# The Conjugacy Problem for Higman's Group

## TL;DR

This paper proves that the conjugacy problem in Higman's group is decidable and provides a polynomial time solution in a generic setting, building on recent advances in solving the word problem efficiently.

## Contribution

It introduces a polynomial time algorithm for the conjugacy problem in Higman's group, extending previous work on the word problem.

## Key findings

- Conjugacy problem in Higman's group is decidable.
- Polynomial time solution (O(n^7)) in a generic setting.
- Builds on recent polynomial-time word problem algorithms.

## Abstract

In 1951, Higman constructed a remarkable group $$H=\left\langle a,b,c,d \, \left| \, b^a = b^2, c^b = c^2, d^c = d^2, a^d = a^2 \right. \right\rangle$$ and used it to produce the first examples of infinite simple groups. By studying fixed points of certain finite state transducers, we show the conjugacy problem in $H$ is decidable (for all inputs). Diekert, Laun and Ushakov have recently shown the word problem in $H$ is solvable in polynomial time, using the power circuit technology of Myasnikov, Ushakov and Won. Building on this work, we show in a strongly generic setting that the conjugacy problem has a $O(n^7)$ polynomial time solution.