LF groups, aec amalgamation, few automorphisms

Abstract

In S. 1 we deal with amalgamation bases, e.g., we define when an a.e.c. $k$ has $(\lambda,\kappa)$-amalgamation which means "many" M in $K^k_\lambda$ are amalgamation bases. We then consider what happens for the class of lf groups. In S. 2 we deal with weak definability of $a \in N\setminus M$ over $M$, for $ K_{exlf}$. In S. 3 we deal with indecomposable members of $K_{exlf}$ and with the existence of universal members of $K^k_\mu$, for $\mu$ strong limit of cofinality $\aleph_0$. Most noteworthy: if $K_{lf}$ has a universal model in $\mu$ then it has a canonical one similar to the special models, (the parallel to saturated ones in this cardinality). In S. 4 we prove "every $G \in K^{lf}_<\lambda$ can be extended to a complete $(\lambda,\theta)$-full G" for many cardinals. In a continuation we may consider "all the cardinals" or at least "almost all the cardinals"; also, we may consider a priori fixing the outer automorphism group.