Fixed-point property of random quotients by plain words

Abstract

We show a fixed-point property of certain random groups for a wide class of CAT(0) spaces. Th e model of random groups under consideration is given as the set of presentations$(S,R)$, where $S$ is a generating set and the set of relation$s$ R is a subset of the set of all plain words of the same length with suitably xed density. Our main theorem says that, with high probability, groups obtained by such presentations have the fixed-point property for all CAT (0) spaces having bounded singularities.