The fixed group of an automorphism of a word hyperbolic group is rational (Q1209129)

The author affirmatively answers a question of \textit{S. Gersten} and \textit{H. Short} [Ann. Math., II. Ser. 134, 125-158 (1991; Zbl 0744.20035)] that, for a word hyperbolic group \(G\) and its automorphism \(\varphi\), the fixed subgroup \(G^ \varphi\) is in fact rational. For instance, for a discrete co-compact \(G\subset \text{Isom}(H^ n)\), it is equivalent to geometric finiteness of \(G^ \varphi\). As shown by Gersten and Short (see above) rationality of a subgroup \(H\) (for any automatic structure \(L\) on \(G\)) is equivalent to the condition that it is quasiconvex, i.e. there is a bound \(N\) such that the \(N\)-neighborhood of \(H\) contains any path defined by an element of \(L\) from 1 to a point of \(H\). Moreover, due to \textit{W. D. Neumann} and \textit{M. Shapiro} [Int. J. Algebra Comput. 2, No. 4, 443-469 (1992; Zbl 0767.20013)], for the case of word hyperbolic groups, both these notions only depend on the equivalence class of the automatic structure. The author's proof of quasiconvexity of \(G^ \varphi\) is based on three geometric Lemmas. It is also shown that, for any quasi-isometric homomorphisms \(\varphi,\psi: G \to G_ 1\) of word hyperbolic groups, \(\{g \in G'\mid \varphi(g) = \psi(g)\}\) is a rational subgroup.