Free amalgamation and automorphism groups

TL;DR

This paper explores free amalgamation in countable structures, establishing conditions under which automorphism groups are universal, with applications to nilpotent Lie algebras and groups, and corrects previous proofs in the literature.

Contribution

It demonstrates that free amalgamation induces a stationary independence relation and provides new examples involving nilpotent Lie algebras and groups, correcting earlier proofs.

Findings

Free amalgamation induces a stationary independence relation.

Automorphism groups of certain structures are universal.

Provides corrected proofs for amalgamation of nilpotent Lie algebras.

Abstract

Let L be a countable elementary language, N be a Fraisse limit. We consider free amalgamation for L-structures where L is arbitrary. If free amalgamation for finitely generated substructures exits in N, then it is a stationary independece relation in the sense of K.Tent and M.Ziegler [TZ12b]. Therefore Aut(N) is universal for Aut(M) for all substructures M of N. This follows by a result of I.M\"uller [Mue13] We show that c-nilpotent graded Lie algebras over a finite field and c-nilpotent groups of exponent p (c < p) with extra predicates for a central Lazard series provide examples. We replace the proof in [Bau04] of the amalgamation of c-nilpotent graded Lie algebras over a field by a correct one.