The centre of a finitely generated strongly verbally closed group is almost always pure

TL;DR

This paper investigates the properties of finitely generated strongly verbally closed groups, revealing that their centers are almost always pure, which has implications for understanding embeddings of complex groups like braid groups and special linear groups.

Contribution

It establishes that the center of such groups is almost always pure, providing new insights into the structure and embedding properties of these groups.

Findings

Centers are almost always pure in finitely generated strongly verbally closed groups

Many interesting groups are not strongly verbally closed due to embedding properties

Implications for the structure of braid groups and special linear groups

Abstract

The assertion in the title implies that many interesting groups (e.g., all non-abelian braid groups or ${\bf SL}_{100}(\mathbb Z)$) are not strongly verbally closed, i.e., they embed into some finitely generated groups as verbally closed subgroups, which are not retracts.