The infinite Fibonacci groups and relative asphericity

TL;DR

This paper classifies when generalized Fibonacci groups are finite or infinite, proving their infiniteness in specific cases using asphericity of related presentations, thus resolving a long-standing question from 1965.

Contribution

It provides a complete classification of finite and infinite generalized Fibonacci groups by establishing asphericity of related presentations.

Findings

F(r,n) is infinite for (7+5k,5) and (8+5k,5) with k ≥ 0

Complete classification of finite F(r,n) groups

Method based on relative asphericity of presentations

Abstract

We prove that the generalised Fibonacci group F(r,n) is infinite for (r,n) in {(7 + 5k,5), (8 + 5k,5)} where k is greater than or equal to 0. This together with previously known results yields a complete classification of the finite F(r,n), a problem that has its origins in a question by J H Conway in 1965. The method is to show that a related relative presentation is aspherical from which it can be deduced that the groups are infinite.