By Steven Zelditch

This paintings is anxious with a couple of twin asymptotics difficulties on a finite-area hyperbolic floor. the 1st challenge is to figure out the distribution of closed geodesics within the unit tangent package deal. The author's effects provide a quantitative shape to Bowen's equidistribution idea: they refine Bowen's theorem a lot because the best geodesic theorem on hyperbolic quotients refines the asymptotic formulation for the variety of closed geodesics of size lower than T. particularly, the writer supplies a fee of equidistribution when it comes to low eigenvalues of the Laplacian. the second one challenge is to figure out the distribution of eigenfunctions (in microlocal feel) within the unit tangent package deal. the most consequence right here (which is required for the equidistribution thought of closed geodesics) is an evidence of a signed and averaged model of the suggest Lindelof speculation for Rankin-Selberg zeta services. the most instrument used here's a generalization of Selberg's hint formulation.

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N}, j = k} is “the” complete graph on n vertices. If n > 0, then Kn has exactly n vertices (each of degree n − 1) and 1/2 · n · (n − 1) edges. 1. : Some graphs Kn,m := {(1, 0), . . , (n, 0), (1, 1), . . , (m, 1)}, {{(j, 0), (k, 1)} | j ∈ {1, . . , n}, k ∈ {1, . . , m}} is “the” complete bipartite graph graph. If n, m > 0, then Kn,m has exactly n + m vertices and 1/2 · n · m edges. 5 (Graph isomorphisms). Let X = (V, E) and X = (V , E ) be graphs. 1 The problem to decide whether two given graphs are isomorphic or not is a difficult problem – in the case of finite graphs, this problem seems to be a problem of high algorithmic complexity, though its exact complexity class is still unknown [68].

2. For which n ∈ N is the group Gn Abelian? 25 (Braid groups**). For n ∈ N the braid group on n strands is defined by Bn := s1 , . . , sn−1 {sj sj+1 sj = sj+1 sj sj+1 | j ∈ {1, . . , n − 2}} {sj sk = sk sj | j, k ∈ {1, . . , n − 1}, |j − k| ≥ 2} . 1. Show that Bn −→ Z sj −→ 1 defines a well-defined group homomorphism. For which n ∈ N is this homomorphism surjective? 2. Show that Bn −→ Sn sj −→ (j, j + 1) 44 2. Generating groups defines a well-defined surjective homomorphism onto the symmetric group Sn .

1 Products and extensions The simplest type of group constructions are direct products and their twisted variants, semi-direct products. 32 2. 1 (Direct product). Let I be a set, and let (Gi )i∈I be a family of groups. The (direct) product group i∈I Gi of (Gi )i∈I is the group whose underlying set is the cartesian product i∈I Gi and whose composition is given by pointwise composition: Gi × i∈I Gi −→ i∈I Gi i∈I (gi )i∈I , (hi )i∈I −→ (gi · hi )i∈I . , homomorphisms to the direct product group are in one-to-one correspondence to families of homomorphisms to all factors.