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Hurwitz groups and finite group actions on hyperbolic 3-manifolds. (English) Zbl 0836.57009

By a result of Hurwitz, the maximal order of an automorphism resp. isometry group \(G\) of a closed Riemann resp. hyperbolic surface of genus \(g > 1\) is \(84(g - 1)\), and the corresponding finite groups \(G\) of this order are nowadays called Hurwitz groups. The Hurwitz groups are exactly the finite quotients of the (2,3,7)-triangle group and have been studied extensively. For isometry groups of closed hyperbolic 3-manifolds, no analogous simple bound is known: Euler-characteristics are 0, and volumes are quite complicated in dimension 3. However, the order of a finite group of homeomorphisms of a 3-dimensional handlebody of genus \(g > 1\) is bounded by \(12(g - 1)\), so using equivariant Heegaard splittings of closed 3-manifolds there is again an extremal case. The corresponding 3- manifolds have been studied in a previous paper [Topology Appl. 43, No. 3, 263-274 (1992; Zbl 0817.57016)]: in the irreducible case, they are either hyperbolic or Seifert fibered.
The main result of the present paper characterizes the finite groups \(G\) occurring in the extremal case, in analogy to the above algebraic characterization of Hurwitz groups.
In a second part of the paper, we study large groups of isometries of hyperbolic 3-manifolds with boundary. More precisely, the following question is discussed: given a Hurwitz action on a closed hyperbolic surface, when does it extend (by isometries) to a compact hyperbolic 3- manifold having the given surface as totally geodesic boundary (and as the only boundary component). We give the first examples of bounding and non-bounding Hurwitz actions (in more recent work, we study in detail the case of the most important class of Hurwitz actions, those of type PSL\((2,q)\)).

MSC:

57M60 Group actions on manifolds and cell complexes in low dimensions
57M50 General geometric structures on low-dimensional manifolds
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)

Citations:

Zbl 0817.57016
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