Nearly-linear solution to the word problem for 3-manifold groups

TL;DR

This paper presents a nearly-linear time algorithm for solving the word problem in 3-manifold groups, significantly improving computational efficiency for these complex algebraic structures.

Contribution

It introduces an $O(n \log^3 n)$ algorithm for the word problem in 3-manifold groups, including non-geometric graph manifolds, using new methods for admissible graphs of groups.

Findings

Word problem for 3-manifold groups solvable in $O(n\log^3 n)$ time

Word problem for admissible graphs of groups solvable in $O(n\log n)$ time

Methods extend to free products, solving their word problem efficiently

Abstract

We show that the word problem for any 3-manifold group is solvable in time $O(n\log^3 n)$. Our main contribution is the proof that the word problem for admissible graphs of groups, in the sense of Croke and Kleiner, is solvable in $O(n\log n)$; this covers fundamental groups of non-geometric graph manifolds. Similar methods also give that the word problem for free products can be solved ``almost as quickly'' as the word problem in the factors.