ON THE EQUIVALENCE OF SEVERAL DEFINITIONS OFCOMPACT INFRA-SOLVMANIFOLDSSHINTARÔ KUROKI† AND LI YU*
Abstract. We show the equivalence of several definitions of compact infrasolvmanifolds that appear in various math literatures.

Infra-solvmanifolds are a speical class of aspherical manifolds studied bymathematicians for a long time. By the literal meaning, an infra-solvmanifold
is a smooth manifold finitely covered by a solvmanifold (which is the quotientof a connected solvable Lie group by a closed subgroup). Infra-solvmanifolds
include all flat Riemannian manifolds and infra-nilmanifolds.
It is known that compact Infra-solvmanifolds are smoothly rigid, i.e. any homotopy equivalence between two compact infra-solvmanifolds is homotopic toa diffeomorphism (see [3] and [8]). Geometrically, a compact infra-solvmanifold
M can be described (see [7, Proposition 3.1]) as a manifold which admits a sequence of Riemannian metric {gn } with uniformly bounded sectional curvatureso that (M, gn ) collapses in the Gromov-Hausdorff sense to a flat orbifold.
In the history of study on this topic, several definitions of compact infrasolvmanifolds were proposed in various math literatures. This is mainly because people look at these manifolds from very different angles (topologically,
algebraically or geometrically). The following are five such definitions.
Def 1: Let G be a connected, simply connected solvable Lie group, K be a
maximal compact subgroup of the group Aut(G) of automorphisms ofG, and Γ be a discrete subgroup of G o K. If [Γ : G ∩ Γ] < ∞ and theaction of Γ on G is cocompact and free, the orbit space Γ\G is called an
infra-solvmanifold modeled on G. See [7, Definition 1.1].
Def 2: A compact infra-solvmanifold is a manifold of the form Γ\G, where G isa connected, simply connected solvable Lie group, and Γ is a torsion-free
discrete subgroup of Aff(G) = GoAut(G) which acts on G cocompactly
Key words and phrases. Infra-solvmanifold, solvable Lie group, homogeneous space, discrete group action, holonomy.
2010 Mathematics Subject Classification. 22E25, 22E40, 22F30, 53C30†
The author is supported in part by the JSPS Strategic Young Researcher Overseas Visits
Program for Accelerating Brain Circulation “Deepening and Evolution of Mathematics andPhysics, Building of International Network Hub based on OCAMI”.
*The author is partially supported by Natural Science Foundation of China (Grant
no.11001120) and the PAPD (priority academic program development) of Jiangsu higher
education institutions.
1and the closure of hol(Γ) in Aut(G) is compact where hol : Aff(G) →Aut(G) is the holonomy projection.
Def 3: A compact infra-solvmanifold is a double coset space Γ\G/K where G isa virtually connected and virtually solvable Lie group, K is a maximal
compact subgroup of G and Γ is a torsion-free, cocompact, discrete
subgroup of G. See [2, Definition 2.10].
A virtually connected Lie group is a Lie group with finitely many
connected components.
Def 4: A compact infra-solvmanifold M is an orbit space of the form M = ∆\S
where S is a connected, simply connected solvable Lie group acted upon
cocompactly by a torsion-free closed subgroup ∆ ⊂ Aff(S) satisfying– the identity component ∆0 of ∆ is contained in the nil-radical of S,– the closure of hol(∆) in Aut(S) is compact.
Def 5: A compact infra-solvmanifold is a manifold of the form ∆\G, where G
is a connected, simply connected solvable Lie group, and ∆ is a torsionfree subgroup of Aff(G) which acts cocompactly on G and the closureof hol(∆) in Aut(G) is compact.
Note that the subgroup ∆ in Def 4 and Def 5 is not necessarily discrete. The
Def 4 is from [3, Definition 1.1] and Def 5 is from [1, Definition 1.1].
The main purpose of this paper is to explain why the above five definitionsof compact infra-solvmanifolds are equivalent. The authors are fully aware thatthe reason is probably known to many people. However, we do not find any
formal proof of this equivalence and feel this phenomenon a little confusing. So
we want to write a proof here for the convenience of future reference. However,
we do not intend to give a completely new treatment of this subject. Our proof
will directly quote some results on infra-solvmanifolds from [1], [3] and [8] anduse many subtle facts in Lie theory. This paper can be treated as an elementary
exposition of compact infra-solvmanifolds and some related concepts. Now let
us start to prove the following main result of the paper.
Proposition 1: The five definitions Def 1 – Def 5 of compact infra-solvmanifolds
are all equivalent.
Proof. We divide our proof into several parts. The framework of the proof isshown below.
Def 1

Def 3

Def 2

Def 4

Def 5
2

Def 1 ⇐⇒ Def 2
Let M be a compact infra-solvmanifold in the sense of Def 1. First of all,any g ∈ Γ can be decomposed as g = kg ug where kg ∈ K ⊂ Aut(G) andug ∈ G. The holonomy projection hol : G o Aut(G) → Aut(G) sends g to kg .
Since hol is a group homomorphism, its image hol(Γ) is a subgroup of K. By
assumption, |hol(Γ)| = [Γ : G ∩ Γ] is finite, so hol(Γ) is compact. In addition,
since G is a connected simply-connected solvable Lie group, G is diffeomorphic
to an Euclidean space. Then by Smith fixed point theorem ([6, Theorem I]), Γacting on G freely implies that Γ is torsion-free. So M satisfies Def 2.
Conversely, if M is a compact infra-solvmanifold in Def 2, then [8, Theorem
3 (a) ⇒ (f)] tells us that there exists a connected, simply-connected solvable
Lie group G0 so that the Γ (which defines M ) can be thought of as discrete
subgroup of G0 o F where F is a finite subgroup of Aut(G0 ). Moreover, there
exists an equivariant diffeomorphism from G to G0 with respect to the actionof Γ (by [8, Theorem 1 and Theorem 2]). Hence
M = Γ\G ∼= Γ\G0 .
diff.

Then [Γ : Γ ∩ G0 ] = |holG0 (Γ)| ≤ |F | is finite.
It remains to show that the action of Γ on G0 is free. Since F is finite, we
can choose an F -invariant Riemannian metric on G0 (in fact we only need F
to be compact). If the action of an element g ∈ Γ on G0 has a fixed point, say
h0 . Let Lh0 be the left translation of G0 by h0 . Then L−1
h0 gLh0 fixes the identity−1
element, which implies Lh0 gLh0 ∈ F . So there exists a compact neighborhood
U of h0 so that g · U = U and g acts isometrically on U with respect to
the Riemannian metric just as the element L−1
h0 gLh0 ∈ F acts around e. So
A = Dg : Th0 U → Th0 U is an orthogonal transformation. Then A is conjugatein O(n) to the block diagonal matrix of the formB(θ1 )
0
..
,

.
0
B(θm )
cos θj − sin θjwhere B(0) = 1, B(π) = −1, and B(θj ) =, 1 ≤ j ≤ m.
sin θj cos θjSince Γ is torsion-free, g is an infinite order element. This implies that atleast one θj is irrational. Then for any v 6= 0 ∈ Th0 G0 , |{An v}n∈Z | = ∞. So
there exists h ∈ U so that |{g n · h}n∈Z | = ∞. Then the set {g n · h}n∈Z ⊂ U has
at least one accumulation point. This contradicts the fact that the orbit spaceΓ\G0 is Hausdorff (since Γ\G0 = M is a manifold).
Combing the above arguments, M is an infra-solvmanifold modeled on G0 in
the sense of Def 1.
Def 1 =⇒ Def 3
3

Let M be an infra-solvmanifold in the sense of Def 1. Let hol(Γ) be theimage of the the holonomy projection hol : Γ → Aut(G). Then hol(Γ) is a
finite subgroup of Aut(G).
e = Gohol(Γ) which is virtually solvable. It is easy to see that hol(Γ)
Define G
e and, Γ is a cocompact, discrete subgroupis a maximal compact subgroup of G∼ee
of G. Then M = Γ\G/hol(Γ). So M satisfies Def 3.
Def 3 =⇒ Def 2
Let M = Γ\G/K be a compact infra-solvmanifold in the sense of Def 3. Since
K is a maximal compact subgroup of G, G/K is contractible. Note that K is
not necessarily a normal subgroup of G. Hence G/K may not directly inherita group structure from G.
Let G0 be the connected component of G containing the identity element.
Then by [4, Theorem 14.1.3 (ii)], K0 = K ∩G0 is connected and K0 is a maximal
compact subgroup of G0 . Moreover, K intersects each connected component of
G and K/K0 ∼= G/G0 . By the classical Lie theory, the Lie algebra of a compact
Lie group is a direct product of an abelian Lie algebra and some simple Liealgebras. Then since the Lie algebra Lie(G) of G is solvable and K is compact,
the Lie algebra Lie(K) ⊂ Lie(G) must be abelian. This implies that K0 is a
torus and hence a maximal torus in G0 .
In addition, since G is virtually solvable, G0 is actually solvable. This isbecause the radical R of G is a normal subgroup of G0 and dim(R) = dim(G0 )(since G is virtually solvable). So G0 /R is discrete. Then since G0 is connected,
G0 must be equal to R.
Let Z(G) be the center of G and define C = Z(G) ∩ K. Then C is clearly anormal subgroup of G. Let G0 = G/C and K 0 = K/C and let ρ : G → G0 be
the quotient map. Since Γ ∩ K = {1}, Γ ∼= ρ(Γ) ⊂ G0 , we can think of Γ as a0
subgroup of G . So we haveM = Γ\G/K ∼(1)= Γ\G0 /K 0 .
Let G00 = ρ(G0 ) be the identity component of G0 . Then G00 is a finite index
solvable normal subgroup of G0 .
Lie(G00 ) = Lie(G0 ) = Lie(G)/Lie(C) = Lie(G0 )/Lie(C).

(2)

Besides, let K00 = K 0 ∩ G00 which is a maximal torus of G00 . We have
Lie(K00 ) = Lie(K 0 ) = Lie(K)/Lie(C) = Lie(K0 )/Lie(C).

(3)

Claim-1: G00 is linear and so G0 is linear.
A group is called linear if it admits a faithful finite-dimensional representation. By [4, Theorem 16.2.9 (b)], a connected solvable Lie group S is linear if
and only if t ∩ [s, s] = {0} where s and t are Lie algebras of S and its maximal
torus TS , respectively. And for a general connected solvable group S, the Liesubalgebra t∩[s, s] is always central in s. So for our G0 and its maximal torus K0 ,4the Lie subalgebra Lie(K0 ) ∩ [Lie(G0 ), Lie(G0 )] is central in Lie(G0 ) = Lie(G).
So Lie(K0 ) ∩ [Lie(G0 ), Lie(G0 )] ⊂ Lie(Z(G)) andLie(K0 ) ∩ [Lie(G0 ), Lie(G0 )] ⊂ Lie(K) ∩ Lie(Z(G)) = Lie(C)Then for the Lie group G00 and its maximal torus K00 , we have
Lie(K0 )Lie(G0 ) Lie(G0 )000
Lie(K0 ) ∩ [Lie(G0 ), Lie(G0 )] =∩,= 0.
Lie(C)
Lie(C) Lie(C)

(4)

(5)

So by [4, Theorem 16.2.9 (b)], G00 is linear. Moreover, suppose V is a faithfulfinite-dimensional representation of G00 . Then R[G0 ] ⊗R[G00 ] V is a faithful finitedimensional representation of G0 where R[G0 ] and R[G00 ] are the group rings of
G0 and G00 over R, respectively. So the Claim-1 is proved.
From the Claim-1 and (1), we can just assume that our group G is linear at
the beginning. Under this assumption, G0 is a connected linear solvable group.
So there exists a simply connected solvable normal Lie subgroup S of G0 sothat G0 = S o K0 and [G0 , G0 ] ⊂ S (see [4, Lemma 16.2.3]). So
Lie(G0 ) = Lie(K0 ) ⊕ Lie(S).
More specifically, we can take S = p−1 (V ) where p : G0 → G0 /[G0 , G0 ] is thequotient map and V is a vector subgroup of the abelian group G0 /[G0 , G0 ] sothat G0 /[G0 , G0 ] ∼= p(K0 ) × V (see the proof of [4, Theorem 16.2.3]). Note thatthe vector subgroup V is not unique, nor is S.
Claim-2: We can choose S to be normal in G and so G ∼= S o K.
Indeed since K is compact, we can choose a metric on Lie(G0 ) which isinvariant under the adjoint action of K. Then we can choose V so that Lie(S) is
orthogonal to Lie(K0 ) in Lie(G0 ). Then because K0 is normal in K, the adjoint
action of K on Lie(G0 ) preserves Lie(K0 ), so it also preserves the orthogonal
complement Lie(S) of Lie(K0 ). This implies that S is preserved under theadjoint action of K.
Let G0 , h1 G0 , · · · , hm G0 be all the connected components of G. Since Kintersects each connected component of G, we can assume that hi ∈ K for all
1 ≤ i ≤ m. Then any element g ∈ G can be written as g = g0 hi for some−1
−1
g0 ∈ G0 and hi ∈ K. So gSg −1 = g0 hi Sh−1
i g0 ⊂ g0 Sg0 ⊂ S. The Claim-2 isproved.
From the semidirect product G = S o K, we can define an injective grouphomomorphism α : G → Aff(S) = S o Aut(S) as follows. For any g ∈ G, we
can write g = sg kg for a unique sg ∈ S and kg ∈ K since S ∩ K = S ∩ K0 = {1}.
Then α(g) : S → S is the composition of the adjoint action of kg on S and theleft translation on S by sg , i.e. α(g) = Lsg ◦ Adkg .
Claim-3: α(Γ)\S is diffeomorphic to the double coset space Γ\G/K.
Notice that each left coset in G/K contains a unique element of S, so we have
G/K = {sK ; s ∈ S}.
5

For any γ ∈ Γ, let γ = sγ kγ where sγ ∈ S and kγ ∈ K. We have
γsK = sγ kγ sK = sγ kγ skγ−1 K = α(γ)(s)K, ∀ s ∈ S.

(6)

So the natural action of Γ on the left coset space G/K can be identified withthe action of α(Γ) ⊂ Aff(S) on S. The Claim-3 is proved.
Let Ad : K → Aut(S) denote the adjoint action of K on S. Since K iscompact and Ad is continuous, Ad(K) ⊂ Aut(S) is also compact. Notice thatα(Γ) is a subgroup of S o Ad(K) ⊂ S o Aut(S). So the closure hol(α(Γ)) of
the holonomy group hol(α(Γ)) in Aut(S) is contained in Ad(K). So hol(α(Γ))
is compact. This implies that Γ\G/K ∼= α(Γ)\S is an infra-solvmanifold in thesense of Def 2.
Remark 1: A simply-connected solvable Lie group is always linear, but for
non-simply-connected solvable Lie groups, this is not always so. A counterexample is the quotient group of the Heisenberg group by an infinite cyclic group(see [5, p.169 Example 5.67]).
It is obviously true that Def 2 =⇒ Def 4 =⇒ Def 5. So it remains to show
Def 5 =⇒ Def 2.
Def 5 =⇒ Def 2
This follows from a nontrivial result in [1]. Indeed, the fundamental groupof the manifold ∆\G in Def 5 is Γ = ∆/∆0 , which is a torision-free virtually
poly-cyclic group (see [2]). It is shown in [1] that such a group Γ determines avirtually solvable real linear algebraic group HΓ which contains Γ as a discreteand Zariski-dense subgroup. HΓ is called the real algebraic hull of Γ in [1].
In addition, we have HΓ = U o T where T is a maximal reductive subgroupof HΓ and U is the unipotent radical of HΓ . The splitting of HΓ gives aninjective group homomorphism α : HΓ → Aff(U ) and a corresponding affine
action of Γ < HΓ on U . It is shown in [1] that MΓ = α(Γ)\U is a compact
infra-solvmanifold whose fundamental group is Γ. MΓ is called the standard
Γ-manifold. Now since α(Γ) is a discrete subgroup of Aff(U ), MΓ is a compact
infra-solvmanifold in the sense of Def 2. Moreover ∆\G is diffeomorphic to MΓ
by [1, Theorem 1.4].
So we finish the proof of Proposition 1.

Remark 2: If we remove the “cocompact” condition in Def 1 – Def 5, we may
get noncompact infra-solvmanifolds, which are vector bundles over some compact infra-solvmanifolds (see [8, Theorem 6]).
6

Acknowledgements
The authors want to thank M. Masuda, J. B. Lee, W. Tuschmann, J. Hillman
and F. Farrell for some helpful information and comments. Besides, the first author would like to thank Y. Karshon for providing him excellent circumstances
to do research.
References[1] O. Baues, Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups,
Topology 43 (2004), no. 4, 903–924.
[2] F. Farrell and L. Jones, Classical aspherical manifolds, CBMS Regional Conference Seriesin Mathematics, 75. Published for the Conference Board of the Mathematical Sciences,
Washington, DC; by the American Mathematical Society, Providence, RI, 1990.
[3] F. Farrell and L. Jones, Compact infrasolvmanifolds are smoothly rigid, Geometry fromthe Pacific Rim (Singapore, 1994), 85–97, de Gruyter, Berlin, 1997.
[4] J. Hilgert and K-H. Neeb, Structure and geometry of Lie groups, Springer monographs inmathematics, 2012.
[5] K. Hofmann and S. A. Morris, The structure of compact groups. A primer for the student— a handbook for the expert, Second revised and augmented edition, de Gruyter Studies
in Mathematics, 25. Walter de Gruyter & Co., Berlin, 2006.
[6] P. A. Smith, Fixed-point theorems for periodic transformations, Amer. J. Math. 63 (1941),
1–8.
[7] W. Tuschmann, Collapsing, solvmanifolds and infrahomogeneous spaces, Differ. Geom.
Appl. 7 (1997), 251–264.
[8] B. Wilking, Rigidity of group actions on solvable Lie groups, Math. Ann. 317 (2) (2000),
195–237.
†

Osaka City University Advanced Mathematical Institute (OCAMI), 3-3-138
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japanand Department of Mathematics, University of Toronto, Room 6290, 40 St.
George Street, Toronto, Ontario, M5S 2E4, Canada
E-mail address: kuroki@scisv.sci.osaka-cu.ac.jp, shintaro.kuroki@utoronto.ca*Department of Mathematics and IMS, Nanjing University, Nanjing, 210093,
P.R.China
E-mail address: yuli@nju.edu.cn

7