A robust family of residually finite groups; spectra of residual finiteness growth, computability properties, and other applications (with an appendix by Arman Darbinyan and Emmanuel Rauzy)

TL;DR

This paper introduces a family of residually finite groups to systematically study residual finiteness growth, explores computability aspects, and applies these results to solve open problems in group theory.

Contribution

It constructs groups with prescribed residual finiteness growth functions and analyzes their computability properties, extending previous theorems and answering open questions.

Findings

Any sufficiently fast-growing function can be realized as an RFG of a residually finite group.

Characterization of decidability of the word problem based on residual finiteness depth functions.

Existence of conjugacy separable groups with decidable word problem and undecidable conjugacy problem.

Abstract

In this paper, we introduce a family of residually finite groups that helps us to systematically study the residual finiteness growth function (RFG) from various perspectives. First, by strengthening results of Bou-Rabee and Seward and also of Bradford, we show that any non-decreasing function $f: \nn \rightarrow \nn$ that satisfies $f(n) > \exp{(\varepsilon n\log{n})}$ for some $\varepsilon>0$ can be realized (up to the standard equivalence) as RFG function of a two-generated residually finite group. Moreover, such a group can be found among solvable groups of derived length $3$; due to what, in a strengthened way, we extend a theorem of Kharlampovich, Miasnikov and Sapir. {Next, we consider computability aspects related to those growth functions. In particular, we characterize the decidability of the word problem in residually finite groups with respect to \emph{individual residual finiteness depth} functions. Then, we give a full description of sufficiently fast {growing} functions that are realizable as RFG for some group \emph{with decidable word problem} in terms of \emph{left-computable functions.} We also show that a Turing degree can be realized via RFG of a group with decidable word problem if and only if it is recursively enumerable. Finally, applying the introduced theoretical framework, we answer several open questions and extend known results. For example, answering a question of Minasyan, by providing a construction, we show the existence of conjugacy separable groups with decidable word problem and undecidable conjugacy problem. Two more applications, including an answer to a question by Nies, can be found in the appendix coauthored with Rauzy.}