Words in Linear Groups, Random Walks, Automata and P-Recursiveness

TL;DR

The paper demonstrates that the sequence counting products of matrices in a finite set that equal the identity matrix is not always P-recursive, answering a question posed by Kontsevich.

Contribution

It proves that the sequence of counts of matrix products in linear groups is not necessarily P-recursive, revealing limitations in the recursive structure of such sequences.

Findings

The sequence $oxed{a_n}$ is not always P-recursive.

Provides a counterexample to a conjecture about P-recursiveness.

Answers a question posed by Kontsevich.

Abstract

Fix a finite set $S \subset {GL}(k,\mathbb{Z})$. Denote by $a_n$ the number of products of matrices in $S$ of length $n$ that are equal to 1. We show that the sequence $\{a_n\}$ is not always P-recursive. This answers a question of Kontsevich.