An L2-quotient algorithm for finitely presented groups on arbitrarily many generators

TL;DR

This paper extends an existing L2-quotient algorithm to finitely presented groups with any number of generators, providing a constructive invariant ring description and an implementation in Magma.

Contribution

It generalizes the L2-quotient algorithm to arbitrary generators and simplifies related invariant theory results.

Findings

Algorithm successfully extended to any number of generators.

Invariant ring description achieved for multiple copies of SL(2, K).

Implementation available in Magma.

Abstract

We generalize the Plesken-Fabia\'nska $\mathrm{L}_2$-quotient algorithm for finitely presented groups on two or three generators to allow an arbitrary number of generators. The main difficulty lies in a constructive description of the invariant ring of $\mathrm{GL}(2, K)$ on $m$ copies of $\mathrm{SL}(2, K)$ by simultaneous conjugation. By giving this description, we generalize and simplify some of the known results in invariant theory. An implementation of the algorithm is available in the computer algebra system Magma.