A metabelian group G acting as automorphism group on a compact Riemann surface of genus g ≥ 2 has order less than or equal to 16(g-1). We calculate for which values of g this bound is achieved and on these cases we calculate a presentation of the group G.
We prove that if the Walsh bipartite map of a regular oriented hypermap is also orientably regular then both and have the same chirality group, the covering core of (the smallest regular map covering ) is the Walsh bipartite map of the covering core of and the closure cover of (the greatest regular map covered by ) is the Walsh bipartite map of the closure cover of . We apply these results to the family of toroidal chiral hypermaps induced by the family of toroidal bipartite maps...
Let be a triangulable compact manifold. We prove that, among closed subgroups of (the identity component of the group of homeomorphisms of ), the subgroup consisting of volume preserving elements is maximal.
Let denote the closed 3-manifold obtained as the connected sum of g copies of S² × S¹, with free fundamental group of rank g. We prove that, for a finite group G acting on which induces a faithful action on the fundamental group, there is an upper bound for the order of G which is quadratic in g, but there does not exist a linear bound in g. This implies then a Jordan-type bound for arbitrary finite group actions on which is quadratic in g. For the proofs we develop a calculus for finite group...
We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. Euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogous statement remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth...
We explore the interior geometry of the CAT(0) spaces , constructed by Croke and Kleiner [Topology 39 (2000)]. In particular, we describe a diffraction effect experienced by the family of geodesic rays that emanate from a basepoint and pass through a certain singular point called a triple point, and we describe the shadow this family casts on the boundary. This diffraction effect is codified in the Transformation Rules stated in Section 3 of this paper. The Transformation Rules have various applications....
The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B 4,4,4ℍ3. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B 4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation...