arXiv subjects

Roman Muchnik

Publications and source records attributed to Roman Muchnik.

4 recordsLinked to original sources

arXiv2011

Boundary unitary representations - irreducibility and rigidity

Let $M$ be compact negatively curved manifold, $Γ=π_1(M)$ and $\tilde{M}$ be its universal cover. Denote by $B =\partial \tilde{M}$ the geodesic boundary of $\tilde{M}$ and by $ν$ the Patterson-Sullivan measure on $X$. In this note we prove that the associated unitary representation of $Γ$ on $L^2(B,ν)$ is irreducible. We also establish a new rigidity phenomenon: we show that some of the geometry of $M$, namely its marked length spectrum, is reflected in this $L^2$-representations.

math.DS↗

arXiv2005

Harmonicity of Gibbs measures

In this paper we extend the construction of random walks with a prescribed Poisson boundary to the case of measures in the class of a generalized Gibbs state. The price for dropping the $α$-quasiconformal assumptions is that we must restrict our attention to CAT($-κ$) groups. Apart from the new estimates required, we prove a new approximation scheme to provide a positive basis for positive functions in a metric measure space.

math.GR↗

arXiv2005

Amenability of Universal 2-Grigorchuk group

We consider the universal Grigorchuk 2-group, i.e., the group such that every Grigorchuk 2-group is a quotient. We show that this group has a nice universal representation in the group of all functions f:{0,1,2}^N --> Aut(T_2), where T_2 is a group of automorphism of the binary tree. Finally, we prove that this universal Grigorchuk 2-group is amenable. The proof is an application of the ``Munchhausen trick'' developed by V. Kaimanovich.

math.GR↗

arXiv2004

Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces

We consider a group G of isometries acting on a (not necessarily geodesic) delta-hyperbolic space X and possessing a radial limit set of full measure within its limit set. For any continuous quasiconformal measure w supported on the limit set, we produce a stationary measure m on G. Moreover the limit set together with w forms a m-boundary and w is harmonic with respect to the random walk induced by m. In the case when X is a CAT(-1) space and G acts cocompactly, for instance, we show that m has finite first moment. This implies that the boundary of X with w is the unique Poisson boundary for m. As a bi-product, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L^1 or sup norm, for the space of uniformly positive lower-semicontinuous functions on a general metric measure space.

math.GR↗