The Baum-Connes conjecture for hyperbolic groups

We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups. Oblatum 20-VI-2001 & 24-VIII-2001?Published online: 15 April 2002 RID="*" ID="*"The second author is partially supported by NSF and MSRI.     

The multiplier conjecture for elementary abelian groups

Applying the method that we presented in [19], in this article we prove: “Let G be an elementary abelian p-group. Let n = dn₁. If d(≠ p) is a prime not dividing n₁, and the order w of d mod p satisfies $ w > \frac{{d^2}}{3} $, then the Second Multiplier Theorem holds without the assumption n₁ > λ, except that only one case is yet undecided: w ≤ d², and $ \frac{{p - 1}}{{2w}} \ge 3 $, and t is a quadratic residue mod p, and t is not congruent to $ x^{\frac{{p - 1}}{{2w}}j} $ (mod p) (1 ≤ j < 2w), where t is an integer meeting the conditions of Second Multiplier Theorem, and x is a primitive root of p.”. © 1994 John Wiley & Sons, Inc.     

The Auslander conjecture for NIL-affine crystallographic groups

Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups in Aff(N)=NAut(N) acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions 1n5 the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic.     

The local theta correspondence for unramified unitary groups

We study the local theta correspondences for dual reductive pairs consisting of quasi-split unitary groups defined over a non-archimedean local field. We construct Howe’s correspondence between the set of spherical representations of the one group and that of the other group by using the Whittaker model.     

On the local case in the Aschbacher theorem for linear and unitary groups

We consider the subgroups H in a linear or a unitary group G over a finite field such that O _r (H) ? Z(G) for some odd prime r. We obtain a refinement of the well-known Aschbacher theorem on subgroups of classical groups for this case.     

Purity conjecture for reductive groups

The following problem was posed by J.-L. Colliot-Th èléne and J.-J. Sansuc in [1, p. 124, Problem 6.4]. Given a local regular ring R and a reductive group scheme G over R determine whether the functor S → H 44-01 (S, G) satisfies the property of purity for R. In this work, we study this problem in a number of interesting particular cases. Namely, let k be a characteristic zero field, and G be one of the following algebraic groups over k: PGL _n , SL_1,A , O(q), SO(q), Spin(q), SL _n /μ _d where d divides n (here, A is a central simple k-algebra). In this paper we prove that the functor R → H _ét¹ (R, G) satisfies the property of purity for the group G and a regular local ring containing the field. In view of this result, it would appear reasonable to suggest that the aforementioned functor possesses the property of purity for an arbitrary connected reductive group G over a zero characteristic field k and an arbitrary regular local ring containing the field k. For groups of the types G ₂ and F ₄ with a trivial g₃ invariant, this conjecture has been proved in [2] and [3]. The problem and conjecture formulated above appear to be an extension of the known conjectures proposed by A. Grothendieck and J.-P. Serre (see [5, Remark 3, pp. 26–27], [6, Remark 1.11.a], and [14, Remark on p. 31]).     

The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups

Let G be a locally compact group with cocompact connected component. We prove that the assembly map from the topological K-theory of G to the K-theory of the reduced C^*-algebra of G is an isomorphism. The same is shown for the groups of k-rational points of any linear algebraic group over a local field k of characteristic zero. Dedicated to the memory of Peter Slodowy     

Lazer-McKenna conjecture: The critical case

We consider an elliptic problem of Ambrosetti-Prodi type involving critical Sobolev exponent on a bounded smooth domain of dimension six or higher. By constructing solutions with many sharp peaks near the boundary of the domain, but not on the boundary, we prove that the number of solutions for this problem is unbounded as the parameter tends to infinity, thereby proving the Lazer-McKenna conjecture in the critical case.     

Multiple commutator formulas for unitary groups

Let (A,Λ) be a formring such that A is quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak’s unitary groups GU(2n, A, Λ), n ≥ 3. For a form ideal (I, Γ) of the form ring (A, Λ) we denote by EU(2n, I, Γ) and GU(2n, I, Γ) the relative elementary group and the principal congruence subgroup of level (I, Γ), respectively. Now, let (I _i , Γ _i ), i = 0,...,m, be form ideals of the form ring (A, Λ). The main result of the present paper is the following multiple commutator formula: [EU(2n, I ₀, Γ ₀),GU(2n, I ₁, Γ ₁), GU(2n, I ₂, Γ ₂),..., GU(2n, I _m , Γ _m )] =[EU(2n, I ₀, Γ ₀), EU(2n, I ₁, Γ ₁), EU(2n, I ₂, Γ ₂),..., EU(2n, I _m , Γ _m )], which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classicallike groups over commutative and finite-dimensional rings.     

Cone-decompositions of the unitary groups

The Lusternik-Schnirelmann category of a space is a homotopy invariant. Cone-decompositions are used for giving upper-bound for Lusternik-Schnirelmann categories of topological spaces. Singhof has determined the Lusternik-Schnirelmann categories of the unitary groups. In this paper I give two cone-decompositions of each unitary group for alternative proofs of Singhof's result. One cone-decomposition is easy. The other is closely related to Miller's filtration and Yokota's cellular decomposition of the unitary groups.     

From acyclic groups to the Bass conjecture for amenable groups

We prove that the Bost Conjecture on the ¹-assembly map for countable discrete groups implies the Bass Conjecture. It follows that all amenable groups satisfy the Bass Conjecture.     

The strong Atiyah conjecture for virtually cocompact special groups

We provide new conditions for the strong Atiyah conjecture to lift to finite group extensions. In particular, we show that fundamental groups of compact special cube complexes satisfy these conditions, so the conjecture holds for finite extensions of these groups.     

The McKay conjecture for exceptional groups and odd primes

Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G → G and G := G ^F , such that the root system is of exceptional type or G is a Suzuki group or Steinberg’s triality group. We show that all irreducible characters of C _G (S), the centraliser of S in G, extend to their inertia group in N _G (S), where S is any F-stable Sylow torus of (G, F). Together with the work in [16] this implies that the McKay conjecture is true for G and odd primes ℓ different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [14]. This research has been supported by the DFG-grant “Die Alperin-McKay-Vermutung für endliche Gruppen” and an Oberwolfach Leibniz fellowship.     

Simple loop conjecture for limit groups

There are noninjective maps from surface groups to limit groups that don’t kill any simple closed curves. As a corollary, there are noninjective all-loxodromic representations of surface groups to SL(2, ?) that don’t kill any simple closed curves, answering a question of Minsky. There are also examples, for any k, of noninjective all-loxodromic representations of surface groups killing no curves with self-intersection number at most k.