# Download Discrete Groups and Geometry by W. J. Harvey, C. Maclachlan PDF

By W. J. Harvey, C. Maclachlan

This quantity encompasses a number of refereed papers provided in honour of A.M. Macbeath, one of many top researchers within the quarter of discrete teams. the topic has been of a lot present curiosity of past due because it includes the interplay of a few various themes similar to workforce conception, hyperbolic geometry, and intricate research.

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Example text

The Riemann-Hurwitz formula gives ) 2 4 sm/ and thus \G\ = 8g - 8 + 4A:, where k = ^-jj, the index in G of the cyclic subgroup generated by XY. In particular, this index k = \G : (^^ /r )| is bounded, whereas the order of G is not (since the order of zs can be arbitrarily large), and therefore the sequence Ns^k-8 is admissible. For example, if the order of K itself is m (chosen to be coprime to 5), then zs generates K and in that case G = KH, so that & = - ^ = if1, which is the index in H of the cyclic subgroup generated by xy.

2,2]) implying that <£>i" is the hyperelliptic involution. Since the hyperelliptic involution is unique, i\$" = \$i<£' and <£" = <£'. 1(c) and (a)(i). Since \$2 does not have the same species as \$1 then the order of 3>i<3>2 is 2 or a multiple of 4. 1 we can suppose that this Symmetries of Riemann surfaces 23 order is 4 or 2. 2 the species of \$2 is — 1. 2 the possible species for \$2 a re 0 , - 1 and +1 if g is even and +2 if g is odd. The existence of Riemann surfaces with involutions having the above species is shown by constructing smooth epimorphisms 0 : T' -* Z2 + Z2 + Z2 - (*i, * 2 , *') or 0 : V -> D4 ~ (\$1, * 2 >, such that ^~ 1 (\$ 2 ) gives the involution that we want, ^~ 1 (\$ 1 ) gives the maximal symmetry and 0~1(^i^t) is the hyperelliptic involution.

For P > 1, Hooley's method works if the classnumber h(—P) of properly primitive binary quadratic forms of discriminant —P satisfies h(—P) = 1. Then, we obtain t2 + Ps2 < (D1D2P)3+£\n\£ , (20) provided that n is not too large compared with D1D2P. For P > 1, h(—P) > 1, a weaker asymptotic formula is under investigation. Section III Suppose now that F is a general hyperbolic Fuchsian group which we may suppose (after a normalization) to act on the unit disk ti = {z G C : \z\ < 1}. Let J7 denote a Dirichlet fundamental region for F; it is, in fact a finite-sided non-Euclidean polygon, the boundary consisting of pairs of equivalent oppositely oriented sides £ and €7 -i which correspond to one another under 7 G F.