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

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 (with ) 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 . 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 in any counterexample is shown to satisfy some strong restrictions. The exponent-sums of and in 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.
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Taxonomy
TopicsMathematics and Applications · graph theory and CDMA systems · Finite Group Theory Research
