# A dichotomy property for locally compact groups

## TL;DR

This paper generalizes Rosenthal's theorem to metrizable locally compact groups, linking the behavior of sequences to the existence of special sets, and explores conditions for Sidon sets and compactness properties in such groups.

## Contribution

It extends Rosenthal's theorem to non-abelian locally compact groups and introduces the notion of $I_0$ sets in this broader context, addressing longstanding open questions.

## Key findings

- Either a sequence has a weak Cauchy subsequence or contains an $I_0$ set subsequence.
- Provides conditions for the existence of Sidon sets in locally compact groups.
- Shows that every locally compact group strongly respects compactness.

## Abstract

We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of $l_1$. For that purpose, we transfer to general locally compact groups the notion of interpolation ($I_0$) set, which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact abelian groups. Thus we prove that for every sequence $\lbrace g_n \rbrace_{n<\omega}$ in a locally compact group $G$, then either $\lbrace g_n \rbrace_{n<\omega}$ has a weak Cauchy subsequence or contains a subsequence that is an $I_0$ set. This result is subsequently applied to obtain sufficient conditions for the existence of Sidon sets in a locally compact group $G$, an old question that remains open since 1974 (see [32] and [20]). Finally, we show that every locally compact group strongly respects compactness extending thereby a result by Comfort, Trigos-Arrieta, and Wu [13], who established this property for abelian locally compact groups.