# The complexity of topological group isomorphism

## TL;DR

This paper investigates the complexity of classifying topological group isomorphisms, showing they are as complex as countable graph isomorphism for certain classes, using Borel reducibility in Polish spaces.

## Contribution

It establishes the complexity level of topological group isomorphism for various classes, connecting it to countable graph isomorphism, and provides bounds for oligomorphic groups.

## Key findings

- Profinite, locally compact, and Roelcke precompact groups have isomorphism complexity equivalent to countable graph isomorphism.
- For oligomorphic groups, the complexity is bounded above but not exactly determined.
- The study employs Borel reducibility on Polish spaces to analyze the classification complexity.

## Abstract

We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound, which is not sharp because the relation is Borel.