Geodesic
The fixed group of an automorphism of a word hyperbolic group is rational.
Inventiones mathematicae, Tome 110 (1992) no. 2, pp. 147-150.

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Mots-clés : automorphisms, fixed subgroups, geometric finiteness, rationality of subgroups, automatic structures, word hyperbolic groups, quasiconvexity, quasi-isometric homomorphisms 
@article{IM_1992__110_2_144046,
     author = {Walter D. Neumann},
     title = {The fixed group of an automorphism of a word hyperbolic group is rational.},
     journal = {Inventiones mathematicae},
     pages = {147--150},
     publisher = {mathdoc},
     volume = {110},
     number = {2},
     year = {1992},
     zbl = {0793.20033},
     url = {http://geodesic.mathdoc.fr/item/IM_1992__110_2_144046/}
}
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Walter D. Neumann. The fixed group of an automorphism of a word hyperbolic group is rational.. Inventiones mathematicae, Tome 110 (1992) no. 2, pp. 147-150. http://geodesic.mathdoc.fr/item/IM_1992__110_2_144046/