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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)

The isometry group of a compact n-dimensional hyperbolic man-ifold 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 Ko-jima [K], respectively. Our proof is non constructive: it uses counting… (More)

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.

In [BGLM] and [GLNP] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x (γ(H)+o(1)) log x/ log log x where γ(H) is an explicit constant computable from the (absolute) root system of H. In this paper we prove that this conjecture is false.… (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.

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)

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

Let C be a one-or 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… (More)

We conjecture that for every dimension n = 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.