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We apply G. Prasad’s volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of SO(1, n). As a result we prove that for any even dimension n there exists a unique compact arithmetic hyperbolic n-orbifold of the smallest volume. We give a formula for the Euler-Poincaré… (More)

We show that if S is a compact Riemann surface of genus g = p+1, where p is prime, with a group of automorphisms G such that |G| ≥ λ(g − 1) for some real number λ > 6, then for all sufficiently large p (depending on λ), S and G lie in one of six infinite sequences of examples. In particular, if λ = 8 then this holds for all p ≥ 17.

The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n ≥ 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n = 3 have been proven by Greenberg [G] and Kojima [K], respectively. Our proof is non constructive: it uses counting… (More)

There are errors in the proof of the uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture and give several remarks on further development. 1.1. Let us recall some notations and basic notions. Following [B] we will assume that n is even. The group of… (More)

We show that degrees of the real fields of definition of arithmetic Kleinian reflection groups are bounded by 35.

We show that for every g ≥ 2 there is a compact arithmetic Riemann surface of genus g with at least 4(g−1) automorphisms, and that this lower bound is attained by infinitely many genera, the smallest being 24.

We study Kazhdan-Lusztig cells and corresponding representations of right-angled Coxeter groups and Hecke algebras associated with them. In case of the infinite groups generated by reflections of hyperbolic plane in the sides of right-angled polygons we obtain a complete explicit description of the left and two-sided cells. In particular, it appears that… (More)

Let C be a oneor two-sided Kazhdan–Lusztig cell in a Coxeter group (W,S), and let Reduced(C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Reduced(C) is regular. In this paper we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon,… (More)

An old but fundamental problem in the arithmetic theory of quadratic forms is the computation of the mass of a lattice L in a quadratic space (V, q) over a number field F . Among the pioneers of its study were Smith, Minkowski and Siegel. After the work of Kneser, Tamagawa and Weil, this problem can be neatly formulated in group theoretic terms. More… (More)

We conjecture that for every dimension n 6=3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n≤ 4 and n = 6 this conjecture follows from the known results. In this paper we show that the conjecture is true for arithmetic hyperbolic n-manifolds of dimension n≥ 30.… (More)