Linear time algorithm for the conjugacy problem in the first Grigorchuk group

TL;DR

This paper presents a linear time algorithm for solving the conjugacy problem in the first Grigorchuk group, enabling efficient conjugacy checks and conjugator finding within this mathematical structure.

Contribution

The paper introduces the first linear time algorithm for the conjugacy problem in the first Grigorchuk group, improving computational efficiency over previous methods.

Findings

Conjugacy problem in the first Grigorchuk group can be solved in linear time.

Deciding if a list contains conjugate elements is linear time.

Conjugators can be found in polynomial time.

Abstract

We prove that the conjugacy problem in the first Grigorchuck group $\Gamma$ can be solved in linear time. Furthermore, the problem to decide if a list of elements $w_1,\ldots,w_k\in\Gamma$ contains a pair of conjugate elements can be solved in linear time. We also show that a conjugator for a pair of conjugate element $u,v\in\Gamma$ can be found in polynomial time.