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Schur ultrafilters and Bohr compactifications of topological groups

Serhii Bardyla, Pavol Zlatoš

TL;DR

The paper develops a unified ultrafilter-based framework to analyze Bohr compactifications and chart groups of topological groups. By leveraging Schur ultrafilters and the Stone-Čech compactification, it introduces a hierarchy of closed congruences Θ, Ξ, Φ, Ψ to realize Bohr and chartifications as concrete quotients, namely $(eta G_d/oldsymbol{ ho},oldsymbol{ rak h}_{oldsymbol{ ho}})$ for appropriate ρ. A key result is the equivalence: a chart group $G$ is a topological group exactly when every Schur ultrafilter on $G$ converges to $1_G$, with related formulations involving dense subgroups of the topological center $oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}}}$. The authors show that $(eta G_d/oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}},oldsymbol{ rak h}_{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}}})}$ is the Bohr compactification of $G$ and that $(eta G_d/oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}},oldsymbol{ rak h}_{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}}})}$ provides a universal chartification; collectively, these results illuminate automatic continuity and the limitations of chart group theory, including existence of chart groups that are not topological. The work thereby connects ultrafilter dynamics with compactifications and group-structure properties in a broad, non-commutative setting.

Abstract

In this paper we investigate Schur ultrafilters on groups. Using the algebraic structure of Stone-Čech compactifications of discrete groups and Schur ultrafilters, we give a new description of Bohr compactifications of topological groups. This approach allows us to characterize chart groups that are topological groups. Namely, a chart group $G$ is a topological group if and only if each Schur ultrafilter on $G$ converges to the unit of $G$.

Schur ultrafilters and Bohr compactifications of topological groups

TL;DR

The paper develops a unified ultrafilter-based framework to analyze Bohr compactifications and chart groups of topological groups. By leveraging Schur ultrafilters and the Stone-Čech compactification, it introduces a hierarchy of closed congruences Θ, Ξ, Φ, Ψ to realize Bohr and chartifications as concrete quotients, namely for appropriate ρ. A key result is the equivalence: a chart group is a topological group exactly when every Schur ultrafilter on converges to , with related formulations involving dense subgroups of the topological center . The authors show that is the Bohr compactification of and that provides a universal chartification; collectively, these results illuminate automatic continuity and the limitations of chart group theory, including existence of chart groups that are not topological. The work thereby connects ultrafilter dynamics with compactifications and group-structure properties in a broad, non-commutative setting.

Abstract

In this paper we investigate Schur ultrafilters on groups. Using the algebraic structure of Stone-Čech compactifications of discrete groups and Schur ultrafilters, we give a new description of Bohr compactifications of topological groups. This approach allows us to characterize chart groups that are topological groups. Namely, a chart group is a topological group if and only if each Schur ultrafilter on converges to the unit of .
Paper Structure (5 sections, 32 theorems, 21 equations)

This paper contains 5 sections, 32 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the main results
  3. Schur ultrafilters
  4. Bohr compactification of topological groups
  5. Chart groups

Key Result

Theorem 2.1

Each compact right topological semigroup contains an idempotent.

Theorems & Definitions (59)

  • Theorem 2.1: Ellis
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Zlatoš
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 49 more