Definable quotients of locally definable groups

TL;DR

This paper investigates conditions under which quotients of locally definable abelian groups are themselves definable, focusing on the roles of type-definable subgroups and divisibility properties.

Contribution

It establishes a connection between the definability of quotients, the existence of the subgroup 0, and the divisibility of the group, providing new insights into the structure of locally definable groups.

Findings

Quotients of locally definable abelian groups can be definable under certain conditions.

The existence of is linked to the definability of quotients.

Divisibility of the group influences the definability of the quotient.

Abstract

We study locally definable abelian groups $\CU$ in various settings and examine conditions under which the quotient of $\CU$ by a discrete subgroup might be definable. This turns out to be related to the existence of the type-definable subgroup $\CU^{00}$ and to the divisibility of $\CU$.