We attack a conjecture of J. Rogawski: any cocompact lattice in SU(2, 1) for which the ball quotient X = B/Γ satisfies b1(X) = 0 and H (X)∩H(X,Q) ≈ Q is arithmetic. We prove the Archimedian suprerigidity for representation of Γ is SL(3,C). Théorie de Simpson et superrigidité des réseaux hyperboliques complexes Résumé Soit Γ ⊂ SU(2, 1) un reseau cocompact et soit X = B/Γ. Nous preuvons: si b1(X)...

We give conditions under which the space of bounded harmonic functions on a Riemannian manifold M is naturally isomorphic to the space of bounded harmonic functions of a Markov chain on a discrete net X M arising from a discretization procedure for the pair (M; X). If, further, M is a regular covering manifold and the net is invariant with respect to the deck transformation group, then the entr...

We consider the Weil–Petersson gradient vector field of renormalized volume on deformation space convex cocompact hyperbolic structures (relatively) acylindrical manifolds. In this paper we prove conjecture that flow has a global attracting fixed point at unique structure $M_{\rm geod}$ with minimum core volume.

The aim of this paper is to establish exponential mixing frame flows for convex cocompact hyperbolic manifolds arbitrary dimension with respect the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula matrix coefficients error term as well equidistribution holonomy closed geodesics. main technical result a spectral bound on transfer operators twisted by , w...

We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices cocompact arithmetic reflection groups.

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...

in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.