New PDF release: Discrete Groups and Geometry

By W. J. Harvey, C. Maclachlan

ISBN-10: 0511565798

ISBN-13: 9780511565793

ISBN-10: 0521429323

ISBN-13: 9780521429320

This quantity incorporates a choice of refereed papers provided in honour of A.M. Macbeath, one of many major researchers within the region of discrete teams. the topic has been of a lot present curiosity of past due because it contains the interplay of a few various themes similar to workforce thought, hyperbolic geometry, and intricate research.

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66 for details) oo (27) ZT{s) = Yl J i t 1 ~ N^o)~s~k] for Re(6) > 1; Pell equations and Fuchsian groups 33 the first product being taken over all the conjugacy classes of F, where (28) 7 /N(y) 1 ~ v- 0 1 0 7o being a generator of the class. In brief, the remarkable fact is that the eigenvalues A are directly related to the zeros s of Zr(s) by the simple relation: (29) A = $(1 -s). It is known that Z-p(s) is an entire function of s which satisfies the Riemann Hypothesis in the sense that all its non-real zeros reside on the line Re(s) = ^.

2 in [7]), Gn-z has a torsion-free subgroup H of finite index. Then by Prop. 6 in [21], cd{C00 x H) < 1 + cd(H) (since cd(Coo) = 1), hence vcd((Coo x G n _ 3 ) < 1 + vcd(Gn-z). Therefore, by [21] Prop. 15, vcd(Gn) < mdLx{vcd(Gn-2), 1 + vcd(Gn-3)}. Euler characteristic of graph products PROPOSITION 45 6. For all n > 1, vcd(Gn) = [-^J = w(Gn). PROOF. The proof is by induction on n. If n = 1, then Gn is cyclic of order 2, so vcd(G\) = 0. For n = 2, Gn is the infinite dihedral group, which has an infinite cyclic subgroup of finite index, so vcd(G2) — 1.

A finitely generated Coxeter group Cp is called aspherical by Pride and Stohr [19] if F contains no triangles A such that CA is finite. In this case the formula of Proposition 3 simplifies to 42 Chiswell where the sum is over all edges e (recall that edges are unoriented and are viewed as unordered pairs of vertices, n is the number of vertices of Y and is the labelling function). This formula is valid without the assumption in [19] that Y has no isolated vertices. It can also be obtained from their results.

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

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