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Spaces with measured walls, the Haagerup property and property (T)
We introduce the notion of a space with measured walls, generalizing the concept of a space with walls due to Haglund and Paulin (Simplicite de groupes d'automorphismes d'espaces courbure negative. Geom. Topol. Monograph 1 (1998), 181-248). We observe that if a locally compact group G acts properly on a space with measured walls, then G has the Haagerup property. We conjecture that the converse holds and we prove this conjecture for the following classes of groups: discrete groups with the Haagerup property, closed subgroups of SO(n, 1), groups acting properly on real trees, SL2(K) where K is a global field and amenable groups.
Analyse harmonique et 1-cohomologie réduite des groupes localement compacts
Vanishing and non-vanishing for the first L
We prove two results on the first L-P-cohomology (H) over bar (1)((p))(Gamma) of a finitely generated group Gamma: 1) If N subset of H subset of Gamma is a chain of subgroups, with N non-amenable and normal in Gamma, then (H) over bar (1)((P))(Gamma) = 0 as soon as (H) over bar (1)((P))(H) = 0. This allows for a short proof of a result of W. Luck: if N < Gamma, N is infinite, finitely generated as a group, and Gamma/N contains an element of infinite order, then (H) over bar (1)((2))(Gamma) = 0. 2) If Gamma acts isometrically, properly discontinuously on a proper CAT(- 1) space X, with at least 3 limit points in theta X, then for p larger than the critical exponent e(Gamma) of Gamma in X, one has (H) over bar (1)((p)) not equal A 0. As a consequence we extend a result of Y Shalom: let G be a cocompact lattice in a rank 1 simple Lie group; if G is isomorphic to Gamma, then e(G) < e(Gamma).
Markov operators on the solvable Baumslag-solitar groups
We consider the solvable Baumslag-Solitar group BSn = < a, b / aba(-1) = b(n)>, for n greater than or equal to 2, and try to compute the spectrum of the associated Markov operators M-S, either for the oriented Cayley graph (S = {a, b}), or for the usual Cayley graph (S = (a(+/-1), b(+/-1)}). We show in both cases that Sp M-S is connected. For S = {a, b} (nonsymmetric case), we show that the intersection of Sp M-S with the unit circle is the set Cn-1 of (n-1)-st roots of 1, and that Sp M-S contains the n - 1 circles {z is an element of C : /z - 1/2w/ = 1/2}, for w is an element of Cn-1, together with the n + 1 curves given by (1/2w(k) - lambda) (1/2w(-k) - lambda) - 1/4exp4 pi i theta = 0, where w is an element of Cn+1, theta is an element of [0, 1]. Conditional on the Generalized Riemann Hypothesis (actually on Artin's conjecture), we show that Sp M-S also contains the circle {z is an element of C : /z/ = 1/2}. This is confirmed by numerical computations for n = 2, 3, 5. For S = {a(+/-1) , b(+/-1)} (symmetric case), we show that Sp M-S = [-1,1] for n odd, and Sp M-S = [-3/4, 1] for n = 2. For n even, at least 4, we only get Sp M-S = [r(n), 1], with -1 < r(n) less than or equal to - sin(2) pi n/2(n+1) We also give a potential application of our computations to the theory of wavelets.