Classifying spaces of Kač-Moody groups

We study the structure of classifying spaces of Kač-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes'T functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kač-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kač-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kač-Moody groups, and centralizers of finite p-subgroups. Received: 15 June 2000 / in final form: 20 September 2001 / Published online: 29 April 2002     

Kernels of maps between classifying spaces

For homomorphisms between groups, one can divide out the kernel to get an injection. Here, we develop a notion of kernels for maps between classifying spaces of compact Lie groups. We show that the kernel is a normal subgroup in a modified sense and prove a generalization of a theorem of Quillen, namely, a mapf:BG→BH _p ^∧ is injective, iff the induced map in mod-p cohomology is finite. Moreover, for compact connected Lie groups, every mapf:BG→BH _p ^∧ factors over a quotient ofG in a modified sense and this factorisation is an injection.     

On profinite groups with finite abelianizations

Profinite groups with finite p-abelianizations arise in various contexts: group theory, number theory and geometry. Using Ph. Furtw?ngler’s transfer vanishing theorem it will be proved that a finitely generated profinite group Ĝ with this property satisfies 〚Ĝ〛) = 0 (Thm. A). As a consequence one finds that a hereditarily just-infinite non-virtually cyclic pro-p group has only one end (Cor. B). Applied to 3-dimensional Poincaré duality groups, Theorem A yields a generalization of A. Reznikov’s theorem on 3-dimensional co-compact hyperbolic lattices violating W. Thurston’s conjecture (Thm. C).     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL ₂(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.     

On the first Hochschild cohomology group of an algebra

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H ¹(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H ¹(A) with the rank and p-rank of the fundamental group π₁(Δ,I) of (Δ,I). We get explicit formulae for dim H ¹(A), when every path in Δ parallel to an arrow belongs to I or when I is homogeneous. Received: 12 April 1999 / Revised version: 9 October 2000     

Integrability of induction cocycles for Kac-Moody groups

We prove that whenever a Kac-Moody group over a finite field is a lattice of its buildings, it has a fundamental domain with respect to which the induction cocycle is L^p for any p ∈ [1;+∞). The proof uses elementary counting arguments for root group actions on buildings. The applications are the possibility to apply some lattice superrigidity, and the normal subgroup property for Kac-Moody lattices.Prépublication de l’Institut Fourier nº 637 (2004); e-mail: http://www-fourier.ujf-grenoble.fr/prepublicatons.html     

Symmetric cohomology of groups in low dimension

We give an explicit characterization for group extensions that correspond to elements of the symmetric cohomology HS ²(G, A). We also give conditions for the map HS ⁿ (G, A) → H ⁿ (G, A) to be injective.     

Overcompleteness of Sequences of Reproducing Kernels in Model Spaces

We give necessary conditions and sufficient conditions for sequences of reproducing kernels (k_Θ(·, λ_n))_{n ≥ 1} to be overcomplete in a given model space K_Θ^p where Θ is an inner function in H^∞, p ∈ (1, ∞), and where (λ_n)_{n ≥ 1} is an infinite sequence of pairwise distinct points of Under certain conditions on Θ we obtain an exact characterization of overcompleteness. As a consequence we are able to describe the overcomplete exponential systems in L² (0, a).     

The K-theory of fields in characteristic p

We show that for a field k of characteristic p, H ⁱ (k,ℤ(n)) is uniquely p-divisible for i≠n (we use higher Chow groups as our definition of motivic cohomology). This implies that the natural map K _n ^M (k)?K _n (k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that K _n ^M (k) and K _n (k) are p-torsion free. As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example K _n (X,ℤ/p ^r )=0 for n>dimX. Another consequence is Gersten’s conjecture with finite coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except possibly the vanishing conjecture. Oblatum 21-I-1998 & 26-VII-1999 / Published online: 18 October 1999     

Weighted integral formulas on manifolds

We present a method of finding weighted Koppelman formulas for (p,q)-forms on n-dimensional complex manifolds X which admit a vector bundle of rank n over X×X, such that the diagonal of X×X has a defining section. We apply the method to ℙ ⁿ and find weighted Koppelman formulas for (p,q)-forms with values in a line bundle over ℙ ⁿ . As an application, we look at the cohomology groups of (p,q)-forms over ℙ ⁿ with values in various line bundles, and find explicit solutions to the -equation in some of the trivial groups. We also look at cohomology groups of (0,q)-forms over ℙ ⁿ ×ℙ ^m with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.     

Denominators of Eisenstein cohomology classes for GL2 over imaginary quadratic fields

We study the arithmetic of Eisenstein cohomology classes for symmetric spaces associated to GL₂ over imaginary quadratic fields. We prove in many cases a lower bound on their denominator in terms of an L-value of a Hecke character providing evidence for a conjecture of Harder that the denominator is given by this L-value. Furthermore, we exibit conditions under which the restriction of the classes to the boundary is integral.     

Correspondance de Jacquet–Langlands explicite II. Le cas de degréégal à la caractéristique résiduelle

Let F be a non-Archimedean locally compact field, and let p be its residual characteristic. Put G=GL _p (F) and let G ^′=D _×, where $D$ is a division algebra with centre F and of degree p ² over F. The Jacquet–Langlands correspondence is a bijection between the discrete series of G and that of G ^′. We describe this explicitly, in terms of Carayol's parametrization of these discrete series. Received: 25 November 1999     

On the non-vanishing of the first Betti number of hyperbolic three manifolds

We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in SL(1,D), where D is a quaternion division algebra defined over a number field E contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbolic three manifolds, with non-torsion first homology group, confirming a conjecture of Waldhausen. The proof uses the characterisation of the image of solvable base change by the author, and the construction of cusp forms with non-zero cusp cohomology by Labesse and Schwermer.Mathematics Subject Classification (2000): 11F75, 22E40, 57M50Revised version: 18 February 2004     

Scale functions on $p$-adic Lie groups

Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ _p , s(x)=s(h) is determined by the semisimple part h of x∈G. For every finite extension K of ℚ _p , the scale functions of G and H:=G(K) are related by s _H ∣ _G =s _G ^{[ K :ℚ} _p ^]. More generally, we clarify the relations between the scale function of a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients. Received: 20 February 1997; Revised version: 18 May 1998     

Riesz transforms on a solvable Lie group of polynomial growth

We study Riesz transforms associated with a sublaplacian H on a solvable Lie group G, where G has polynomial volume growth. It is known that the standard second order Riesz transforms corresponding to H are generally unbounded in L^p(G). In this paper, we establish boundedness in L^p for modified second order Riesz transforms, which are defined using derivatives on a nilpotent group G_N associated with G. Our method utilizes a new algebraic approach which associates a distinguished choice of Cartan subalgebra with the sublaplacian H. We also obtain estimates for higher derivatives of the heat kernel of H, and give a new proof (without the use of homogenization theory) of the boundedness of first order Riesz transforms. Our results can be generalized to an arbitrary (possibly non-solvable) Lie group of polynomial growth.     

A note on restriction maps in the cohomology¶of S-arithmetic groups

We prove injectivity results for restriction maps in the cohomology of S-arithmetic groups: the results proved are valid for cohomology with both characteristic 0 and characteristic p coefficients. Received: 12 March 1999 / Revised version: 10 July 2000     

Bounded cohomology of lattices in higher rank Lie groups

We prove that the natural map H_b ²(Γ)?H²(Γ) from bounded to usual cohomology is injective if Γ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for Γ: the stable commutator length vanishes and any C¹–action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating H_b ^•(Γ) to the continuous bounded cohomology of the ambient group with coefficients in some induction module. Received July 14, 1998 / final version received January 7, 1999     

On the cohomology of orbit space of free ℤ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-actions on lens spaces

Let G = ℤ _p , p an odd prime, act freely on a finite-dimensional CW-complex X with mod p cohomology isomorphic to that of a lens space L ^2m−1(p; q ₁, …, q _m ). In this paper, we determine the mod p cohomology ring of the orbit space X/G, when p ² ∤ m.     

A hasse principle for function fields over pac fields

LetK be a perfect pseudo-algebraically closed field and letF be an extension ofK of relative transcendence degree 1. It is shown that the restriction map Res: Br(F)→Π_p Br(F _p ^h ) is injective, where p ranges over all non-trivialK-places ofF, andF _p ^h is the corresponding henselization. Conversely, the validity of this Hasse principle for all such extensionsF implies a weaker version of pseudo-algebraic closedness. As an application we determine the finitely generated pro-p closed subgroups of the absolute Galois group ofK(t).