Alessandro Sisto
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View article: Short hierarchically hyperbolic groups II: Quotients and the Hopf property for Artin groups
Short hierarchically hyperbolic groups II: Quotients and the Hopf property for Artin groups Open
View article: Power quotients of surface groups and mapping class groups
Power quotients of surface groups and mapping class groups Open
Let $Γ$ be the fundamental group of a closed, orientable, hyperbolic surface $S$. The $n$-power quotient, $Γ(n)$, is the quotient of $Γ$ by the $n$th powers of simple closed curves. We prove an analogue of the Dehn--Nielsen--Baer theorem f…
View article: Bounded cohomology, quotient extensions, and hierarchical hyperbolicity
Bounded cohomology, quotient extensions, and hierarchical hyperbolicity Open
We call a central extension bounded if its Euler class is represented by a bounded cocycle. We prove that a bounded central extension of a hierarchically hyperbolic group (HHG) is still a HHG; conversely if a central extension is a HHG, th…
View article: Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture
Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture Open
We show that colorable hierarchically hyperbolic groups (HHGs) admit asymptotically CAT(0) metrics, that is, roughly, metrics where the CAT(0) inequality holds up to sublinear error in the size of the triangle. We use the asymptotically CA…
View article: Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks
Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks Open
A mixed equation in a group $G$ is given by a non-trivial element $w (x)$ of the free product $G \ast \mathbb{Z}$, and a solution is some $g\in G$ such that $w(g)$ is the identity. For $G$ acylindrically hyperbolic with trivial finite radi…
View article: Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity (with an appendix by Jacob Russell)
Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity (with an appendix by Jacob Russell) Open
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 hyperbolic…
View article: Uniform models and short curves for random 3-manifolds
Uniform models and short curves for random 3-manifolds Open
We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splitting with asymptotic probability 1. These constructions provid…
View article: On uniqueness of coarse median structures
On uniqueness of coarse median structures Open
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 ma…
View article: (Non‐)existence of Cannon–Thurston maps for Morse boundaries
(Non‐)existence of Cannon–Thurston maps for Morse boundaries Open
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.
View article: Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups
Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups Open
We prove that most Artin groups of large and hyperbolic type are Hopfian, meaning that every self-epimorphism is an isomorphism. The class covered by our result is generic, in the sense of Goldsborough-Vaskou. Moreover, assuming the residu…
View article: The congruence subgroup property for mapping class groups and the residual finiteness of hyperbolic groups
The congruence subgroup property for mapping class groups and the residual finiteness of hyperbolic groups Open
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 hyperbol…
View article: Hyperbolic actions of higher rank lattices come from rank-one factors
Hyperbolic actions of higher rank lattices come from rank-one factors Open
We study actions of higher rank lattices $\Gamma
View article: Corrigendum to “Morse boundaries of proper geodesic metric spaces”
Corrigendum to “Morse boundaries of proper geodesic metric spaces” Open
We introduce refined Morse gauges to correct the proof and statement of Lemma 2.10 of [Groups Geom. Dyn. 11, 1281–1306 (2017)] written by the first author.
View article: Morse subsets of injective spaces are strongly contracting
Morse subsets of injective spaces are strongly contracting Open
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 metri…
View article: Effective hyperbolization and length bounds for Heegaard splittings
Effective hyperbolization and length bounds for Heegaard splittings Open
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 …
View article: Nearly-linear solution to the word problem for 3-manifold groups
Nearly-linear solution to the word problem for 3-manifold groups Open
We show that the word problem for any 3-manifold group is solvable in time $O(n\log^3 n)$. Our main contribution is the proof that the word problem for admissible graphs of groups, in the sense of Croke and Kleiner, is solvable in $O(n\log…
View article: Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms
Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms Open
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 ad…
View article: Drilling hyperbolic groups
Drilling hyperbolic groups Open
Given a hyperbolic group $G$ and a maximal infinite cyclic subgroup $\langle g \rangle$, we define a drilling of $G$ along $g$, which is a relatively hyperbolic group pair $(\widehat{G}, P)$. This is inspired by the well-studied procedure …
View article: (Non-)existence of Cannon-Thurston maps for Morse boundaries
(Non-)existence of Cannon-Thurston maps for Morse boundaries Open
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.
View article: On the Čech cohomology of Morse boundaries
On the Čech cohomology of Morse boundaries Open
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…
View article: Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity
Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity Open
We show that for any lattice Veech group in the mapping class group Mod(S) of a closed surface S , the associated \pi_{1}S -extension group is a hierarchically hyperbolic group. As a consequence, we prove that any such extension group is q…
View article: Maps between relatively hyperbolic spaces and between their boundaries
Maps between relatively hyperbolic spaces and between their boundaries Open
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-isomet…
View article: Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity
Induced quasi-isometries of hyperbolic spaces, Markov chains, and acylindrical hyperbolicity Open
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 hyperbolic…
View article: A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces
A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces Open
The combinatorial hierarchical hyperbolicity criterion is a very useful way of constructing new hierarchically hyperbolic spaces (HHSs). We show that, conversely, HHSs satisfying natural assumptions (satisfied, for example, by mapping clas…
View article: Stable cubulations, bicombings, and barycenters
Stable cubulations, bicombings, and barycenters Open
We prove that the hierarchical hulls of finite sets of points in mapping\nclass groups and Teichm\\"uller spaces are stably approximated by a CAT(0) cube\ncomplexes, strengthening a result of Behrstock-Hagen-Sisto. As applications, we\npro…
View article: Central extensions and bounded cohomology
Central extensions and bounded cohomology Open
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 satisfies Property QITB (quasi-isometrically tr…
View article: Extensions of Veech groups I: A hyperbolic action
Extensions of Veech groups I: A hyperbolic action Open
Given a lattice Veech group in the mapping class group of a closed surface , this paper investigates the geometry of , the associated ‐extension group. We prove that is the fundamental group of a bundle with a singular Euclidean‐by‐hyperbo…
View article: Some examples of separable convex‐cocompact subgroups
Some examples of separable convex‐cocompact subgroups Open
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 prio…
View article: Hyperbolic Heegaard splittings and Dehn twists
Hyperbolic Heegaard splittings and Dehn twists Open
We consider the family of Heegaard splittings of genus $g$ at least three which are defined via a glueing map that is the $n$-th power of the Dehn twist along a curve that satisfies a natural topological assumption, namely pared acylindric…
View article: On the Čech cohomology of Morse boundaries
On the Čech cohomology of Morse boundaries Open
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 mo…