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The word problem and growth of groups

Ievgen Bondarenko

TL;DR

This work investigates how the time complexity of the word problem WP_G for finitely generated groups relates to group growth and to automaton-based constructions. It establishes a sharp characterization: WP_G ∈ DTIME_1(n log n) if and only if G is virtually nilpotent, and it shows that WP_G cannot be solved in time o(n log γ(n)) where γ is G’s growth function, linking growth rates to computational difficulty. For automaton groups, the authors derive precise bounds: bounded automata yield WP_G ∈ DTIME_*(n log n), strongly contracting groups yield WP_G ∈ DTIME_*(n), and polynomial automata of degree d yield WP_G ∈ DTIME_*(n (log n)^{(d+1)^2}); the Grigorchuk group is a central example, with WP_G ∈ DTIME_*(n) but not in subexponential real-time bounds. The paper also provides a nearly complete picture for virtually nilpotent groups and outlines a framework (via contraction properties and crossing sequences) that explains how automaton structure governs WP_G complexity, offering both concrete results and several open questions on linear-time and real-time solvability.

Abstract

Let $\mathrm{WP}_G$ denote the word problem in a finitely generated group $G$. We consider the complexity of $\mathrm{WP}_G$ with respect to standard deterministic Turing machines. Let $\mathrm{DTIME}_k(t(n))$ be the complexity class of languages solved in time $O(t(n))$ by a Turing machine with $k$ tapes. We prove that $\mathrm{WP}_G\in\mathrm{DTIME}_1(n\log n)$ if and only if $G$ is virtually nilpotent. We relate the complexity of the word problem and the growth of groups by showing that $\mathrm{WP}_G\not\in \mathrm{DTIME}_1(o(n\logγ(n)))$, where $γ(n)$ is the growth function of $G$. We prove that $\mathrm{WP}_G\in\mathrm{DTIME}_k(n)$ for strongly contracting automaton groups, $\mathrm{WP}_G\in\mathrm{DTIME}_k(n\log n)$ for groups generated by bounded automata, and $\mathrm{WP}_G\in\mathrm{DTIME}_k(n(\log n)^d)$ for groups generated by polynomial automata. In particular, for the Grigorchuk group, $\mathrm{WP}_G\not\in\mathrm{DTIME}_1(n^{1.7674})$ and $\mathrm{WP}_G\in\mathrm{DTIME}_1(n^2)$.

The word problem and growth of groups

TL;DR

This work investigates how the time complexity of the word problem WP_G for finitely generated groups relates to group growth and to automaton-based constructions. It establishes a sharp characterization: WP_G ∈ DTIME_1(n log n) if and only if G is virtually nilpotent, and it shows that WP_G cannot be solved in time o(n log γ(n)) where γ is G’s growth function, linking growth rates to computational difficulty. For automaton groups, the authors derive precise bounds: bounded automata yield WP_G ∈ DTIME_*(n log n), strongly contracting groups yield WP_G ∈ DTIME_*(n), and polynomial automata of degree d yield WP_G ∈ DTIME_*(n (log n)^{(d+1)^2}); the Grigorchuk group is a central example, with WP_G ∈ DTIME_*(n) but not in subexponential real-time bounds. The paper also provides a nearly complete picture for virtually nilpotent groups and outlines a framework (via contraction properties and crossing sequences) that explains how automaton structure governs WP_G complexity, offering both concrete results and several open questions on linear-time and real-time solvability.

Abstract

Let denote the word problem in a finitely generated group . We consider the complexity of with respect to standard deterministic Turing machines. Let be the complexity class of languages solved in time by a Turing machine with tapes. We prove that if and only if is virtually nilpotent. We relate the complexity of the word problem and the growth of groups by showing that , where is the growth function of . We prove that for strongly contracting automaton groups, for groups generated by bounded automata, and for groups generated by polynomial automata. In particular, for the Grigorchuk group, and .
Paper Structure (8 sections, 17 theorems, 13 equations)

This paper contains 8 sections, 17 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Time complexity and the word problem
  3. The word problem and growth of groups
  4. The word problem in virtually nilpotent groups
  5. The word problem in automaton groups
  6. Automaton groups
  7. The word problem in contracting automaton groups
  8. The word problem in groups of polynomial automata

Key Result

Theorem 1

Let $G$ be a finitely generated group. Then $\mathrm{WP}_G\in \mathrm{DTIME}_1(n\log n)$ if and only if $G$ is virtually nilpotent.

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 3.1
  • Theorem 4
  • Proposition 5
  • proof
  • Theorem 6
  • proof
  • Corollary 6.1
  • ...and 21 more