Finite maximal orientation reversing group actions on handlebodies and 3-manifolds
The author considers a closed 3-manifold \(M\) (orientable or not) with a finite group \(G\) of diffeomorphisms or piecewise linear (PL) homeomorphisms acting on \(M\) (\(M\) is a \(G\)-manifold). As the (equivariant) Heegaard decomposition \(M= V\cup_\partial W\) of \(M\) into two handlebodies \(V\) and \(W\) of genus \(g\) which are invariant under the \(G\)-action, i.e. \(G\) maps both bodies of the decomposition to itself. \(G\)-invariant triangulations guarantee such a genus \(g\). Here the non-exceptional cases \(g>1\) are considered only. Looking for \(G\) of maximal possible order \(24(g-1)\), the author classifies first the so-called minimal handlebody orbifolds to obtain the 8 corresponding maximal handlebody groups, i.e. the following 8 free products with amalgamation: \[ \overline D_2 *_{\overline Z_2}\overline D_3,\quad\overline D_3*_{\overline Z_3}\overline A_4,\quad\overline D_4*_{\overline Z_4}\overline S_4,\quad\overline D_5*_{\overline Z_5}\overline A_5; \] \[ D_{2*2} *_{\overline Z_2}\overline D_3,\quad D_{2*3}*_{\overline Z_3}\overline A_4,\quad D_{2*4}*_{\overline Z_4}\overline S_4,\quad D_{2*5}*_{\overline Z_5}\overline A_5. \] Then it follows Theorem 2. Any reducible strongly maximally symmetric \(G\)-manifold is obtained by doubling a maximal \(G\)-action of order \(24(g- 1)\) on a handlebody of genus \(g>1\). Any irreducible strongly maximally symmetric \(G\)-manifold is spherical, Euclidean or hyperbolic and is obtained as a regular covering of one of the nine minimal tetrahedral orbifolds \(T(m,n)\), i.e. \(S^3\), \(E^3\), \(H^3\) are factorized by a freely acting normal subgroup of the nine Coxeter groups \(C(m,n)\), respectively. Here \((m,n)= (2,3)\), \((2,4)\), \((2,5)\), \((3,3)\), \((3,4)\), \((3,5)\) for spherical, \((4,4)\) for Euclidean, \((4,5)\), \((5,5)\) for hyperbolic cases, respectively.
scientific article; zbMATH DE number 1370612
scientific article