We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K0 of the category of “φ-endomorphisms of finitely generated free RΠ(G,X)-modules”. We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.

For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/\tau$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to abelian $p$-completed $BP_*BP$-comodules are concentrated in even degrees. We prove $\mathcal{D}^b({{BP}_*{B...

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...

in this paper the periodic solutions of fourth order delay differential equation of the form $ddddot{x}(t)+adddot{x}(t)+f(ddot{x}(t-tau(t)))+g(dot{x}(t-tau(t)))+h({x}(t-tau(t)))=p(t)$  is investigated. some new positive periodic criteria are given.

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.

Let $X$ be an infinite set, equipped with a topology $tau$. In this paper we studied the relationship between $tau$, and ultrafilters on $X$. We can discovered, among other thing, some relations of the Robinson's compactness theorem, continuity and the separation axioms. It is important also, aspects of communication between mathematical concepts.