Volumes of Discrete Groups and Topological Complexity of Homology Spheres

We address two fundamental and well-known problems of Gromov and Lyndon: \demo{Problem A} (Gromov, see [5]). Consider a category $M_n$ of closed manifolds of dimension $n$ with nonzero-degree ways as morphisms. Study a partial order $M \ge N \Leftrightarrow Mor (M, N) \neq \phi$. For which $N$ the degrees of maps $f: M \to N$ are bounded for all $M$? \demo{Problem B} (Lyndon, [12], problem 13). Extend and relate the theories of deficiency, the rate of growth and the Euler-Poincar\'e characteristic. In particular, what influence does the deficiency have on the structure of an infinite group?Comment: Plain TE

Simpson's Theory and Superrigidity of Complex Hyperbolic Lattices

We attack a conjecture of J. Rogawski: any cocompact lattice in $S U (2, 1)$ for which the ball quotient $X = B^2 / \Gamma$ satisfies $b_1 (X) = 0$ and H^{1, 1} (X) \cap H^2 (X, \bbq) \approx \bbq is arithmetic. We prove the Archimedian suprerigidity for representation of $\Gamma$ is S L (3, \bbc).Comment: 6 pages, plain TE

Sharp weak type estimates for weights in the class Ap1,p2A_{p_1, p_2}

We get sharp estimates for the distribution function of nonnegative weights, which satisfy so called $A_{p_1, p_2}$ condition. For particular choices of parameters $p_1$, $p_2$ this condition becomes an $A_p$-condition or Reverse H\"{o}lder condition. We also get maximizers for these sharp estimates. We use the Bellman technique and try to carefully present and motivate our tactics. As an illustration of how these results can be used, we deduce the following result: if a weight $w$ is in $A_2$ then it self-improves to a weight, which satisfies a Reverse H\"{o}lder condition.Comment: Submitted to Revista Matematica Iberoamerican

continious cohomology of groups of volume-preserving and symplectic diffepmorphisms, measurable transfer and higher asymtotic cycles

I construct the real counterparts (which I call Borel-Bott classes) of the R/Z classes constructed in "Characteristic classes in symplectic topology", to appear, in the cohomology of volume-preserving and symplectomorhisms of a compact (symplectic) manifold.I show that, for the symplectic action of the mapping class group in the moduli space of stable vector bundles over a Riemann surface, the restriction of the first constructed class from the symplectomorphism group gives a generator for the second (bounded) cohomology of the mapping class group.Comment: amste

Rationality of secondary classes

We prove the Bloch conjecture : c_2(E) \in H^4_\cald (X,\bbz(2)) is torsion for holomorphic rank two vector bundles $E$ with an integrable connection over a complex projective variety $X$. We prove also the rationality of the Chern-Simons invariant of compact arithmetic hyperbolic three-manifolds. We give a sharp higher-dimensional Milnor inequality for the volume regulator of all representations to $PSO(1,n)$ of fundamental groups of compact $n$-dimensional hyperbolic manifolds, announced in our earlier paper

Hakenness and b_1

Three great theorems of Thurston read: Haken manifolds are hyperbolic; big ramified coverings are hyperbolic; big surgeries are hyperbolic. Recent developments indicate that the later two theorems are essentially a corollary of the first, that is there are much more Haken manifolds than expected by Thurston. In fact Freedman showed very recently that big ramified coverings are Haken. A version of his proof with various improvements was obtained by Cooper-Long and Cooper-Long-Reid. The two main contributions of the present paper are the following. I give an analytic proof of Freedman's result and the improved version of Cooper-Long-Reid. This proof is based on fundamentally different approach than Freedman's and is 10 times shorter, but uses the full forse of the hyperbolization theorem. Secondly, I prove that ANY ramified covering of a tight knot is Haken. Morover any knot becomes tight after a big surgery. So any ramified covering of a big surgery is Haken. Many other results of the paper are better seen from the introduction. The paper uses many different techniques and may be difficult to read for a beginner.Comment: AMSTEX, preprint MPI (August 1997, revised January, 1998

Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory

We consider restrictions along closed geodesics and geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann surface. We obtain a non-trivial bound on the L^2-norm of such restrictions as the eigenvalue tends to infinity. We use methods from the theory of automorphic functions and in particular the uniqueness of invariant functionals on irreducible unitary representations of PGL(2,R).Comment: An updated version of the text not intended for a publication for being obsolete. A remark on a bound for a period added

Limiting cycles and periods of Maass forms

We consider (generalized) periods of Maass forms along non-closed geodesics having a closed geodesic as the limit set

Microlocal lifts of eigenfunctions on hyperbolic surfaces and trilinear invariant functionals

S. Zelditch introduced an equivariant version of a pseudo-differential calculus on a hyperbolic Riemann surface. We recast his construction in terms of trilinear invariant functionals on irreducible unitary representations of PGL(2,R). This allows us to use certain properties of these functionals in the study of the action of pseudo-differential operators on eigenfunctions of the Laplacian on hyperbolic Riemann surfaces.Comment: 24 page

Yamabe Spectra

We study the set of volumes of constant scalar curvature one metrics on an atoroidal three-manifold.The infinum of this set is believed to be attained at a hyperbolic metric. We prove that the supremum of this set is always infinity. The technique is: minimal surfaces, Thurston norm in homology and new conformal invariants.Comment: Plain TEX, 13 pages. The auxilliary files: vanilla.sty, definiti.tex and mathchar.tex should be available from dg-ga, or may be sent by the autho