Topology of Asymptotic Cones and S-machine

Abstract

Sapir, Birget and Rips showed how to construct groups from Turing machines. To achieve such a construction they introduced the notion of S-machine. Then considering a simplified S-machine Sapir and Olshanskii showed how to construct a group such that each of its asymptotic cone is non-simply connected. Still using the notion S-machine, they constructed a group with two asymptotic cone non-homeomorphic. In this paper we show that each asymptotic cone of a group constructed following the whole method of Sapir, Birget and Rips is not simply connected.