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Korean Mathematical Society JKMS

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J. Korean Math. Soc. 2013; 50(2): 425-444

Printed March 1, 2013

https://doi.org/10.4134/JKMS.2013.50.2.425

Copyright © The Korean Mathematical Society.

Some hyperbolic space forms with few generated fundamental groups

Alberto Cavicchioli, Emil Moln\'{a}r, and Agnese I. Telloni

Universit\`a di Modena e Reggio E., Budapest University of Technology and Economics, Universit\`a di Modena e Reggio E.

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We construct some hyperbolic hyperelliptic space forms wh\-ose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane reflections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic $3$--manifold with one cusp.

Keywords: hyperbolic 3--manifold, hyperelliptic involution, cyclic branched covering, Heegaard diagram, hyperbolic orbifold, fundamental group, isometry group

MSC numbers: Primary 57M12, 57N10; Secondary 57M05, 57M50

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