Compressed decision problems in hyperbolic groups

TL;DR

This paper demonstrates that certain decision problems in hyperbolic groups, such as the compressed word and conjugacy problems, are solvable efficiently, while the compressed knapsack problem is NP-complete.

Contribution

It establishes polynomial-time algorithms for the compressed word and conjugacy problems and proves NP-completeness of the compressed knapsack problem in hyperbolic groups.

Findings

Polynomial-time solvability of the compressed word problem in hyperbolic groups

Polynomial-time solvability of the compressed simultaneous conjugacy problem in hyperbolic groups

NP-completeness of the compressed knapsack problem in infinite hyperbolic groups

Abstract

We prove that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups. In such problems, group elements are input as words defined by straight line programs defined over a finite generating set for the group. We prove also that, for any infinite hyperbolic group $G$, the compressed knapsack problem in $G$ is ${\mathsf{NP}}$-complete.