# Constructive Membership Tests in Some Infinite Matrix Groups

## TL;DR

This paper develops algorithms and heuristics to express elements of certain infinite matrix groups as products of generators, achieving efficient representations for randomly generated elements.

## Contribution

It introduces new algorithms and heuristics for constructive membership testing in SLn(Z) and Sp2n(Z), improving expression length efficiency.

## Key findings

- Product expressions are competitive with input lengths for random elements.
- Algorithms successfully express arbitrary elements as products of generators.
- Heuristics enhance the efficiency of constructive membership tests.

## Abstract

We describe algorithms and heuristics that allow us to express arbitrary elements of SLn (Z) and Sp2n (Z) as products of generators in particular "standard" generating sets. For elements obtained experimentally as random products, it produces product expressions whose lengths are competitive with the input lengths.