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Geometric finiteness and rationality. (English) Zbl 0806.57003
The paper under review answers positively one of the main questions raised in a paper of S. M. Gersten and H. B. Short [Ann. Math., II. Ser. 134, 125-158 (1991; Zbl 0744.20035)]. To state the result, let us recall briefly (and informally) a few definitions. (The paper is concise, and for background and definitions, the reader is referred to the paper of Gersten and Short.)
Let \(A\) be a finite set and \(A^*\) the free monoid generated by \(A\). A subset of \(A^*\) is said to be a regular language if it is the set of words recognized by a finite state automaton. An automatic structure for a group \(G\) is given by a pair \((A,L)\) where \(A\) is a finite set which is a semi-group generator of \(G\), and \(L\) a regular language in \(A^*\), with the property that the question whether two elements of \(L\) representing elements of \(G\) at distance \(\leq 1\) in the Cayley graph can be decided by a finite state automaton. The notion of an automatic group is due to Thurston et al. A subset \(S\subset G\) is said to be \(L\)- rational if the complete inverse image of \(S\) in \(L\) is a regular subset.
The paper considers geometrically finite groups, that is, discrete subgroups of isometries of the hyperbolic space \(H^ n\) such that the quotient by \(G\) of the Nielsen convex hull of the action is compact (for this, and other equivalent definitions, see Thurston’s Princeton lecture notes). We can state now the main theorem of this paper: Let \(G\) be a geometrically finite group without parabolics, and let \((A,L)\) be any rational structure on \(G\) which is automatic. Then, a subgroup \(H\) of \(G\) is \(L\)-rational if and only if \(H\) is geometrically finite.
In his proof of this theorem, the author establishes also the following result, which concerns negatively curved groups (i.e. hyperbolic groups in the sense of Gromov): Let \(G\) be a negatively curved group. A subgroup \(H\) of \(G\) is \(L\)-rational with respect to any automatic structure \((A,L)\) on \(G\) if and only if the embedding of \(H\) in \(G\) is a quasi- isometry (with respect to any Cayley-graph metric). Finally, the author makes new conjectures.

MSC:
57M60 Group actions on manifolds and cell complexes in low dimensions
20F65 Geometric group theory
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
20M35 Semigroups in automata theory, linguistics, etc.
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