arXiv subjects
Publications and source records attributed to Giorgio Mangioni.
We generalise the Milnor-Schwarz lemma to inverse monoids acting on presheaves of geodesic metric spaces. We provide two proofs of this fact: one only uses elementary techniques, inspired by the arguments for group actions on metric spaces; the other involves a version of the Vietoris-Rips complex, and builds on work of Chung-Martínez-Szakács.
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, then the extension is bounded, and under a further mild assumption the quotient is commensurable to a HHG. Motivated by questions on hierarchical hyperbolicity of quotients of mapping class groups, we therefore consider the general problem of determining when a quotient of a bounded central extension is still bounded, which we prove to be equivalent to an extendability problem for quasihomomorphisms. Finally, we show that quotients of the 4-strands braid group by suitable powers of a pseudo-Anosov are HHG, and in fact bounded central extensions of some HHG. We also speculate on how to extend the previous result to all mapping class groups.
We prove that the outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group (HHG), under mild orientability conditions on the associated JSJ decomposition. This is done by proving that a finite-index subgroup is a central extension of a product of orbifold mapping class groups, and the extension has bounded Euler class. Our theorem is sharp: we exhibit a surface amalgam whose fundamental group has full outer automorphism group which is not a HHG.
We reduce a strong version of the twist conjecture for Artin groups to Artin groups whose defining graphs have no separating vertices. This produces new examples of Artin groups satisfying the conjecture, and sheds more light on the isomorphism problem for Artin groups. Along the way we also prove a combination result for the ribbon property for vertices.
We introduce a construction that simultaneously yields cusped spaces of relatively hyperbolic groups, and spaces quasi-isometric to Teichmueller metrics. We use this to study Dehn-filling-like quotients of various groups, among which mapping class groups of punctured spheres. In particular, we show that the mapping class group of a five-holed sphere (resp. the braid group on four strands) has infinite hyperbolic quotients (strongly) not isomorphic to hyperbolic quotients of any other given sphere mapping class group (resp. any other braid group). These quotients are obtained by modding out suitable large powers of Dehn twists, and we further argue that the corresponding quotients of the extended mapping class group have trivial outer automorphism groups. We obtain these results by studying torsion elements in the relevant quotients.
We propose a new model for random quotients of groups using independent random walks. In this model, we show that random quotients of acylindrical hyperbolic groups asymptotically almost surely remain acylindrically hyperbolic. Our main tools relate the theories of spinning families and projection complexes to random walks. In the presence of a hierarchical hyperbolic structure on the group, we leverage the fine control of projections to show that this structure is preserved in the quotient asymptotically almost surely. The same techniques yield that random quotients of a non-elementary hyperbolic group (relative to any finite collection of finitely generated peripheral subgroups) are asymptotically almost surely hyperbolic (relative to commensurable peripheral subgroups). Finally, we also prove that any two groups that are both acylindrically and hierarchically hyperbolic have a common quotients which is itself acylindrically and hierarchically hyperbolic. This produces "exotic" hierarchically hyperbolic groups with strong fixed point properties, such as Kazhdan's property (T).
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 class groups) admit a combinatorial HHS structure. This can be useful in constructions of new HHSs, and also our construction clarifies how to apply the combinatorial HHS criterion to suspected examples. We also uncover connections between HHS notions and lattice theory notions.
We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups of finite-type surfaces, that is, those subgroups coming from the embeddings of closed subsurfaces. In the second half of the paper, we study under which hypotheses our amalgamation procedure preserves several notions of convexity, such as hierarchical quasiconvexity (as introduced by Behrstock, Hagen, and Sisto) and strong quasiconvexity (every quasigeodesic with endpoints on the subset lies in a uniform neighbourhood). This answers a question of Russell, Spriano, and Tran.
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 residual finiteness of certain hyperbolic groups with an explicit presentation, we get that all large and hyperbolic type Artin groups are residually finite. We also show that most quotients of the five-holed sphere mapping class group are hierarchically hyperbolic, up to taking powers of the normal generators of the kernels. The main tool we use to prove both results is a Dehn-filling-like procedure for short hierarchically hyperbolic groups (these also include e.g. non-geometric 3-manifolds, and triangle- and square-free RAAGs).
We prove that the mapping class group of a sphere with five punctures admits uncountably many coarsely equivariant coarse median structures. The same is shown for right-angled Artin groups whose defining graphs are connected, triangle- and square-free, and have at least three vertices. Remarkably, in the latter case, the coarse median structures we produce are not induced by cocompact cubulations. To obtain the above results, we develop the theory of short hierarchically hyperbolic groups (HHG), which also include Artin groups of large and hyperbolic type, graph manifold groups, and extensions of Veech groups. We develop tools to modify their hierarchical structure, including using quasimorphisms to construct quasilines that serve as coordinate spaces, and this is where the abundance of coarse median structures comes from. These techniques are of independent interest, and are used in a follow-up paper with Alessandro Sisto to study quotients of short HHG. In the process, we also clarify a proof of Hagen, Martin, and Sisto on hierarchical hyperbolicity of Artin groups of large and hyperbolic type.
We prove that an Artin group splits over infinite cyclic subgroups if and only if its defining graph has a separating vertex, and explicitly construct a JSJ decomposition over infinite cyclic subgroups for all Artin groups. We then use these facts to show that, if two Artin groups are isomorphic, then they have the same set of parabolics supported on "big chunks", that is, maximal subgraphs without separating vertices. We also deduce acylindrical hyperbolicity for the automorphism groups of many Artin groups, partially answering a question of Genevois in the case of Artin groups. As a consequence, we produce new families of Artin groups with the R-infinity property.
We prove several rigidity properties for random quotients of mapping class groups of surfaces, namely whose kernel is normally generated by the n-th steps of finitely many independent random walks. Firstly, we generalise a celebrated theorem of Ivanov's: every automorphism of the corresponding quotient of the curve graph is induced by a mapping class. Next, we show that, if a finitely generated group is quasi-isometric to a random quotient, then the two groups are weakly commensurable. This uses techniques from the world of hierarchically hyperbolic groups: indeed, in the process we clarify a proof of Behrstock, Hagen, and Sisto on the quasi-isometric rigidity of mapping class groups, which might possibly be applied to other hierarchically hyperbolic groups. Finally, we show that the automorphisms groups of our quotients, as well as their abstract commensurators, coincide with the groups themselves. Our results hold for a wider family of quotients, namely those whose kernel act by sufficiently large translations on the curve graph. This class also includes quotients by suitable powers of a pseudo-Anosov element.
We study quotients of mapping class groups of punctured spheres by suitable large powers of Dehn twists, showing an analogue of Ivanov's theorem for the automorphisms of the corresponding quotients of curve graphs. Then we use this result to prove quasi-isometric rigidity of these quotients, answering a question of Behrstock, Hagen, Martin, and Sisto in the case of punctured spheres. Finally, we show that the automorphism groups of our quotients of mapping class groups are "small", as are their abstract commensurators. This is again an analogue of a theorem of Ivanov about the automorphism group of the mapping class group. In the process we develop techniques to extract combinatorial data from a quasi-isometry of a hierarchically hyperbolic space, and use them to give a different proof of a result of Bowditch about quasi-isometric rigidity of pants graphs of punctured spheres.