Computing $H$-equations with 2-by-2 integral matrices

TL;DR

This paper develops an explicit algorithm to determine algebraic relations over subgroups in PSL_2(Z) using 2-by-2 integral matrices, highlighting the problem's solvability for size 2 but not for larger matrices.

Contribution

It introduces a novel algorithm for solving H-equations in PSL_2(Z) with 2x2 matrices, extending the understanding of algebraic relations in this context.

Findings

Algorithm solves the problem for 2x2 matrices

Problem becomes unsolvable for matrices of size 4 or larger

Provides a characterization of algebraic over subgroup relations

Abstract

We study the transference through finite index extensions of the notion of equational coherence, as well as its effective counterpart. We deduce an explicit algorithm for solving the following algorithmic problem about size two integral invertible matrices: ''given $h_1,\ldots, h_r; g\in \operatorname{PSL}_2(\mathbb{Z})$, decide whether $g$ is algebraic over the subgroup $H=\langle h_1,\ldots ,h_r\rangle \leqslant \operatorname{PSL}_2(\mathbb{Z})$ (i.e., whether there exist a non-trivial $H$-equation $w(x)\in H*\langle x\rangle$ such that $w(g)=1$) and, in the affirmative case, compute finitely many such $H$-equations $w_1(x),\ldots ,w_s(x)\in H*\langle x\rangle$ further satisfying that any $w(x)\in H*\langle x\rangle$ with $w(g)=1$ is a product of conjugates of $w_1(x),\ldots ,w_s(x)$''. The same problem for square matrices of size 4 and bigger is unsolvable.