Preprints
-
A rigidity framework for Roe-like algebras
(with Diego Martínez)
Preprint (2024)
arXivAbstract: We give a complete solution to the problem of C*-rigidity for (extended) locally compact metric spaces. Namely, we show that (stable) isomorphisms among Roe-like algebras always give rise to coarse equivalences. To do so, we construct a framework to study rigidity problems for Roe-like algebras of countably generated coarse spaces. This allows us to provide a unified proof of C*-rigidity for Roe algebras, algebras of operators of controlled propagation, and algebras of quasi-local operators. As further applications, we prove that the outer automorphism groups of all of these algebras are isomorphic to the group of coarse equivalences of the starting coarse space. -
On a coarse invertibility spectrum for coarse groups
(with Leo Schäfer)
Preprint (2024)
arXivAbstract: 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. -
Dynamical propagation and Roe algebras of warped spaces
(with Tim de Laat and Jeroen Winkel)
Preprint (2023)
arXivAbstract: 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 $L^{\infty}X \rtimes_{alg} \Gamma$ 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. -
Roe algebras of coarse spaces via coarse geometric modules
(with Diego Martínez)
Preprint (2023)
arXivAbstract: We provide a construction of Roe (\cstar{})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 prove several key 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.
A monograph
An Invitation to Coarse Groups
(with Arielle Leitner)Springer Lecture Notes in Mathematics, 2339, 2023
Abstract: 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; Out(Fn); topological group actions.
Publications
-
A Markovian and Roe-algebraic approach to asymptotic expansion in measure
(with Kang Li and Jiawen Zhang)
Banach Journal of Mathematical Analysis, 17, no. 74, 2023
Published version, arXivAbstract: 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. -
A new construction of CAT(0) cube complexes
(with Robert Kropholler)
Algebraic & Geometric Topology, 22, no. 7, 3327–3375, 2022
Published version, arXivAbstract: 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. -
Embedding cube complexes into products of trees and fly maps
(with Robert Kropholler)
Appendix to the paper: 'Hyperbolic groups with almost finitely presented subgroups' by R. Kropholler.
Groups, Geometry, and Dynamics, 16, no. 1, 153-178, 2022
Published version, arXivAbstract: This appendix is devoted to justify the construction of fly maps used in Section 2.1. Many parts of this appendix are well known; we include them to give the reader a better understanding of how fly maps are defined -
On the structure of asymptotic expanders
(with Ana Khukhro, Kang Li and Jiawen Zhang)
Advances in Mathematics, 393, 2021
Published version, arXivAbstract: 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 Lp-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. -
Asymptotic expansion in measure and strong ergodicity
(with Kang Li and Jiawen Zhang)
Journal of Topology and Analysis, 2021
Published version, arXivAbstract: 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. -
Remarks on partitions into expanders
Aequationes mathematicae, 95, no. 4, 751-759, 2021
Published version, arXivAbstract: 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. -
Discrete fundamental groups of Warped Cones and expanders
Mathematische Annalen, 373, no. 1-2, 355–396, 2019
Published version, arXivAbstract: 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. -
Superexpanders from group actions on compact manifolds
(with Tim de Laat)
Geometriae Dedicata, 200, no. 1, 287–302, 2019
Published version, arXivAbstract: 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. -
Measure expanding actions, expanders and warped cones
Transactions of the AMS, 371, no. 3, 287–302, 2019
Published version, arXivAbstract: 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. -
Fundamental groups as limits of discrete fundamental groups
Bulletin of the LMS, 50, no. 5, 801-810, 2018
Published version, arXivAbstract: In this note we investigate to what extent the fundamental group of a metric space can be described as the inverse limit of its discrete fundamental groups. We show that some mild conditions suffice to imply the existence of an isomorphism and we provide a list of counterexamples to possible weakenings of these hypotheses.
Other things I wrote
PhD thesis: Geometry of actions, expanders and warped cones, University of Oxford (2018)