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

TL;DR

The paper proves that the word problem for fundamental groups of closed connected oriented 3-manifolds can be solved in $O(n\log^3 n)$ on a multi-tape Turing machine. The main advance is an $O(n\log n)$ algorithm for the word problem in admissible graphs of groups, encompassing graph-manifold groups, together with linear-time solutions for central extensions of hyperbolic groups and for free-product constructions via middle derivations. The approach relies on a careful decomposition given by the geometrisation theorem and introduces middle derivations to balance global reductions with local, edge-wise checks, all while employing storage schemes tailored to the TM model. Collectively, these components yield a nearly-linear-time WP framework for a broad class of 3-manifold groups and related graph-of-groups constructions, with practical implications for algorithmic 3-manifold topology and geometric group theory.

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.

Figures (1)

• Figure 1: A middle derivation. Transitioning between any two given vertex spaces requires crossing an edge labelled by some $t\in E(\Gamma)$.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.2
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.3
• proof
• Lemma 3.1
• proof