Equalizers and coactions of groups. (Q2778630)

Let \(G,H\) be groups and let \(p\colon G*H\to G\), \(q\colon G*H\to H\) be the natural projection maps from the free product of \(G\) and \(H\). If \(f\colon G\to H\) is a homomorphism then the equalizer \({\mathcal E}_f\) of \(f\) is the subgroup \(\{w\mid w\in G*H,\;fp(w)=q(w)\}\).NEWLINENEWLINE This paper is concerned with the study of \({\mathcal E}_f\) and related concepts. It is shown that \({\mathcal E}_f\) is the free product of \(\ker f\) with a free group. The authors also study coactions: if \(f\colon G\to H\) then a (right) coaction relative to \(f\) of \(H\) on \(G\) is a homomorphism \(s\colon G\to G*H\) such that \(p\circ s\) is the identity map on \(G\) and \(q\circ s=f\). Such maps have been studied in algebraic topology. It is determined precisely when a group \(G\) admits a coaction relative to \(f\).NEWLINENEWLINE Homomorphisms of coactions are also studied, continuing an investigation begun in an earlier paper of the authors [Proc. R. Soc. Edinb., Sect. A, Math. 131, No. 2, 225-239 (2001; Zbl 0984.20013)].