Díaz Sánchez, Raquel and Garijo, Ignacio and Hidalgo, Rubén A. and Gromadzki, G. (2010) Structure of Whittaker groups and applications to conformal involutions on handlebodies. Topology and its Applications, 157 (15). pp. 2347-2361. ISSN 0166-8641
Restricted to Repository staff only until 31 December 2020.
The geometrically finite complete hyperbolic Riemannian metrics in the interior of a handlebody of genus g, having injectivity radius bounded away from zero, are exactly those produced by Schottky groups of rank g; these are called Schottky structures. A Whittakergroup of rank g is by definition a Kleinian groupK containing, as an index two subgroup, a Schottky groupΓ of rank g. In this case, K corresponds exactly to a conformalinvolution on the handlebody with Schottky structure given by Γ. In this paper we provide a structural description of Whittakergroups and, as a consequence of this, we obtain some facts concerning conformalinvolutions on handlebodies. For instance, we give a formula to count the type and the number of connected components of the set of fixed points of a conformalinvolution of a handlebody with a Schottky structure in terms of a group of automorphisms containing the conformalinvolution.
|Uncontrolled Keywords:||Group actions in low dimensions; Fuchsian groups and their generalizations|
|Subjects:||Sciences > Mathematics > Geometry|
I. Agol, Tameness of hyperbolic 3-manifolds, arXiv:math.GT/0405568, 2004.
L. Bers, Automorphic forms for Schottky groups, Adv. in Math. 16 (1975) 332–361.
D. Calegari, D. Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc. 19 (2) (2006) 385–446
G. Gromadzki, R.A. Hidalgo, B. Maskit, Symmetries of handlebodies with Schottky structures, preprint, 2008.
R.A. Hidalgo, The mixed elliptically fixed point property for Kleinian groups, Ann. Acad. Sci. Fenn. 19 (1994) 247–258.
R.A. Hidalgo, Cyclic extensions of Schottky uniformizations, Ann. Acad. Sci. Fenn. 29 (2004) 329–344
R.A. Hidalgo, Dihedral groups are of Schottky type, Revista Proyecciones 18 (1) (1999) 23–48.
R.A. Hidalgo, On Γ -hyperelliptic Schottky groups, Notas Soc. Mat. Chile (1) 8 (1989) 27–36.
J. Kalliongis, A. Miller, Equivalence and strong equivalence of actions on handlebodies, Trans. Amer. Math. Soc. 308 (2) (1988) 721–745
J. Kania-Bartoszynska, Involutions on 2-handlebodies, in: Transformation Groups, Poznań, 1985, in: Lecture Notes in Math., vol. 1217, 1986, pp. 151–166
L. Keen, On hyperelliptic Schottky groups, Ann. Acad. Sci. Fenn. Math. 5 (1980) 165–174.
L. Keen, J. Gilman, The geometry of two generator groups: Hyperelliptic handlebodies, Geometriae Dedicata 110 (2005) 159–190.
S. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983) 235–265.
P. Koebe, Über die Uniformisierung der Algebraischen Kurven II, Math. Ann. 69 (1910) 1–81.
B. Maskit, Kleinian Groups, GMW, Springer-Verlag, 1987.
B. Maskit, A characterization of Schottky groups, J. Anal. Math. 19 (1967) 227–230.
B. Maskit, On Klein's combination theorem IV, Trans. Amer. Math. Soc. 336 (1993) 265–294.
B. Maskit, A theorem on planar covering surfaces with applications to 3-manifolds, Ann. of Math. (2) 81 (1965) 341–355.
W.H. Meeks III, S.-T. Yau, Topology of three-dimensional manifolds and the embedding problem in minimal surface theory, Ann. of Math. (2) 112 (1980) 441–484.
A. Pantaleoni, R. Piergallini, Involutions of 3-dimensional handlebodies, arXiv:0806.0904v2.
M. Reni, B. Zimmermann, Extending finite group actions from surfaces to handlebodies, Proc. Amer. Math. Soc. 124 (9) (1996) 2877–2887
B. Zimmermann, Über Homöomorphismen n -dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen (On homeomorphisms of n -dimensional handlebodies and on finite extensions of Schottky groups), Comment. Math. Helv. 56 (3) (1981) 474–486 (in German).
|Deposited On:||21 Jun 2012 10:52|
|Last Modified:||06 Feb 2014 10:30|
Repository Staff Only: item control page