arXiv subjects
Publications and source records attributed to Federico Vigolo.
We prove that if a complete Riemannian surface $(\Sigma,d_\Sigma)$ is quasi-isometric to some bounded degree graph $G$, then $\Sigma$ admits a triangulation whose 1-skeleton is quasi-isometric to it when equipped with the simplicial metric. We study several variants of the problem, and identify the right condition making it an if and only if statement.
We investigate the action of the automorphism group of an acylindrically hyperbolic group G on its space of homogeneous quasimorphisms, and identify its kernel with the subgroup of "strongly commensurating" automorphisms. We deduce that if G has no non-trivial finite normal subgroups then it has sufficiently many quasimorphisms to recognize whether an automorphism is inner. As consequences, we show that Out(G) acts faithfully on the kernel of the comparison map in bounded cohomology and it embeds in (several) groups of coarse automorphisms.
We prove a conjecture of Roe by constructing unified warped cones that violate the coarse Baum-Connes conjecture. Interestingly, the reason for this is probably not what Roe expected, as the obstruction arises in odd rather than even degree.
We prove that uniformly locally finite metric spaces with isomorphic Roe algebras must be coarsely equivalent. As an application, we also prove that the outer automorphism group of the Roe algebra of a metric space of bounded geometry is canonically isomorphic to the group of coarse equivalences of the space up to closeness.
We introduce a coarse algebraic invariant for coarse groups and use it to differentiate various coarsifications of the group of integers. This lets us answer two questions posed by Leitner and the second author. The invariant is obtained by considering the set of exponents n such that taking n-th powers defines a coarse equivalence of the coarse group.
We provide a construction of Roe (C*-)algebras of general coarse spaces in terms of coarse geometric modules. This extends the classical theory of Roe algebras of metric spaces and gives a unified framework to deal with either uniform or non-uniform Roe algebras, algebras of operators of controlled propagation, and algebras of quasi-local operators; both in the metric and general coarse geometric setting. The key new definitions are that of coarse geometric module and coarse support of operators between coarse geometric modules. These let us construct natural bridges between coarse geometry and operator algebras. We then study the general structure of Roe-like algebras, and investigate several structural properties, such as admitting Cartan subalgebras or computing their intersection with the compact operators. Lastly, we prove that assigning to a coarse space the K-theory groups of its Roe algebra(s) is a natural functorial operation.
In this memoir we develop a framework to study rigidity problems for Roe-like C*-algebras of countably generated coarse spaces. The main goal is to give a complete and self-contained solution to the problem of C*-rigidity for proper (extended) metric spaces. Namely, we show that (stable) isomorphisms among Roe algebras always give rise to coarse equivalences. The material is organized as to provide a unified proof of C*-rigidity for Roe algebras, algebras of operators of controlled propagation, and algebras of quasi-local operators. We also prove a more refined C*-rigidity statement which has several additional applications. For instance, we can put the correspondence between coarse geometry and operator algebras in a categorical framework, and we prove that the outer automorphism groups of these C*-algebras are all isomorphic to the group of coarse equivalences of the coarse space.
Given a non-singular action $\Gamma \curvearrowright (X,\mu)$, we define the $*$-algebra $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and only depends on the measure class of $\mu$. We prove that the algebraic crossed product $L^{\infty}X \rtimes_{\rm alg} \Gamma$ surjects onto $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ and that this surjection is a $\ast$-isomorphism whenever the action is essentially free. As a consequence, we canonically characterize ergodicity and strong ergodicity of the action in terms of structural properties of $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ and its closure. We also use these techniques to describe the Roe algebra of a warped space in terms of the Roe algebra of the (non-warped) space and the group action. We apply this result to Roe algebras of warped cones.
In this monograph we lay the foundation for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms "up to uniformly bounded error". In the first part of this work, we develop the theory of coarse homomorphisms, quotients, and subgroups, and prove that coarse versions of the Isomorphism Theorems hold true. We also initiate the study of coarse actions and show how they relate to the fundamental observation of Geometric Group Theory. In the second part we explore a selection of specialized topics, such as the study of coarse group structures on set-groups, groups of coarse automorphisms and spaces of controlled maps. Here the main aim is to show how the theory of coarse groups connects with classical subjects. These include: number theory; the study of bi-invariant metrics on groups; quasimorphisms and stable commutator length; ${\rm Out}(F_n)$; topological group actions.
In this paper, we conduct further studies on geometric and analytic properties of asymptotic expansion in measure. More precisely, we develop a machinery of Markov expansion and obtain an associated structure theorem for asymptotically expanding actions. Based on this, we establish an analytic characterisation for asymptotic expansion in terms of the Druţu-Nowak projection and the Roe algebra of the associated warped cones. As an application, we provide new counterexamples to the coarse Baum-Connes conjecture.
In this paper, we use geometric tools to study the structure of asymptotic expanders and show that a sequence of asymptotic expanders always admits a "uniform exhaustion by expanders". It follows that asymptotic expanders cannot be coarsely embedded into any $L^p$-space, and that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we provide uncountably many new counterexamples to the coarse Baum--Connes conjecture. These appear to be the first counterexamples that are not directly constructed by means of spectral gaps. Finally, we show that vertex-transitive asymptotic expanders are actually expanders. In particular, this gives a $C^*$-algebraic characterisation of expanders for vertex-transitive graphs.
We introduce the notion of coupled link cube complex (CLCC) as a means of constructing interesting cocompactly cubulated groups. CLCCs are defined locally, making them a useful tool when precise control over the links is required. In this paper we study some general properties of CLCCs, such as their (co)homological dimension and criteria for hyperbolicity. Some examples of fundamental groups of CLCCs are RAAGs, RACGs, surface groups and some manifold groups. As immediate applications of our criteria we produce a number of cubulated 3 and 4 manifolds with hyperbolic fundamental group.
In this paper, we introduce and study a notion of asymptotic expansion in measure for measurable actions. This generalises expansion in measure and provides a new perspective on the classical notion of strong ergodicity. Moreover, we obtain structure theorems for asymptotically expanding actions, showing that they admit exhaustions by domains of expansion. As an application, we recover a recent result of Marrakchi, characterising strong ergodicity in terms of local spectral gaps. We also show that homogeneous strongly ergodic actions are always expanding in measure and establish a connection between asymptotic expansion in measure and asymptotic expanders by means of approximating spaces.
In this note we give a short proof that graphs having no linearly small Følner sets can be partitioned into a union of expanders. We use this fact to prove a partition result for graphs admitting linearly small maximal Følner sets and we deduce that a family of such graphs must contain a family of expanders. We also show that the existence of partitions into expanders is a quasi-isometry invariant.
In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type $FP_3$.
In this paper we compute the discrete fundamental groups of warped cones. As an immediate consequence, this allows us to show that there exist coarsely simply-connected expanders and superexpanders. This also provides a strong coarse invariant of warped cones and implies that many warped cones cannot be coarsely equivalent to any box space.
We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat unified approach to construct expanders. We also show that the graphs we obtain are uniformly quasi-isometric to the level sets of warped cones. This way we can also prove non-embeddability results for the latter and restate an old conjecture of Gamburd-Jakobson-Sarnak.
It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the corresponding warped cone does not coarsely embed into this Banach space. Combining this with non-embeddability results for warped cones by Nowak and Sawicki, which relate the non-embeddability of a warped cone to a spectral gap property of the underlying action, we provide new examples of expanders that do not coarsely embed into any Banach space with nontrivial type. Moreover, we prove that these expanders are not coarsely equivalent to a Lafforgue expander. In particular, we provide infinitely many coarsely distinct superexpanders that are not Lafforgue expanders. In addition, we prove a quasi-isometric rigidity result for warped cones.