Groups with ALOGTIME-hard word problems and PSPACE-complete compressed word problems

TL;DR

This paper establishes that the word problems for various complex groups are computationally hard, with some being rac{ ext{NC}^1}{ ext{hard}} and their compressed versions rac{ ext{PSPACE}}{ ext{complete}}, highlighting significant complexity bounds.

Contribution

It provides new lower bounds on the complexity of word problems for a broad class of non-solvable groups, including free, Grigorchuk's, and Thompson's groups, especially for compressed versions.

Findings

Word problems are rac{ ext{NC}^1}{ ext{-hard}} for many non-solvable groups.

Compressed word problems are rac{ ext{PSPACE}}{ ext{-complete}} for certain groups.

Results establish fundamental complexity bounds for these algebraic problems.

Abstract

We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk's group and Thompson's groups, we prove that their word problem is $\mathsf{NC}^1$-hard. For some of these groups (including Grigorchuk's group and Thompson's groups) we prove that the compressed word problem (which is equivalent to the circuit evaluation problem) is $\mathsf{PSPACE}$-complete.