Group Embeddings with Algorithmic Properties

TL;DR

This paper demonstrates that any countable group with a solvable word problem can be embedded into a 2-generated group that preserves solvability and has manageable complexity, with explicit estimates provided.

Contribution

It introduces a method for embedding computable groups into 2-generated groups while maintaining solvability and complexity bounds.

Findings

Every countable solvable group embeds into a 2-generated group with solvable word problem.

The embedding preserves the solvability of the membership problem.

Explicit time and space complexity estimates are provided for the word and membership problems.

Abstract

We show that every countable group H with solvable word problem (=computable group) can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We also give estimates of time and space complexity of the word problem in G and of the membership problem for H < G.