Warren Dicks and Peter Kropholler, Free groups and almost equivariant maps. Bull. London Math. Soc. 27 (1995), 319-326.
 


Addenda

January 24, 2000

Remark. As is implicit in the proof of Corollary 2.4, a map between $G$-sets is almost G-stable if and only if it is an almost $G$-map.
On page 322, the paragraph at lines 8-15 can be replaced with the following longer, but slightly more intuitive, argument.

Proof of Lemma 1.1. Let T = (V,E,i,t) be a tree, let u, v, w denote vertices of T, and consider the pairing

<-,->:V x V -> E \vee {*},

where < v,w> is defined to be the first edge in the T-geodesic from v to w, with the understanding that this is * if v=w. As was noted by Euler, the map

<-,w>: V -> E \vee {*}

is bijective. We see that < v,u> \ne < v,w> if and only if u \ne w, and v is one of the vertices in the T-geodesic from u to w (which includes both u and w). Hence the two maps <-,u> and <-,w> in (V, E \vee {*}) are almost equal. Now, if G acts on T, then, for all g in G, < vg,wg> = < v,w>g so

<-, w>^g = <-,wg> =_a <-, w>.

Hence <-,w> is an almost G-map (or almost G-stable).

In summary, for each G-tree T = (V,E,i,t), we have shown that each vertex w of T determines a bijective almost G-map

<-,w>:V -> E \vee {*}.

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