On Rigidity of Roe algebras

TL;DR

This paper investigates whether the isomorphism class of Roe algebras determines the coarse geometry of the underlying metric spaces, establishing rigidity results for large classes of spaces and groups.

Contribution

It proves that for broad classes of spaces, isomorphic Roe algebras imply coarse equivalence, demonstrating a form of C*-rigidity in large-scale geometry.

Findings

Roe algebra isomorphism implies coarse equivalence for many spaces.

For certain groups, isomorphic Roe algebras imply quasi-isometry.

The results support the rigidity of Roe algebras as invariants of large-scale geometry.

Abstract

Roe algebras are C*-algebras built using large-scale (or 'coarse') aspects of a metric space (X,d). In the special case that X=G is a finitely generated group and d is a word metric, the simplest Roe algebra associated to (G,d) is isomorphic to the reduced crossed product C*-algebra l^\infty(G)\rtimes G. Roe algebras are 'coarse invariants', in the sense that if X and Y are coarsely equivalent metric spaces, then their Roe algebras are isomorphic. Motivated in part by the coarse Baum-Connes conjecture, we ask if there is a converse to the above statement: that is, if X and Y are metric spaces with isomorphic Roe algebras, must X and Y be coarsely equivalent? We show that for very large classes of spaces the answer to this question, and some related questions, is 'yes'. This can be thought of as a 'C*-rigidity result': it shows that the Roe algebra construction preserves a large amount of information about the space, and is thus surprisingly 'rigid'. As an example of our results, in the group case we have that if G and H are finitely generated elementary amenable, hyperbolic, or linear, groups such that the crossed products l^\infty(G)\rtimes G and l^\infty(H)\rtimes H are isomorphic, then G and H are quasi-isometric.