On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

TL;DR

This paper addresses the isomorphism problem for certain classes of groups, providing algorithms for constructing classifying spaces and reducing the problem for biautomatic groups to a more manageable case.

Contribution

It introduces algorithms for constructing classifying spaces of automatic groups and solves the isomorphism problem for central extensions of hyperbolic groups and biautomatic groups.

Findings

Isomorphism problem solvable for central extensions of hyperbolic groups

Algorithm for constructing finite models of classifying spaces

Reduction of isomorphism problem for biautomatic groups to finite center case

Abstract

We show that the isomorphism problem is solvable in the class of central extensions of word-hyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an algorithm that, given an arbitrary finite presentation of an automatic group $\Gamma$, will construct explicit finite models for the skeleta of $K(\Gamma,1)$ and hence compute the integral homology and cohomology of $\Gamma$.