Riemann surface transformation groups

We show that every action of a finite dihedral group on a closed orientable surface F \mathcal F extends to a 3-dimensional handlebody V \mathcal V , with ∂ V = F \partial \mathcal V=\mathcal F . In the case of a finite abelian group G G , we give necessary and sufficient conditions for a G G -action on a surface to extend to a compact 3 3 -manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order 84 ( g − 1 ) 84(g-1) of P S L ( 2 , 27 ) PSL(2,27) on a surface F \mathcal F of genus g = 118 g=118 does not extend to any compact 3-manifold M M with ∂ M = F \partial M=\mathcal F , thus resolving the only case of Hurwitz actions of type P S L ( 2 , q ) PSL(2,q) of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137–151).

In this paper we survey recent advances in probabilistic group theory and related topics, with particular emphasis on Fuchsian groups. Combining a character-theoretic approach with probabilistic ideas we study the spaces of homomorphisms from Fuchsian groups to symmetric groups and to finite simple groups, and obtain a whole range of applications. In particular we show that a random homomorphism from a Fuchsian group to the alternating group An is surjective with probability tending to 1 as n → ∞, and this implies Higman's conjecture that every Fuchsian group surjects to all large enough alternating groups. We also show that a random homomorphism from a Fuchsian group of genus ≥ 2 to any finite simple group G is surjective with probability tending to 1 as |G| → ∞. These results can be viewed as far-reaching extensions of Dixon's conjecture, establishing a similar result for free groups. Other applications concern counting branched coverings of Riemann surfaces, studying the subgroup growth of Fuchsian groups, and computing the dimensions of representation varieties of Fuchsian groups over fields of arbitrary characteristic. A main tool in our proofs are results of independent interest which we obtain on the representation growth of symmetric groups and groups of Lie type. These character-theoretic results also enable us to analyze random walks in symmetric groups and in finite simple groups, with certain conjugacy classes as generating sets, and to determine the precise mixing time. Most of the results outlined here were proved in recent joint works by Liebeck and myself, but we shall also describe related results by other authors.

We show that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.

Tests are described for determining whether certain functions are the boundary values of analytic functions taking the upper half plane into itself. In one theorem the functions are real-valued and defined on an open interval and a single point. In other theorems, the functions are defined on a real interval taking values in an arc γ in the upper half plane. In every case these results are closely related to Loewner's theorem in which the putative boundary values are required to be real-valued. As in Loewner's theory, the quadratic inequalities are finite sums involving at most the value of the function and its first derivative at points. These results are closely related to the Pick-Nevanlinna inequalities for functions defined on a sequence of points in the upper half plane; the Pick-Nevanlinna inequalities characterize those functions that are restrictions of analytic functions taking the upper half plane into itself.

The goal of this paper is to construct infinite-dimensional Lie algebras by using infinite product identities and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. We expect the Lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory (yet to be fully constructed) with symmetries given by the monster simple group.\n¶\nWe introduce vector-valued modular functions attached to families of modular functions of different levels, and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half-planes. We recast this result using the Borcherds–Harvey–Moore singular theta-lift and show that the vector-valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps.\n¶\nFor each element of the monster simple group, we construct an infinite-dimensional Lie algebra, such that its denominator formula is an infinite product expansion of the automorphic function arising from that element’s McKay–Thompson series. These Lie algebras have the unusual property that their simple roots and all root multiplicities are known. We show that under certain hypotheses, characters of groups acting on these Lie algebras form functions on the upper half-plane that are either constant or invariant under a genus zero congruence group.

Rational Period Functions and Parabolic Cohomology

According to a theorem of Belyi, a smooth projective algebraic curve is defined over a number field if and only if there exists a non-constant element of its function field ramified only over 0, 1 and ∞. The existence of such a Belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the Poincaré upper half plane by a subgroup of finite index in a Fuchsian triangle group. On the other hand, Fuchsian triangle groups arise in many contexts, such as in the theory of hypergeometric functions and certain triangular billiard problems, which would appear at first sight to have no relation to the Galois problems that motivated the above discovery of Belyi. In this note we review several results related to Belyi's theorem and we develop certain aspects giving examples. For preliminary accounts, see the preprint [Wo1], the conference proceedings article [BauItz] and the “Comptes Rendus” note [CoWo2].

Let H |r denote the upper half plane H with r additional odd (anticommuting) coordinates. It admits a transitive super action of a certain super Lie group 𝒢. First, we define the spaces of super automorphic and cusp forms on H |r for an ordinary lattice Γ of 𝒢, give an asymptotic formula for their dimensions for high weight and show how to embed Γ \\ H |r into the super projective space with the help of super automorphic forms. For also involving the odd directions of 𝒢 we introduce local super deformation of lattices in 𝒢 and show that for high weight the spaces of super automorphic and cusp forms are stable under such local super deformations.

Let G be a finite group. A gap G-module V is a finite dimensional real G-representation space satisfying the following two conditions: (1) The following strong gap condition holds: dimVP>2dimVH for all P<H≤G such that P is of prime power order, which is a sufficient condition to define a Gsurgery obstruction group and a G-surgery obstruction. (2)V has only one H-fixed point 0 for all large subgroups H, namely H∊L(G). A finite group G not of prime power order is called a gap group if there exists a gap Gmodule. We discuss the question when the direct product K×L is a gap group for two finite groups K and L. According to [5], if K and K×C2 are gap groups, so is K×L. In this paper, we prove that if K is a gap group, so is K×C2. Using [5], this allows us to show that if a finite group G has a quotient group which is a gap group, then G itself is a gap group. Also, we prove the converse: if K is not a gap group, then K×D2n is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.

We give explicit representation of Neumann boundary value problem for Bitsadze equation on the upper half plane. We will also give solution of the inhomogeneous polyanalytic equation arising from Neumann and (n-1) Dirichlet boundary conditions on the upper half plane H.

Let $\{\alpha _k\}_{k=1}^\infty$ be an arbitrary sequence of complex numbers in the upper half plane. We generalize the orthogonal rational functions $\phi _n$ based upon those points and obtain the Nevanlinna measure, together with the Riesz and Poisson kernels, for Caratheodory functions $F(z)$ on the upper half plane. Then, we study the relation between ORFs and their functions of the second kind as well as their interpolation properties. Further, by using a linear transformation, we generate a new class of rational functions and state the necessary conditions for guaranteeing their orthogonality.

In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.

Let G be a finite group and X a space on which G acts. In this chapter we will describe a cohomological analysis of X which involves H*(G; \({\Bbb F}\) p) in a fundamental way. First developed by Borel and then by Quillen, this approach is the natural generalization of classical Smith Theory. After reviewing the basic constructions and a few examples, we will apply these techniques to certain complexes defined from subgroups of a group G, first introduced by K. Brown and D. Quillen. Using results due to Brown and P. Webb we will show how these complexes provide a systematic method for approaching the cohomology of simple groups which will be discussed later on. One of the aims of this chapter is to expose the reader to the important part played by group cohomology in the theory of finite transformation groups. By no means is this a complete account; in addition it requires a different background than the previous chapters have. We recommend the texts [AP], [Brel] as excellent sources for those wanting to learn more about group actions. For a more recent survey, we suggest [AD].KeywordsFinite GroupSpectral SequenceSimplicial ComplexEquivariant CohomologyHomotopy EquivalentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A griffith crack at the interface of two bonded dissimilar elastic half planes in which body forces are acting

Let $G$ be a group. An element $g$ is c.d. in $G$ if and only if given any element $h$ of $G$, either it is conjugate to $h$ or there is a homomorphism $\xi$ from $G$ onto a finite group such that $\xi (g)$ is not conjugate to $\xi (h)$. Following A. Mostowski, a group is conjugacy separable or c.s. if and only if every element of the group is c.d. Let $F$ be a Fuchsian group, i.e. let $F$ be presented as \[ F = ({S_1}, \ldots ,{S_n},{a_1}, \ldots ,{a_{2r}},{b_1}, \ldots ,{b_t};S_{{1^1}}^e = \cdots = S_{{n^n}}^e = {S_1} \ldots {S_n}{a_1} \ldots {a_{2r}}a_1^{ - 1} \ldots a_{2r}^{ - 1}{b_1} \ldots {b_t} = 1).\] In this paper, we show that every element of infinite order in $F$ is c.d. and if $t \ne 0$ or $r \ne 0$, $F$ is c.s.