Fra\"iss\'e limits of limit groups

TL;DR

This paper introduces a modified Fra"iss"e framework to construct limits for classes of groups like nonabelian limit groups and elementary free groups, revealing new structural insights.

Contribution

It extends Fra"iss"e theory to new classes of groups and identifies their limits, including Lyndon's exponential completions as Fra"iss"e limits.

Findings

Fra"iss"e limits exist for nonabelian limit groups

Finitely generated elementary free groups have Fra"iss"e limits

Lyndon's $ ext{Z}[t]$-exponential completions are Fra"iss"e limits

Abstract

We modify the notion of a Fra\"iss\'e class and show that various interesting classes of groups, notably the class of nonabelian limit groups and the class of finitely generated elementary free groups, admit Fra\"iss\'e limits. Furthermore, we rediscover Lyndon's $\Z[t]$-exponential completions of countable torsion-free CSA groups, as Fra\"iss\'e limits with respect to extensions of centralizers. Dedicated to the memory of Charles Sims.