A counterexample to the easy direction of the geometric Gersten conjecture
aa r X i v : . [ m a t h . G R ] A p r A counterexample to the easy direction of the geometric Gerstenconjecture.
David Bruce CohenAugust 13, 2018
Abstract
For finitely generated groups H and G , equipped with word metrics, a translation-like actionof H on G is a free action such that each element of H acts by a map which has finite distancefrom the identity map in the uniform metric. For example, if H is a subgroup of G , then righttranslation by elements of H yields a translation-like action of H on G . Whyte asked whether agroup having no translation-like action by a Baumslag-Solitar group must be hyperbolic, wherethe free abelian group of rank 2 is understood to be a Baumslag-Solitar group. We show thatthe converse question has a negative answer, and in particular the fundamental group of a closedhyperbolic 3-manifold admits a translation-like action by the free abelian group of rank 2. A metric space X is said to be uniformly discrete if it has a minimum distance, meaninginf { d ( x, y ) : x, y ∈ X ; x = y } > , and said to have bounded geometry if for all r >
0, there is some N r > r -ballhas cardinality at most N r . If X satisfies both of these conditions, it is said to be a UDBG space[13, § X is the vertex set of a connected graph of bounded degree, equipped with themetric that assigns length 1 to each edge, then X is a UDBG space. In [13], Whyte introduced thefollowing notion. Definition. [13, Definition 6.1] Let X be a UDBG space. A translation-like action of a group H on X is a free action by maps at a finite distance from the identity. That is, the action satisfies thefollowing rules. • For x ∈ X and h ∈ H , if h · x = x , then h = 1 H . • For all h ∈ H , the set { d ( x, h · x ) : x ∈ X } is boundedWe will mostly be interested in the case where H is finitely generated and the UDBG space X is a finitely generated group G equipped with a word metric. In this case, a translation-like actionof H on G is just a vertex-surjective embedding of a disjoint union of copies of a Cayley graph of H into a Cayley graph of G (since an orbit of a translation-like action of H on G embeds the Cayleygraph of H into some Cayley graph of G ). 1 ranslation-like actions generalize subgroups. If H is a finitely generated subgroup of G ,then H acts translation-like on G via h · g = gh − for h ∈ H and g ∈ G . Many properties which pass to subgroups of G also pass to groups which acttranslation-like on G . For instance, Jeandel [8, Theorem 3] has shown that if G has no weakly aperi-odic subshift of finite type, then the same is true for finitely presented groups acting translation-likeon G . Whyte used this idea to give “geometric” versions of several famous conjectures about howthe properties of G constrain its subgroups [13, § The Geometric Von Neumann-Day Conjecture.
The Von Neumann-Day conjecture (dis-proven by Olshanskii [10]) asserts that a group G should be nonamenable if and only if G containsa free subgroup. Whyte used translation-like actions to formulate and prove a geometric version ofthis conjecture—namely, that G is nonamenable if and only if Z ∗ Z acts translation-like on G [13,Theorem 6.1]. The Geometric Burnside Problem.
The Burnside problem (answered in the negative byGolod and Shafarevich [7]) asks whether every infinite finitely generated group contains a Z -subgroup. The geometric Burnside problem asks whether every infinite, finitely generated groupadmits a translation-like action of Z . Seward [12, Theorem 1.4] proved that the answer to thisquestion is yes. The Geometric Gersten Conjecture.
Recall that for nonzero integers m, n , the Baumslag-Solitar group BS ( m, n ) is the group presented by h a, b | ab m a − = b n i , and in particular BS (1 , ∼ = Z . It is known that these groups do not embed in hyperbolic groups. The Gersten conjecture [2,Q 1.1]—usually attributed to Gromov—roughly states that for a group satisfying some finitenessproperties, hyperbolicity should be equivalent to having no Baumslag-Solitar subgroup. We donot know whether Gersten actually asked this question, although [6] asks whether every finitelypresented subgroup of a hyperbolic group must be hyperbolic. [3] showed that this was false, andhence that the Gersten conjecture is false for finitely presented groups (weaker versions remainopen).The geometric Gersten conjecture states being hyperbolic is equivalent to having no translation-like action by any BS ( m, n ). In point of fact, Whyte only asked about the “hard” direction—whether a group which is not hyperbolic must admit a translation-like action of a Baumslag-Solitargroup—and only for 2-dimensional groups. By an observation of Jeandel[8, § § Theorem 1.1.
Let G be the fundamental group of a closed hyperbolic 3-manifold. Then Z actstranslation-like on G . Proof of Theorem 1.1.
Let G be the fundamental group of a closed hyperbolic 3-manifold. We will prove Theorem 1.1 byshowing that G is bilipschitz to a UDBG space which admits a translation-like action of Z —thefollowing lemma says that this is sufficient. Lemma 2.1. If H acts translation-like on X , and X is bilipschitz to X then H acts translation-like on X .Proof. Let ψ : X → X be a bilipschitz map. We define a translation-like action of H on X byconjugating the action as follows. For x ∈ X , take h · x = ψ ( h · ψ − ( x )) . It is clear that this is a free action, and it is translation-like because d ( x, h · x ) ≤ Lip( ψ ) d ( ψ − ( x ) , h · ψ − ( x )) . Lemma 2.2.
There exists a UDBG space X such that Z acts translation-like on X and X isbilipschitz to G .Proof. Consider the set of points X = { (2 c a, c b, c ) : a, b, c ∈ Z } in the upper half space model of H . The reader may verify that this is indeed a UDBG space(the shortest distance is log(2) and it is not hard to see that the size of r -balls in X is roughlyexponential in r ).To define a translation-like action of Z on X , let the generators e , e of Z act by e · (2 c a, c b, c ) = (2 c ( a + 1) , c b, c )and e · (2 c a, c b, c ) = (2 c a, c ( b + 1) , c ) . These maps commute, each moves points by a distance of 1, and the Z -action they induce is clearlyfree, so it is translation-like.Observe that X is quasi-isometric to H because every point of H lies within a bounded distanceof X ⊂ H . Thus, by the Svarc-Milnor theorem, X is quasi isometric to G . By [13, Theorem 2],any quasi-isometry between nonamenable UDBG spaces is at a bounded distance from a bilipschitzmap, so X is bilipschitz to G .Combining Lemma 2.1 and Lemma 2.2, we have proved Theorem 1.1 We close with three questions. 3 ther Baumslag-Solitar groups.
Do any hyperbolic groups admit translation-like actions ofBaumslag-Solitar groups BS ( m, n ) with m ≥ Other hyperbolic groups.
Which hyperbolic groups admit translation-like actions of Z ? Jiang[9]has recently observed one may use results of [1] to show that Z cannot act translation-like on freegroups, and it appears [1, Proposition 4.1] that this technique may be used to rule out translation-like actions of Z on hyperbolic surface groups, but we have no idea whether such actions exist onhyperbolic one-relator groups or on random groups. Gromov-Furstenberg for returns of the horospherical flow in a hyperbolic 3-manifold. (See [4] for context). Let Γ be a cocompact lattice in
P SL (2; C ), let H be an ǫ -neighborhood ofsome horosphere H in H , let ∗ ∈ H , and consider the intersection O = (Γ · ∗ ) ∩ H . If we equip O with the metric inherited from H , then O is quasi isometric to H ∩ (Γ · B ǫ ( ∗ )), where B ǫ ( ∗ )denotes the ǫ ball around ∗ in H . From Ratner’s theorem [11], it then follows (with some thought)that O is quasi isometric to Z . Must O be bilipschitz to Z ? This was our original attempt atfinding a translation-like action of Z on Γ. We wish to thank Kevin Whyte, Benson Farb, and Andy Putman for conversations. We wish tothank Yongle Jiang for sharing an early draft of his preprint with us. This work has been supportedby NSF award 1502608.
References [1] Itai Benjamini, Oded Schramm, and ´Ad´am Tim´ar. On the separation profile of infinite graphs.
Groups, Geometry, and Dynamics , 6(4):639–658, 2012.[2] Mladen Bestvina. Questions in geometric group theory. Problem list, 2004.[3] Noel Brady. Branched coverings of cubical complexes and subgroups of hyperbolic groups.
Journal of the London Mathematical Society , 60(02):461–480, 1999.[4] Dmitri Burago and Bruce Kleiner. Rectifying separated nets.
Geometric and FunctionalAnalysis , 12(1):80–92, 2002.[5] David Carroll and Andrew Penland. Periodic points on shifts of finite type and commensura-bility invariants of groups.
New York J. Math , 21:811–822, 2015.[6] SM Gersten. Subgroups of word hyperbolic groups in dimension 2.
Journal of the LondonMathematical Society , 54(2):261–283, 1996.[7] Evgeniy Solomonovich Golod. On nil-algebras and finitely approximable p-groups.
IzvestiyaRossiiskoi Akademii Nauk. Seriya Matematicheskaya , 28(2):273–276, 1964.[8] Emmanuel Jeandel. Translation-like actions and aperiodic subshifts on groups. arXiv preprintarXiv:1508.06419 , 2015. 49] Yongle Jiang. Translation-like actions yield regular maps. arXiv preprint arXiv:1703.09253 ,2017.[10] Aleksandr Yur’evich Ol’shanskii. On the problem of the existence of an invariant mean on agroup.
Russian Mathematical Surveys , 35(4):180–181, 1980.[11] Marina Ratner. On raghunathan’s measure conjecture.
Annals of Mathematics , 134(3):545–607, 1991.[12] Brandon Seward. Burnside’s problem, spanning trees and tilings.
Geometry & Topology ,18(1):179–210, 2014.[13] Kevin Whyte. Amenability, bilipschitz equivalence, and the von neumann conjecture.