**SEMINARIO TEORÍA DE GRUPOS UAM-ICMAT**

** Fecha y hora: Miercoles 5 de Abril, 10:30**

Lugar: Aula 420, Módulo 17, Facultad de Ciecnias, UAM

`Speaker: Álvaro Martínez Pérez, `

`Universidad de Castilla la Mancha.`

** Título: Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs**

** Abstract: **

** Given any Riemannian n-manifold $M$, the Cheeger isoperimetric constant of $M$ is defined as $h(M)=inf_A frac{Vol_{n-1}(partial A)}{Vol_n(A)}$,**

** where $A$ ranges over all non-empty bounded open subsets of $M$, and $Vol_k(B)$ denotes the k-dimensional Riemannian volume of the set $B$. **

** Given any graph $Gamma$, let $d_Gamma$ be the usual metric where every edge has length 1. The combinatorial Cheeger isoperimetric constant of $Gamma$ is defined to be **

** $h(M)=inf_A frac{|partial A|}{|A|},$ where A ranges over all non-empty finite subsets of vertices in $Gamma$, $partial A = {v in Gamma , | , d_Gamma (v, A) = 1}$ and $|A|$ denotes the cardinality of A. **

** A Riemannian manifold or graph $X$ satisfies the (Cheeger) isoperimetric inequality if $h(X) > 0$. (Let us mention that if $Gamma$ is a connected uniform graph, then $h(Gamma) > 0$ if and only if $Gamma$ is non-amenable.) **

** In this talk we analyze the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). Joint work with José M. Rodríguez, UC3M.**