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Computable topological groups

Heer Tern Koh, Alexander Melnikov, Keng Meng Ng

TL;DR

The paper develops a robust framework for computable presentations of topological groups, clarifying how several competing notions relate in Polish groups. It proves that for locally compact or abelian Polish groups, computable topological presentation and right-c.e. Polish presentation are equivalent, with the metric can be chosen left-invariant; in the locally compact case, effective local compactness yields an effectively proper left-invariant metric. The work also provides an effective Birkhoff-Kakutani metrization and a method to recover a dense computable sequence from a point-free presentation, yielding a right-c.e. Polish presentation under suitable completeness. Through carefully constructed counterexamples, the authors delineate the boundaries between notions, including discrete and profinite cases, and connect the theory to existing results in computable algebra, computable topology, and Pontryagin duality. Overall, the results advance a cohesive theory of computable topological groups, with implications for effective classification and dualities in broader classes of groups.

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.

Computable topological groups

TL;DR

The paper develops a robust framework for computable presentations of topological groups, clarifying how several competing notions relate in Polish groups. It proves that for locally compact or abelian Polish groups, computable topological presentation and right-c.e. Polish presentation are equivalent, with the metric can be chosen left-invariant; in the locally compact case, effective local compactness yields an effectively proper left-invariant metric. The work also provides an effective Birkhoff-Kakutani metrization and a method to recover a dense computable sequence from a point-free presentation, yielding a right-c.e. Polish presentation under suitable completeness. Through carefully constructed counterexamples, the authors delineate the boundaries between notions, including discrete and profinite cases, and connect the theory to existing results in computable algebra, computable topology, and Pontryagin duality. Overall, the results advance a cohesive theory of computable topological groups, with implications for effective classification and dualities in broader classes of groups.

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.
Paper Structure (25 sections, 21 theorems, 16 equations)

This paper contains 25 sections, 21 theorems, 16 equations.

Table of Contents

  1. Intorduction
  2. A brief overview
  3. Notions of computable presentability
  4. Results
  5. Connections to other notions in the literature.
  6. The two main definitions
  7. Preliminaries
  8. Computable topological spaces
  9. Computable topological groups
  10. Realtivization
  11. Effectively metrized spaces and groups
  12. Effective compactness and effective local compactness
  13. Effective local compactness
  14. Effectively proper metrics
  15. A unified generalization: represented spaces
  16. ...and 10 more sections

Key Result

Theorem 1.1

For a Polish group $G$ that is either abelian or locally compact, the following are equivalent: Furthermore, in $(2)$ the metric can be taken left-invariant. (Or right-invariant.)

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1: see, e.g., Definition 2.1 of KudKorTop of Definition 4 of comptop
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 46 more