Computable topological groups

TL;DR

This paper explores various definitions of computability for topological groups, establishing equivalences in broad classes like locally compact Polish and abelian Polish groups, and linking these to effective presentability concepts.

Contribution

It compares multiple notions of computability for topological groups, proving their equivalence in certain classes and connecting them with existing effective presentation frameworks.

Findings

Two definitions of computability are equivalent for locally compact Polish and abelian Polish groups.

Every such computable topological group admits a right-c.e. presentation with a left-invariant metric.

In effectively locally compact groups, an effectively proper left-invariant metric can be constructed.

Abstract

We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare these results with similar results in computable topology. Most of these definitions can be separated by counter-examples. Remarkably, we prove that two such definitions are equivalent for locally compact Polish and abelian Polish groups. More specifically, we prove that in these broad classes of groups, every computable topological group admits a right-c.e.~(upper semi-computable) presentation with a left-invariant metric, and a computable dense sequence of points. In the locally compact case, we also show that if the group is additionally effectively locally compact, then we can produce an effectively proper left-invariant metric.