Generalised Triangle Groups of Type (2,4,2)

TL;DR

This paper investigates a specific case of Rosenberger's conjecture on generalized triangle groups, providing partial results and computational evidence that restrict the form of potential counterexamples.

Contribution

It offers new restrictions on the word W in potential counterexamples for the (2,4,2) case of Rosenberger's conjecture, including parity and length bounds.

Findings

Exponent-sums of x and y in W are even and odd, respectively.

Free-product length of W must be at least 68, with computational evidence suggesting at least 196.

Results support Rosenberger's conjecture by limiting possible counterexamples.

Abstract

A conjecture of Rosenberger says that a group of the form $\langle x,y|x^p=y^q=W(x,y)^r=1\rangle$ (with $r>1$) is either virtually solvable or contains a non-abelian free subgroup. This note is an account of an attack on the conjecture in the case $(p,q,r)=(2,4,2)$. The results obtained are only partial, but nevertheless provide strong evidence in support of the conjecture in the case in question, in that the word $W$ in any counterexample is shown to satisfy some strong restrictions. The exponent-sums of $x$ and $y$ in $W$ must be even and odd respectively, while its free-product (or syllable) length must be at least 68. There is also a report of computer investigations which yield a stronger lower bound of 196 for the free-product length.