On the degree of rapid decay

A finitely generated group \G equipped with a word-length is said to satisfy property RD if there are $C, s\geq 0$ such that, for all non-negative integers $n$, we have $\|a\|\leq C (1+n)^s \|a\|_2$ whenever a\in\C\G is supported on elements of length at most $n$. We show that, for infinite \G, the degree $s$ is at least 1/2.Comment: 6 pages, final versio

Spectral morphisms, K-theory, and stable ranks

We give a brief account of the interplay between spectral morphisms, K-theory, and stable ranks in the context of Banach algebras.Comment: 12 pages; to appear in the Proceedings of the Workshop on Noncommutative Geometry (Fields Institute, Toronto 2008

Homotopical stable ranks for Banach algebras

The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants. This paper studies the two homotopical stable ranks, viz. their general properties as well as specific examples and computations. The picture that emerges is that of a strong affinity between the homotopical stable ranks, and a marked contrast with the dimensional ones.Comment: 23 pages, final versio

Unimodular graphs and Eisenstein sums

Motivated in part by combinatorial applications to certain sum-product phenomena, we introduce unimodular graphs over finite fields and, more generally, over finite valuation rings. We compute the spectrum of the unimodular graphs, by using Eisenstein sums associated to unramified extensions of such rings. We derive an estimate for the number of solutions to the restricted dot product equation $a\cdot b=r$ over a finite valuation ring. Furthermore, our spectral analysis leads to the exact value of the isoperimetric constant for half of the unimodular graphs. We also compute the spectrum of Platonic graphs over finite valuation rings, and products of such rings - e.g., $\mathbb{Z}/(N)$. In particular, we deduce an improved lower bound for the isoperimetric constant of the Platonic graph over $\mathbb{Z}/(N)$.Comment: V2: minor revisions. To appear in the Journal of Algebraic Combinatoric

Two applications of strong hyperbolicity

We present two analytic applications of the fact that a hyperbolic group can be endowed with a strongly hyperbolic metric. The first application concerns the crossed-product C*-algebra defined by the action of a hyperbolic group on its boundary. We construct a natural time flow, involving the Busemann cocycle on the boundary. This flow has a natural KMS state, coming from the Hausdorff measure on the boundary, which is furthermore unique when the group is torsion-free. The second application is a short new proof of the fact that a hyperbolic group admits a proper isometric action on an $\ell^p$-space, for large enough $p$.Comment: 8 pages; final version (February 2017), to appear in Kyoto Journal of Mathematic