Triangles of Baumslag-Solitar Groups

TL;DR

This paper investigates the structure of triangles of Baumslag-Solitar groups, showing that many such configurations collapse to finite, solvable groups under certain divisibility conditions, extending known examples and analyzing their quotients.

Contribution

It generalizes the collapse phenomenon of Baumslag-Solitar group triangles and characterizes conditions for non-developability and finiteness.

Findings

Many triangles of Baumslag-Solitar groups are finite and solvable.

Developability fails when parameters divide their partners, with some exceptions.

Provides detailed information on finite quotients of these groups.

Abstract

Our main result is that many triangles of Baumslag-Solitar groups collapse to finite groups, generalizing a famous example of Hirsch. A triangle of Baumslag-Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of class<4. We obtain a lot of information about finite quotients, even when we cannot determine developability.