (−1)n−r+1 · 1 = (−1)n−r+2 . So the two signs are equal
and therefore the two forms coincide.
This implies
dη = 0
So η is closed and the conditions of Proposition 2.3 are fulfilled. Consequently we get the lower bound
n+2
n+1
FH
n (l) < l n .
H

n+1
Together with the upper bound FH
n (l) 4 l
H

n+2
n

from [3] this proves Theorem 1.

References

[1] José Burillo. Lower bounds of isoperimetric functions for nilpotent groups. In Geometric and computational perspectives on infinite groups (Minneapolis, MN and New Brunswick, NJ, 1994), volume 25
of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 1–8. Amer. Math. Soc., Providence,
RI, 1996.
[2] Herbert Federer and Wendell H. Fleming. Normal and integral currents. Ann. of Math. (2), 72:458–
520, 1960.
[3] Moritz Gruber. Filling Invariants of Stratified Nilpotent Lie Groups. arXiv:1507.04871v3.
[4] Moritz Gruber. Large Scale Geometry of Stratified Nilpotent Lie Groups. Dissertation, 2016.
[5] Stefan Wenger. A short proof of Gromov’s filling inequality. Proc. Amer. Math. Soc., 136(8):2937–
2941, 2008.
[6] Robert Young. Homological and homotopical higher-order filling functions. Groups Geom. Dyn.,
5(3):683–690, 2011.
[7] Robert Young. Filling inequalities for nilpotent groups through approximations. Groups Geom. Dyn.,
7(4):977–1011, 2013.
[8] Robert Young. High-dimensional fillings in Heisenberg groups. J. Geom. Anal., 26(2):1596–1616,
2016.

Moritz Gruber
Fakultät für Mathematik
Karlsruhe Institute of Technology
Englerstrae 2
76131 Karlsruhe
Germany
E-mail address: moritz.gruber@kit.edu

