arXiv subjects
Publications and source records attributed to Alessandro Sisto.
We show that any product of bushy hyperbolic spaces has a unique coarse median structure, and that having a unique coarse median structure is a property closed under relative hyperbolicity. As a consequence, in contrast with the case of mapping class groups, there are non-hyperbolic pants graphs that have unique coarse median structures.
We show that quasi-isometries of (well-behaved) hierarchically hyperbolic groups descend to quasi-isometries of their maximal hyperbolic space. This has two applications, one relating to quasi-isometry invariance of acylindrical hyperbolicity, and the other a linear progress result for Markov chains. The appendix, by Jacob Russell, contains a partial converse under the (necessary) condition that the maximal hyperbolic space is one-ended.
Assuming that every hyperbolic group is residually finite, we prove the congruence subgroup property for mapping class groups of hyperbolic surfaces of finite type. Under the same assumption, it follows that profinitely equivalent hyperbolic 3-manifolds are commensurable.
We show that the Morse boundary exhibits interesting examples of both the existence and non-existence of Cannon-Thurston maps for normal subgroups, in contrast with the hyperbolic case.
We study actions of higher rank lattices $Γ<G$ on hyperbolic spaces, and we show that all such actions satisfying mild properties come from the rank-one factors of $G$. In particular, all non-elementary actions on an unbounded hyperbolic space are of this type.
We consider 3-manifolds given as Heegaard splittings $M=H^-\cup_ΣH^+$ with the aim to describe the hyperbolic metric of $M$ under topological conditions on the splitting guaranteeing that the manifold is hyperbolic. In particular, given a suitable "sufficiently incompressible" curve $γ\subsetΣ$, we show (without appealing to Geometrization) that $M$ is hyperbolic and we compute the length of $γ$ in terms of the projection coefficients of the disk sets, up to a uniform multiplicative error.
We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). Under residual finiteness assumptions, we construct many non-elementary hyperbolic quotients of mapping class groups. Using these quotients, we reduce questions of Reid and Bridson-Reid-Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.
Given a lattice Veech group in the mapping class group of a closed surface $S$, this paper investigates the geometry of $Γ$, the associated $π_1S$--extension group. We prove that $Γ$ is the fundamental group of a bundle with a singular Euclidean-by-hyperbolic geometry. Our main result is that collapsing "obvious" product regions of the universal cover produces an action of $Γ$ on a hyperbolic space, retaining most of the geometry of $Γ$. This action is a key ingredient in the sequel where we show that $Γ$ is hierarchically hyperbolic and quasi-isometrically rigid.
We show that for any lattice Veech group in the mapping class group $\mathrm{Mod}(S)$ of a closed surface $S$, the associated $π_1 S$--extension group is a hierarchically hyperbolic group. As a consequence, we prove that any such extension group is quasi-isometrically rigid.
We consider cusped hyperbolic $n-$manifolds, and compute Čech cohomology groups of the Morse boundaries of their fundamental groups. In particular, we show that the reduced Čech cohomology with real coefficients vanishes in dimension at most $n-3$ and does not vanish in dimension $n-2$. A similar result holds for relatively hyperbolic groups with virtually nilpotent peripherals and Bowditch boundary homeomorphic to a sphere; these include all non-uniform lattices in rank$-1$ simple Lie groups.
We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups.
We introduce Property (NL), which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. In other words, such a group $G$ can only admit elliptic or horocyclic hyperbolic actions, and consequently its poset of hyperbolic structures $\mathcal{H}(G)$ is trivial. It turns out that many groups satisfy this property; and we initiate the formal study of this phenomenon. Of particular importance is the proof of a dynamical criterion in this paper that ensures that groups with rich actions on compact Hausdorff spaces have Property (NL). These include many Thompson-like groups, such as $V, T$ and even twisted Brin--Thompson groups, which implies that every finitely generated group quasi-isometrically embeds into a finitely generated simple group with Property (NL). We also study the stability of the property under group operations and explore connections to other fixed point properties. In the appendix (by Alessandro Sisto) we describe a construction of cobounded actions on hyperbolic spaces starting from non-cobounded ones that preserves various properties of the initial action.
Behrstock, Hagen, and Sisto classified 3-manifold groups admitting a hierarchically hyperbolic space structure. However, these structures were not always equivariant with respect to the group. In this paper, we classify 3-manifold groups admitting equivariant hierarchically hyperbolic structures. The key component of our proof is that the admissible groups introduced by Croke and Kleiner always admit equivariant hierarchically hyperbolic structures. For non-geometric graph manifolds, this is contrary to a conjecture of Behrstock, Hagen, and Sisto and also contrasts with results about CAT(0) cubical structures on these groups. Perhaps surprisingly, our arguments involve the construction of suitable quasimorphisms on the Seifert pieces, in order to construct actions on quasi-lines.
We show that a quasi-geodesic in an injective metric space is Morse if and only if it is strongly contracting. Since mapping class groups and, more generally, hierarchically hyperbolic groups act properly and coboundedly on injective metric spaces, we deduce various consequences relating, for example, to growth tightness and genericity of pseudo-Anosovs/Morse elements. Moreover, we show that injective metric spaces have the Morse local-to-global property and that a non-virtually-cyclic group acting properly and coboundedly on an injective metric space is acylindrically hyperbolic if and only it contains a Morse ray.
Reid asked whether all convex-cocompact subgroups of mapping class groups are separable. Using a construction of Manning-Mj-Sageev, we give examples of separable convex-cocompact subgroups that are free of arbitrary finite rank, while prior examples seem to all be virtually cyclic.
The divergence of a group is a quasi-isometry invariant defined in terms of pairs of points and lengths of paths avoiding a suitable ball around the identity. In this paper we study "random divergence'', meaning the divergence at two points chosen according to independent random walks or Markov chains; the Markov chains version can be turned into a quasi-isometry invariant. We show that in many cases, such as for relatively hyperbolic groups, mapping class groups, and right-angled Artin groups, the divergence at two randomly chosen points is with high probability equivalent to the divergence of the group. That is, generic points realise the largest possible divergence.
We propose the study of Markov chains on groups as a "quasi-isometry invariant" theory that encompasses random walks. In particular, we focus on certain classes of groups acting on hyperbolic spaces including (non-elementary) hyperbolic and relatively hyperbolic groups, acylindrically hyperbolic 3-manifold groups, as well as fundamental groups of certain graphs of groups with edge groups of subexponential growth. For those, we prove a linear progress result and various applications, and these lead to a Central Limit Theorem for random walks on groups quasi-isometric to the ones we consider.
It was shown by Gersten that a central extension of a finitely generated group is quasi-isometrically trivial provided that its Euler class is bounded. We say that a finitely generated group $G$ satisfies Property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of $G$ is bounded. We exhibit a finitely generated group $G$ which does not satisfy Property QITB. This answers a question by Neumann and Reeves, and provides partial answers to related questions by Wienhard and Blank. We also prove that Property QITB holds for a large class of groups, including amenable groups, right-angled Artin groups, relatively hyperbolic groups with amenable peripheral subgroups, and 3-manifold groups. Finally, we show that Property QITB holds for every finitely presented group if a conjecture by Gromov on bounded primitives of differential forms holds as well.