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42
General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
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Cited by 31 (18 self)
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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
On dense free subgroups of Lie groups
 J. Algebra
"... Abstract. We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G, and that any finitely generated dense subgroup in a con ..."
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Cited by 23 (11 self)
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Abstract. We give a method for constructing dense and free subgroups in real Lie groups. In particular we show that any dense subgroup of a connected semisimple real Lie group G contains a free group on two generators which is still dense in G, and that any finitely generated dense subgroup in a connected nonsolvable Lie group H contains a dense free subgroup of rank ≤ 2 · dimH. As an application, we obtain a new and elementary proof of a conjecture of Connes and Sullivan on amenable actions, which was first proved by Zimmer. 1.
The congruence subgroup problem
, 2003
"... This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the nonspecialists and avoids technical details. ..."
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Cited by 13 (0 self)
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This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the nonspecialists and avoids technical details.
MurnaghanKirillov theory for supercuspidal representations of tame GL_N
, 2000
"... Let F be a nonarchimedean local field and G a connected reductive group defined over F with Lie algebra g. This paper exploits the formalism of Moy and Prasad to sharpen and extend familiar harmonic analysis results for G = G(F). We show that the Gorbits of the MoyPrasad filtration lattices are a ..."
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Cited by 11 (11 self)
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Let F be a nonarchimedean local field and G a connected reductive group defined over F with Lie algebra g. This paper exploits the formalism of Moy and Prasad to sharpen and extend familiar harmonic analysis results for G = G(F). We show that the Gorbits of the MoyPrasad filtration lattices are asymptotic to the set of nilpotent elements in g = g(F). For g we define Gdomains in terms of the filtration lattices and explore their properties. We then show that the domain where the local expansion for Ginvariant distributions on g is valid behaves well with respect to parabolic induction. For GLn (F) we show that, subject to certain tameness restrictions, the character of a supercuspidal representation can be expressed on a surprisingly large set as its formal degree times the Fourier transform of an elliptic orbital integral.
The homology of special linear groups over polynomial rings
 Ann. Sci. École Norm. Sup
, 1997
"... Abstract. We study the homology of SLn(F [t, t −1]) by examining the action of the group on a suitable simplicial complex. The E 1 –term of the resulting spectral sequence is computed and the differential, d 1, is calculated in some special cases to yield information about the lowdimensional homolo ..."
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Cited by 8 (8 self)
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Abstract. We study the homology of SLn(F [t, t −1]) by examining the action of the group on a suitable simplicial complex. The E 1 –term of the resulting spectral sequence is computed and the differential, d 1, is calculated in some special cases to yield information about the lowdimensional homology groups of SLn(F [t, t −1]). In particular, we show that if F is an infinite field, then H2(SLn(F [t, t −1]), Z) = K2(F [t, t −1]) for n ≥ 3. We also prove an unstable analogue of homotopy invariance in algebraic K–theory; namely, if F is an infinite field, then the natural map SLn(F) → SLn(F [t]) induces an isomorphism on integral homology for all n ≥ 2. Since Quillen’s definition of the higher algebraic K–groups of a ring [15], much attention has been focused upon studying the (co)homology of linear groups. There have been some successes—Quillen’s computation [14] of the mod l cohomology of GLn(Fq), Soulé’s results [18] on the cohomology of SL3(Z)—but few explicit calculations have been completed. Most known results concern the stabilization of the homology of linear groups. For example, van der Kallen [11], Charney [7], and others have proved quite general stability theorems for GLn of a ring. Also, Suslin [19] proved that if F is an infinite field, then the natural map Hi(GLm(F)) − → Hi(GLn(F)) is an isomorphism for i ≤ m. Other noteworthy results include Borel’s computation of the stable cohomology of arithmetic groups [1], [2], the computation of H • (SLn(F), R) for F a number field by Borel and Yang [3], and Suslin’s isomorphism [20] of H3(SL2(F)) with the indecomposable part of K3(F). This paper is concerned with studying the homology of linear groups defined over the polynomial rings F [t] and F [t, t−1]. One motivation for this is an attempt to find unstable analogues of the fundamental theorem of algebraic K–theory [15]: If R is a regular ring, then there are natural isomorphisms
Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series
 Adv. Math
"... Abstract. We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p–adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result re ..."
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Cited by 5 (2 self)
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Abstract. We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p–adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p–adic type.
KAZHDAN AND HAAGERUP PROPERTIES IN ALGEBRAIC GROUPS OVER LOCAL FIELDS
, 2005
"... Abstract. Given a Lie algebra s, we call Lie salgebra a Lie algebra endowed with a reductive action of s. We characterize the minimal sLie algebras with a nontrivial action of s, in terms of irreducible representations of s and invariant alternating forms. As a first application, we show that if g ..."
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Cited by 4 (4 self)
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Abstract. Given a Lie algebra s, we call Lie salgebra a Lie algebra endowed with a reductive action of s. We characterize the minimal sLie algebras with a nontrivial action of s, in terms of irreducible representations of s and invariant alternating forms. As a first application, we show that if g is a Lie algebra over a field of characteristic zero whose amenable radical is not a direct factor, then g contains a subalgebra which is isomorphic to the semidirect product of sl2 by either a nontrivial irreducible representation or a Heisenberg group (this was essentially due to Cowling, Dorofaeff, Seeger, and Wright). As a corollary, if G is an algebraic group over a local field K of characteristic zero, and if its amenable radical is not, up to isogeny, a direct factor, then G(K) has Property (T) relative to a noncompact subgroup. In particular, G(K) does not have Haagerup’s property. This extends a similar result of Cherix, Cowling and Valette for connected Lie groups, to which our method also applies. We give some other applications. We provide a characterization of connected Lie groups all of whose countable subgroups have Haagerup’s property. We give an example of an arithmetic lattice in a connected Lie group which does not have Haagerup’s property, but has no infinite subgroup with relative Property (T). We also give a continuous family of pairwise nonisomorphic connected Lie groups with Property (T), with pairwise nonisomorphic (resp. isomorphic) Lie algebras.
Nondiscrete Euclidean buildings for the Ree and Suzuki groups
, 2008
"... We call a nondiscrete Euclidean building a BruhatT.its space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of nondiscrete Euclidean buildings introduced and studied by Bruhat ..."
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Cited by 4 (2 self)
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We call a nondiscrete Euclidean building a BruhatT.its space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of nondiscrete Euclidean buildings introduced and studied by Bruhat and T.its. We give the complete classification of BruhatT.its spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type B2, F4 or G2 associated with a Ree or Suzuki group endowed with the usual root datum. (In the B2 and G2 cases, this fixed point set is a building of rank one; in the F4 case, it is a generalized octagon whose Weyl group is not crystallographic.) We also show that each of these BruhatT.its spaces has a natural embedding in the unique BruhatT.its space whose building at infinity is the corresponding ambient building.