Inner ultrahomogeneous groups

TL;DR

This paper introduces and analyzes inner ultrahomogeneous groups, including Hall's universal group, providing criteria for various properties and exploring their model-theoretic complexity and examples.

Contribution

It defines the class of inner ultrahomogeneous groups, establishes criteria for their properties, and explores their model-theoretic complexity and examples.

Findings

Inner ultrahomogeneous groups include Hall's universal group.

Groups of infinite exponent are not $eth_0$-saturated and have complex theories.

Finite exponent groups have uniformly bounded exponents.

Abstract

We define and study the class of inner ultrahomogeneous groups, which includes Hall's universal group and the universal locally recursively presentable group. We provide simple criteria for ample generic automorphisms, straight maximality, uniform simplicity and divisibility (all of which apply to both Hall's universal group and the universal locally recursively presentable group). We show that such groups of infinite exponent are not $\aleph_0$-saturated, their theories are not small, not rosy and have TP$_2$+SOP+IP$_n$ for all $n$. This strengthens and generalises known results about ample generic automorphisms and unstability of Hall's universal group. We also show that the exponents of finite exponent inner ultrahomogeneous groups are uniformly bounded, and we provide a series of examples of inner ultrahomogeneous groups.