The combinatorial structure of cocompact discrete hyperbolic groups.

*(English)*Zbl 0606.57003Combinatorial group theory began with Dehn’s study [M. Dehn, Math. Ann. 69, 137-168 (1910); ibid. 71, 116-144 (1912); ibid. 72, 413-420 (1912)] of the fundamental group of the closed 2-dimensional manifold admitting a hyperbolic structure. At first Dehn freely used arguments from hyperbolic geometry, but rapidly he and others moved in the direction of results which could be stated and proved in purely combinatorial and algebraic terms.

W. P. Thurston [see ”Geometry and topology of 3-manifolds”, Princeton Univ., Notes (1978); ”On the geometry and dynamics of diffeomorphisms of surfaces. I”, preprint] has recently shown that large classes of groups of interest to topologists, while not obviously amenable to attack by standard methods of combinatorial group theory, nevertheless are discrete hyperbolic groups. His result suggests the value of a return to geometric considerations in combinatorial group theory.

We show that Dehn’s principal combinatorial theorems - in particular his solutions to the word and conjugacy problems for hyperbolic surface groups - have simple geometric reinterpretations, and that these solutions, as reinterpreted, are true for all cocompact, discrete hyperbolic groups. We also show that the global combinatorial structure of such groups is particularly simple in the sense that the Cayley group graphs have descriptions by linear recursion.

W. P. Thurston [see ”Geometry and topology of 3-manifolds”, Princeton Univ., Notes (1978); ”On the geometry and dynamics of diffeomorphisms of surfaces. I”, preprint] has recently shown that large classes of groups of interest to topologists, while not obviously amenable to attack by standard methods of combinatorial group theory, nevertheless are discrete hyperbolic groups. His result suggests the value of a return to geometric considerations in combinatorial group theory.

We show that Dehn’s principal combinatorial theorems - in particular his solutions to the word and conjugacy problems for hyperbolic surface groups - have simple geometric reinterpretations, and that these solutions, as reinterpreted, are true for all cocompact, discrete hyperbolic groups. We also show that the global combinatorial structure of such groups is particularly simple in the sense that the Cayley group graphs have descriptions by linear recursion.

##### MSC:

57M05 | Fundamental group, presentations, free differential calculus |

20F05 | Generators, relations, and presentations of groups |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

57N65 | Algebraic topology of manifolds |

57R19 | Algebraic topology on manifolds and differential topology |

20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |

20F65 | Geometric group theory |

22E40 | Discrete subgroups of Lie groups |