A Topological Approach to the Nilpotent Multipliers of a Free Product

TL;DR

This paper extends the understanding of nilpotent multipliers of free products of groups using topological methods, generalizing previous results to all c-nilpotent multipliers and identifying conditions for their decomposition.

Contribution

It introduces a topological approach to analyze c-nilpotent multipliers of free products, broadening prior results to include all c≥1 and specific classes of groups.

Findings

M^{(c)}(G*H) is isomorphic to M^{(c)}(G) plus M^{(c)}(H) under certain conditions.

Extension of previous results from second to c-nilpotent multipliers.

Application of topological interpretation to group invariants.

Abstract

In this paper, using the topological interpretation of the Baer invariant of a group $G$, $\mathcal{V}M(G)$, with respect to an arbitrary variety $\mathcal{V}$, we extend a result of Burns and Ellis (Math. Z. 226 (1997) 405-428) on the second nilpotent multiplier of a free product of two groups to the $c$-nilpotent multipliers, for all $c\geq 1$. In particular, we show that $M^{(c)}(G\ast H)\cong M^{(c)}(G)\oplus M^{(c)}(H)$ when $G$ and $H$ are finite groups with some conditions or when $G$ and $H$ are two perfect groups.