Non-commutative lattice problems

TL;DR

This paper explores the computational complexity of subgroup-related problems in various groups, providing polynomial time algorithms for key problems in nilpotent, surface, Coxeter, and free groups.

Contribution

It introduces polynomial time algorithms for subgroup problems in several classes of groups, extending computational lattice problem techniques to non-commutative group settings.

Findings

Polynomial algorithms for subgroup element proximity in nilpotent groups

Efficient computation of shortest non-trivial subgroup elements in certain groups

Polynomial time geodesic computation in free group subgroups

Abstract

We consider several subgroup-related algorithmic questions in groups, modeled after the classic computational lattice problems, and study their computational complexity. We find polynomial time solutions to problems like finding a subgroup element closest to a given group element, or finding a shortest non-trivial element of a subgroup in the case of nilpotent groups, and a large class of surface groups and Coxeter groups. We also provide polynomial time algorithm to compute geodesics in given generators of a subgroup of a free group.