# Strongly bounded groups of various cardinalities

## TL;DR

This paper constructs strongly bounded groups across various cardinalities, including countable ones, by embedding any infinite group into a larger strongly bounded group, thus expanding the known examples and answering a longstanding question.

## Contribution

It introduces new strongly bounded groups of smaller cardinalities, including , and shows any infinite group can embed into a strongly bounded group with minimal cardinal increase.

## Key findings

- Existence of strongly bounded groups of cardinality 
- Any infinite group embeds into a strongly bounded group of at most two cardinalities larger
- Answer to Yves de Cornulier's question about small cardinality strongly bounded groups

## Abstract

Strongly bounded groups are those groups for which every action by isometries on a metric space has orbits of finite diameter. Many groups have been shown to have this property, and all the known infinite examples so far have cardinality at least $2^{\aleph_0}$. We produce examples of strongly bounded groups of many cardinalities, including $\aleph_1$, answering a question of Yves de Cornulier [4]. In fact, any infinite group embeds as a subgroup of a strongly bounded group which is, at most, two cardinalities larger.