on rigidity of roe algebras.pdf

On Rigidity of Roe Algebras J′an ˇSpakula ?and Rufus Willett August 20, 2013 Mathematisches Institut, Universit ¨at M ¨unster, Einsteinstr. 62, 48149 M ¨unster, Germany E-mail:jan.******@uni- 2565 McCarthy Mall, University of Hawai‘i at M ˉanoa, Honolulu, HI 96822, USA E-mail:******@ Abstract Roe algebras areC ?-algebras built using large-scale (or ‘coarse’) as- pects of a metric space (X, d). In the special case thatX= Γ is a ?nitely generated group anddis a word metric, the simplest Roe algebra associ- ated to (Γ, d) is isomorphic to the crossed productC ?-algebral ∞(Γ)o rΓ. Roe algebras are coarse invariants, meaning that ifXandYare coarsely equivalent metric spaces, then their Roe algebras are isomor- phic. Motivated in part by the coarse Baum-Connes conjecture, we show that the converse statement is true for a very large classes of spaces. 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 Γ and Λ are ?nitely generated amenable, hyperbolic, or linear, groups such that the crossed productsl ∞(Γ)o rΓ andl ∞(Λ)o rΛ are isomorphic, then Γ and Λ are quasi-isometric. MSC:primary 46L85, 51K05. Keywords:coarse geometry, coarse Baum-Connes conjecture. 1 Introduction This piece asks to what extent certain mutativeC ?-algebras associated to a metric spaceX, called Roe algebras, capture the large scale geometry of X. As such it forms part of the mutative geometry program of Connes [9].?Corresponding author 1 For us, ‘large scale geometry’ will mean those aspects of metric geometry that are preserved under coarse equivalence as in the following de?nition. De?nition metric spaces. A (not necessarily continuous) mapf:X→Yis said to beuniformly expansiveif for allR >0 there exists S >0 such that ifx 1, x 2∈Xsatisfyd(x 1, x 2)≤R, thend(f(x