Logspace and compressed-word computations in nilpotent groups

TL;DR

This paper develops efficient algorithms for key problems in finitely generated nilpotent groups, achieving logarithmic space and polynomial time solutions for normal forms, membership, conjugacy, and subgroup presentations, including compressed-word variants.

Contribution

It introduces logspace algorithms and compressed-word solutions for classical problems in nilpotent groups, enhancing computational efficiency and uniformity.

Findings

Normal forms computed in logarithmic space

Membership and conjugacy problems solved efficiently

Compressed-word problems addressed in polynomial time

Abstract

For finitely generated nilpotent groups, we employ Mal'cev coordinates to solve several classical algorithmic problems efficiently. Computation of normal forms, the membership problem, the conjugacy problem, and computation of presentations for subgroups are solved using only logarithmic space and quasilinear time. Logarithmic space presentation-uniform versions of these algorithms are provided. Compressed-word versions of the same problems, in which each input word is provided as a straight-line program, are solved in polynomial time.