Outer automorphisms of hyperbolic groups with property (T)

Outer automorphisms of hyperbolic groups with property (T)

Rudolf Zeidler April 23, 2013

In this expository note, we present a result due to Frédéric Paulin which implies that a finitely generated hyperbolic group with Kazhdan’s property (T) has finite outer automorphism group.

Statement and context

For a groupG, theouter automorphism group, denoted by Out(G), is the group of auto- morphisms ofGmodulo the normal subgroup of inner automorphisms.

The result we aim to present is the following.

Theorem 1([Pau91]). LetGbe a finitely generated hyperbolic group. If the outer automor- phism groupOut(G)is infinite, there exists a non-trivial (i.e. without a global fixed point) isometric action ofGon anR-tree.

Corollary 2. A finitely generated hyperbolic group with Kazhdan’s property (T) has finite outer automorphism group.

Examples of hyperbolic groups with property (T) are lattices in Sp(1, n) (the isom- etry group of quaternionic hyperbolic space), and some examples of (T)-groups ob- tained fromŻuk’s spectral criterion (in particular, see the construction of Ballmann–

Świa¸tkowski [BŚ97, Section 4]).

Note that there are hyperbolic groups with infinite outer automorphism group, for instance the free group, and more generally, fundamental groups of compact orientable surfaces.

Also, there exist groups with property (T) and infinite outer automorphism group [Cor07; OW07], an example of which we sketch below.

Example 3([Cor07],[BHV08, Exercise 1.8.19]). Letn>3 and consider the semi-direct product

G:= (M_n(Z),+)oSL_n(Z),

where SL_n(Z) acts by left matrix multiplication on the additive group ofn×n-matrices M_n(Z). ThenGhas Kazhdan’s property (T), cf. [BHV08, Corollary 1.4.16, Exercise 1.8.7].

To see that Out(G) is infinite, note that SL_n(Z) also acts by right matrix multiplication on M_n(Z), and this induces an action of SL_n(Z) on G. — More precisely, for each γ∈SL_n(Z), let

φ_γ:G→G, φ_γ((M, s)) = (Mγ⁻¹, s),

and observe that φ_γ is a group homomorphism. Now suppose that φ_γ is an inner automorphism, that isφ_γ(g) = (M₀, s₀)g(M₀, s₀)⁻¹ for some fixed (M₀, s₀)∈G and all g∈G. Then we have,

(Mγ⁻¹,id_n) =φ_γ((M,id_n)) = (M₀, s₀)(M,id_n)(M₀, s₀)⁻¹= (s₀M,id_n) ∀M∈M_n(Z), where the last equality follows because (M_n(Z),+) is abelian. However, this implies that γ⁻¹=s₀∈SL_n(Z) commutes with all matrices, whenceγ∈ {±id_n}. We conclude that the homomorphism SL_n(Z)→Aut(G), γ 7→φ_γ factors to an embedding PSL_n(Z),→Out(G), which proves that Out(G) is infinite.

Furthermore, there are hyperbolic groups with finite outer automorphism group but without property (T). — Let us recall the classicalMostow rigiditytheorem (for an exposition of a proof see [Roe03, Chapter 8]).

Theorem 4(Mostow rigidity). LetM andN be closed hyperbolic¹n-manifolds withn>3.

Then any homotopy equivalenceM→^' N is homotopic to an isometryM → N.

Corollary 5. For a closed hyperbolic manifold M of dimension at least 3, we have that Out(π₁(M))is finite.

Proof. By the Cartan–Hadamard theorem,M is aK(π₁(M),1) Eilenberg–MacLane space.

Thus, Out(π₁(M)) is isomorphic to the group of homotopy equivalencesM→^' Mmodulo free homotopy. If Out(π₁(M)) were infinite, by Theorem 4, there would exist an infinite sequence of pairwise non-homotopic isometriesM→ M. However, this is impossible on a compact manifold.

Note that the fundamental group of a closed hyperbolic manifold is a-T-menable (e.g.

because real hyperbolic space admits measured walls [CDH10, Example 3.7]), whence it does not have property (T).

More generally, the following result due to Thurston and Gromov holds.

Theorem 6([Gro87, §5, 5.4.A]). LetM be a closed aspherical manifold of dimension at least 3 with hyperbolic fundamental group. ThenOut(π₁(M))is finite.

1A Riemannian manifold is calledhyperbolicif it has constant sectional curvatureK≡ −1.

Non-parabolic actions on hyperbolic spaces

For the rest of this note, we closely follow Paulin [Pau91].

Let (X, d) be some proper geodesic hyperbolic space and π: G→Isom(X), g 7→π_g some group action by isometries. LetS=S⁻¹be a finite generating subset ofG, and define a function,

l_π,S: X→R_>0, l_π,S(x) = max

s∈S d(x, π_s(x)).

For allg∈G, x∈Xwe have,

(1) d(x, π_g(x))6l_π,S(x)|g|_S, where|g|

Sis the word length ofgwith respect toS.

For simplicity, we will sometimes writeg·xinstead ofπ(g)(x), andl_S instead ofl_π,S (if it is clear from context which action we use).

We denote the Gromov boundary ofX by ∂∞X, for a definition and discussion of which we refer to [BH99, III.H, 3]. Every quasi-isometry f : X →X gives rise to an homeomorphism∂∞f : ∂∞X→∂∞X. In particular,Gacts by homeomorphisms on∂∞X.

We call the action ofGon (X, d)parabolicif the induced action on ∂∞X has a global fixed point. A finitely generated hyperbolic group is calledelementaryif the action on a Cayley graph of itself is parabolic. (In fact, this only the case if the group is virtually cyclic.)

Lemma 7. If the action is not parabolic, then the functionl_S: X→R_>0is proper. In this case, the minimuml_S^min:= min_x∈Xl_S(x)is attained, andl_S^min= 0iffthere is a point inXfixed by all ofG.

Proof. Assume by contraposition that l_S is not proper. Then there exists a se- quence (x_n)_n∈N ⊂ X tending to infinity (in the locally compact sense) such that L:= sup_n∈Nl_S(x_n) <∞. Since X∪∂∞X is compact, we may assume — after passing to a subsequence — thatx_n→x∞∈∂∞X. By (1), we haved(g·x_n, x_n)6L|g|_Sfor alln∈N, and henceg·x∞=x∞for allg∈G. Thus, the action is parabolic.

To see the second claim, observe that a non-negative proper functionX→Ralways attains its minimum, and clearly, a pointx∈Xis a global fixed point iffl_S(x) = 0.

Let Cay(G, S) the Cayley graph ofGwith respect toS, and consider it as a geodesic metric space by declaring each edge to have length 1. Note that each automorphism ϕ ∈ Aut(G) induces an isometric action π: G→Cay(G, S), where π(g) : Cay(G, S)→ Cay(G, S) is defined by left-multiplication withϕ(g) for eachg∈G. Note that ifGis non- elementary, then the induced action on Cay(G, S) is not parabolic for each automorphism ofG. We write [ϕ] for the element of Out(G) represented byϕ∈Aut(G).

Lemma 8. Suppose thatG is non-elementary hyperbolic, and let (ϕ_n)_n∈N ⊂Aut(G) be a sequence of automorphisms. For each n ∈ N, let π_n: G → Isom(X) the action on X :=

Cay(G, S)induced by the automorphismϕ_n, and defineλ_n:=l_π^min

n,S.

If the sequence(λ_n)_n∈Nis bounded, then the set{[ϕ_n]|n∈N} ⊆Out(G)is finite.

Proof. Assume thatR:= sup_n∈Nλ_n<∞. If{[ϕ_n]|n∈N}were infinite, we can assume that [ϕ_n],[ϕ_m] forn,m. For eachn∈N, we choose•

n∈Xsuch thatλ_n=l_πn,S(•

n), and g_n∈G⊆Xsuch thatd(g_n,•_n)<1. For eachs∈S, we have,

d(g_n, ϕ_n(s)g_n)< d(•_n, π_n(s)(•_n)) + 26R+ 2, and sod

1_G, g_n⁻¹ϕ_n(s)g_n

< R+ 2 for alls∈S,n∈N. SinceS andB_R+2(1_G)∩Gare finite sets, existm,nsuch that,

g_m⁻¹ϕ_m(s)g_m=g_n⁻¹ϕ_n(s)g_n, ∀s∈S.

As S is a generating set, we conclude [ϕ_m] = [ϕ_n] ∈ Out(G), which contradicts our assumption.

Asymptotic cones

Unlike our source, we will not employ equivariant Gromov–Hausdorfftopologies for the proof. Instead we use ultralimits and asymptotic cones, but otherwise it is exactly the same as in [Pau91].

Letω be a non-principal ultrafilter onNand (X_n, d_n,•_n)_n∈N a sequence of pointed metric spaces. We define theultraproductby

X_ω=











(x_n)_n∈N∈Y

n∈N

X_n

sup

n∈N

d_n(x_n,•_n)<∞









 ,

∼,

where (x_n)_n∼(y_n)_nifd_ω((x_n)_n,(y_n)_n) := lim_ωd_n(x_n, y_n) = 0. Thend_ω induces a metric on X_ω. If each (X_n, d_n) is aδ_n-hyperbolic geodesic space with lim_n→∞δ_n= 0, thenX_ω is a 0-hyperbolic geodesic space, i.e. anR-tree.

For everyn∈N, letπ_n: G→Isom(X_n) be an isometric group action, and assume that sup_n∈Nd_n(•

n, π_n(g)(•

n))<∞for every g∈G. Then there is an isometric group action π_ω:G→Isom(X_ω) defined byπ_ω(g)((x_n)_n) = (π_n(g)(x_n))_n.

If we apply this construction to a sequence of the form (X,_λ¹

nd,•_n), where (X, d) is some fixed metric space with a sequence (•_n)_n⊂X, and (λ_n)_n⊂R_>0some sequence with λ_n→ ∞asn→ ∞, then we call the resulting spaceX_ωanasymptotic coneof (X, d).

Proof of Theorem 1. We consider the case thatGis non-elementary and choose a finite symmetric generating setS⊆G. We assume that there is an infinite sequence (ϕ_n)_n∈N⊂ Aut(G) such that [ϕ_n],[ϕ_m]∈Out(G) for alln,m. For the rest of this proof, we use the notation from Lemma 8. We conclude that the sequence (λ_n)_n∈N is unbounded, whence we may assume that λ_n → ∞ as n→ ∞. We construct an asymptotic cone X_ωby applying the above construction to the sequence (X,_λ¹

nd,•_n), where•_n∈Xis a point which minimizesl_πn,S. SinceGis hyperbolic, it follows thatX_ωis anR-tree. The sequence of actions (π_n)_nsatisfiesd(•_n, π_n(g)(•_n))6l_πn,S(•_n)|g|_S=λ_n|g|_S for alln∈N, g ∈G. Therefore, the sequence ₁

λ_nd(•_n, π_n(g)(•_n))

n∈N is bounded for all g ∈ G, and there is an induced actionπ_ω:G→Isom(X_ω).

Now suppose that the actionπ_ωhas a global fixed pointx∈X_ω. — For a sequence (x_n)_n∈Nrepresentingx, this means,

limω

d(x_n, π_n(g)(x_n))

λ_n = 0, ∀g∈G.

In particular, there existsn₀∈Nsuch thatd(x_n0, π_n0(s)(x_n0))< ^λ₂ⁿ⁰ for alls∈S. Hence it follows that l_πn

0,S(x_n0) < ^λ₂ⁿ⁰, but on the other hand, we have by definitionλ_n0 6 l_πn

0,S(x_n0), a contradiction.

References

[BH99] M. R. Bridson and A. Haefliger.Metric spaces of non-positive curvature.

Vol. 319. Grundlehren der Mathematischen Wissenschaften. Berlin:

Springer-Verlag, 1999.

[BHV08] B. Bekka, P. de la Harpe, and A. Valette.Kazhdan’s property (T). Vol. 11. New Mathematical Monographs. Cambridge: Cambridge University Press, 2008.

doi:10.1017/CBO9780511542749.

[BŚ97] W. Ballmann and J.Świ

atkowski.‘ OnL²-cohomology and property (T) for automorphism groups of polyhedral cell complexes. Geom. Funct. Anal. 7.4 (1997), 615–645.doi:10.1007/s000390050022.

[CDH10] I. Chatterji, C. Druţu, and F. Haglund.Kazhdan and Haagerup properties from the median viewpoint. Adv. Math. 225.2 (2010), 882–921.doi:

10.1016/j.aim.2010.03.012.

[Cor07] Y. de Cornulier.Finitely presentable, non-Hopfian groups with Kazhdan’s property (T) and infinite outer automorphism group. Proc. Amer. Math. Soc.

135.4 (2007), 951–959.doi:10.1090/S0002-9939-06-08588-1.

[Gro87] M. Gromov.Hyperbolic groups. In:Essays in group theory. Vol. 8. Math. Sci.

Res. Inst. Publ. New York: Springer, 1987, 75–263.

[OW07] Y. Ollivier and D. T. Wise.Kazhdan groups with infinite outer automorphism group. Trans. Amer. Math. Soc. 359.5 (2007), 1959–1976 (electronic).doi: 10.1090/S0002-9947-06-03941-9.

[Pau91] F. Paulin.Outer automorphisms of hyperbolic groups and small actions on R-trees. In:Arboreal group theory (Berkeley, CA, 1988). Vol. 19. Math. Sci. Res.

Inst. Publ. New York: Springer, 1991, 331–343.doi: 10.1007/978-1-4612-3142-4_12.

[Roe03] J. Roe.Lectures on coarse geometry. Vol. 31. University Lecture Series.

Providence, RI: American Mathematical Society, 2003.