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

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.