Maximal nonparabolic subgroups of the modular group

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46221/1/208_2005_Article_BF01457079.pd

On Extendability of Group Actions on Compact Riemann Surfaces

The question of whether a given group G which acts faithfully on a compact Riemann surface X of genus g 2 is the full group of automorphisms of X (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group G in terms of a concrete partial presentation for G associated with the relevant branching data, using Singerman's list of signatures of Fuchsian groups which are not nitely maximal. By way of illustration, the results are applied to the special case where G is a non-cyclic abelian group

Double Coverings Of Klein Surfaces By A Given Riemann Surface

Let X be a Riemann surface. Two coverings p1 : X → Y1 and p2 : X → Y2 are said to be equivalent if p2 =’p1 for some conformal homeomorphism ’: Y1 → Y2. In this paper we determine, for each integer g¿2, the maximum number R(g) of inequivalent rami>ed coverings between compact Riemann surfaces X → Y of degree 2; where X has genus g. Moreover, for in>nitely many values of g, we compute the maximum number U(g) of inequivalent unrami>ed coverings X → Y of degree 2 where X has genus g and admits no rami>ed covering. For the remaining values of g, the computation of U(g) relies on a likely conjecture on the number of conjugacy classes of 2-groups. We also extend these results to double coverings X → Y , where. Y is now a proper Klein surface. In the language of algebraic geometry, this means we calculate the number of real forms admitted by the complex algebraic curve X . c 2002 Elsevier Science B.V. All rights reserved

Chiral Polytopes and Suzuki Simple Groups

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