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.

Show description

Read Online or Download Discrete Groups and Geometry PDF

Best group theory books

The Classification of Three-Dimensional Homogeneous Complex Manifolds

This publication presents a category of all third-dimensional complicated manifolds for which there exists a transitive motion (by biholomorphic differences) of a true Lie team. this implies homogeneous complicated manifolds are thought of an identical in the event that they are isomorphic as complicated manifolds. The class relies on equipment from Lie team thought, complicated research and algebraic geometry.

Buchsbaum Rings and Applications: An Interaction Between Algebra, Geometry and Topology

Da die algebraische Geometrie wesentlich vom Fundamentalsatz der Algebra ausgeht, den guy nur deshalb in der gewohnten aUgemeinen shape aussprechen kann, weil guy dabei die Vielfachheit der Losungen in Betracht zieht, so mull guy auch bei jedem Resultat der algebra is chen Geometrie beim Zuriickschreiten die gemeinsame QueUe wiederfinden.

Group Theory and Quantum Mechanics

The German variation of this ebook seemed in 1932 less than the name "Die gruppentheoretische Methode in der Quantenmechanik". Its objective was once, to give an explanation for the elemental notions of the idea of teams and their Representations, and the appliance of this thought to the Quantum Mechanics of Atoms and Molecules.

Additional info for Discrete Groups and Geometry

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.

Download PDF sample

Rated 4.22 of 5 – based on 43 votes