Geometric finiteness and rationality.

*(English)*Zbl 0806.57003The 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.

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.

Reviewer: A.Papadopoulos (Strasbourg)

##### 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. |

##### Keywords:

rational subgroup; regular language; finite state automaton; automatic structure; Cayley graph; automatic group; geometrically finite groups; discrete subgroups of isometries of the hyperbolic space; negatively curved groups; hyperbolic groups
PDF
BibTeX
XML
Cite

\textit{G. A. Swarup}, J. Pure Appl. Algebra 86, No. 3, 327--333 (1993; Zbl 0806.57003)

Full Text:
DOI

##### References:

[1] | Bestvina, M.; Feighn, M., A combination theorem for negatively curved groups, J. differential geometry, (1990), also:, to appear., Preprint · Zbl 0724.57029 |

[2] | Bonahon, F., Bouts des varietes hyperboliques de dimension 3, Ann. of math., 124, 71-158, (1986) · Zbl 0671.57008 |

[3] | Bowdich, B., Geometric finiteness for hyperbolic groups, () |

[4] | Cannon, J.W.; Epstein, D.B.A.; Holt, D.F.; Patterson, M.S.; Thurston, W.P., Word processing and group theory, Warwick preprint, (1990) |

[5] | Coornaert, M.; Delzant, T.; Papadopoulos, A., Notes sur LES groupes hyperboliques de Gromov, () |

[6] | Gersten, S.; Short, H., Rational subgroups of biautomatic groups, Ann. of math., 134, 125-158, (1991) · Zbl 0744.20035 |

[7] | Ghys, E.; de la Harpe, P., Sur LES groupes hyperboliques d’aprés Gromov, Progress in mathematics, Vol. 83, (1990), Birkhäuser Boston, MA · Zbl 0731.20025 |

[8] | Gromov, M., Hyperbolic groups, (), Math. Sci. Res. Inst. Publ., No. 8 · Zbl 0634.20015 |

[9] | Maskit, B., Kleinian groups, (1987), Springer Berlin · Zbl 0144.08203 |

[10] | Morgan, J., Thurston’s uniformization theorem, (), 37-126 |

[11] | Susskind, P.D.; Swarup, G.A., Limit sets of geometrically finite hyperbolic groups, Amer. J. math., 114, 233-250, (1992), Preprint, also: · Zbl 0791.30039 |

[12] | Tatsuoka, K., Finite volume hyperbolic groups are automatic, (1990), Preprint |

[13] | Thurston, W.P., Geometry and topology of 3-manifolds, () · Zbl 0324.53031 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.