Residual Properties of Free Products

Abstract

Let be a class of groups. We give sufficient conditions ensuring that a free product of residually groups is again residually , and analogous conditions are given for LEgroups. As a corollary, we obtain that the class of residually amenable groups and the one of locally embeddable into amenable (LEA) groups are closed under taking free products.Moreover, we consider the pro-topology and we characterize special HNN extensions and amalgamated free products that are residually , where is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable.

Between free and direct products of groups

“…It is easy to see that direct products preserve LEF, and it is known that free products do so too, see Corollary 1.6 in [1]. We obtain the following weaker form of permanence.…”

“…Then G ∨ H = Alt(p + q − 1) still satisfies this property, because its 2-Sylow subgroups have solubility length ≥ 2 (see Theorem 3 in [11]). It follows that (G ∨ H) ∨ J is the alternating group on 1 2 (p + q − 1)! + r − 1 elements, and likewise G ∨ (H ∨ J) on 1 2 (r + q − 1)!…”

“…It follows that (G ∨ H) ∨ J is the alternating group on 1 2 (p + q − 1)! + r − 1 elements, and likewise G ∨ (H ∨ J) on 1 2 (r + q − 1)! + p − 1 elements.…”

Amenability versus non‐exactness of dense subgroups of a compact group

“…Therefore, is LEF; recall our discussion in Section 2. (Compare with arguments in . )…”

Hydra group doubles are not residually finite

“…Remark 2. For any property P, if H is not P-separable, then Γ is not residually P. Berlai has shown that for P the properties of solvability and amenability, so long as G is residually P, failure to be P-separable is the only obstruction to Γ being residually P [2].…”