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The Poisson Boundary Of The Mapping Class Group
, 1995
"... . A theory of random walks on the mapping class group and its nonelementary subgroups is developed. We prove convergence of sample paths in the Thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the Pois ..."
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Cited by 54 (3 self)
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. A theory of random walks on the mapping class group and its nonelementary subgroups is developed. We prove convergence of sample paths in the Thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the Poisson boundary of random walks. The methods are based on an analysis of the asymptotic geometry of Teichmuller space and of the contraction properties of the action of the mapping class group on the Thurston boundary. We prove, in particular, that Teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees. Using our analysis of the mapping class group action on the Thurston boundary we prove that no nonelementary subgroup of the mapping class group can be a lattice in a higher rank semisimple Lie group. Contents 0. Introduction 1. Asymptotic properties of Teichmuller space 1.1. The space of projective measured foliations 1.2. The mapping class group 1.3. Teichmu...
Boundaries of hyperbolic groups
 CONTEMPORARY MATHEMATICS
"... In this paper we survey the known results about boundaries of wordhyperbolic groups. ..."
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Cited by 13 (0 self)
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In this paper we survey the known results about boundaries of wordhyperbolic groups.
FURSTENBERG ENTROPY REALIZATIONS FOR VIRTUALLY FREE GROUPS AND LAMPLIGHTER GROUPS
"... Abstract. Let (G, µ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε> 0 such that the Furstenberg entropy of any (G, µ)stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have prop ..."
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Cited by 7 (6 self)
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Abstract. Let (G, µ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε> 0 such that the Furstenberg entropy of any (G, µ)stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have property (T). For these groups, we construct stationary actions with arbitrarily small, positive entropy. This construction involves building and lifting spaces of lamplighter groups. For some classical lamplighters, these spaces realize a dense set of entropies
Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces
 Geom. Funct. Anal
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Matrix random products with singular harmonic measure
"... Abstract. Any Zariski dense countable subgroup of SL(d, R) is shown to carry a nondegenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the proje ..."
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Cited by 6 (0 self)
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Abstract. Any Zariski dense countable subgroup of SL(d, R) is shown to carry a nondegenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in R d in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and Lyapunov exponents going to infinity.
THE CHOQUETDENY EQUATION IN A BANACH SPACE
, 2006
"... Abstract. Let G be a locally compact group and π a representation of G by weakly* continuous isometries acting in a dual Banach space E. Given a probability measure µ on G we study the ChoquetDeny equation π(µ)x = x, x ∈ E. We prove that the solutions of this equation form the range of a projection ..."
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Cited by 4 (2 self)
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Abstract. Let G be a locally compact group and π a representation of G by weakly* continuous isometries acting in a dual Banach space E. Given a probability measure µ on G we study the ChoquetDeny equation π(µ)x = x, x ∈ E. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula ” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law µ. The relation between the space of solutions of the ChoquetDeny equation in E and the space of bounded harmonic functions can be understood in terms of a construction resembling the W ∗crossed product and coinciding precisely with the crossed product in the special case of the ChoquetDeny equation in the space E = B(L 2 (G)) of bounded linear operators on L 2 (G). Other general properties of the ChoquetDeny equation in a Banach space are also discussed.
FURSTENBERG ENTROPY REALIZATIONS FOR VIRTUALLY FREE GROUPS AND LAMPLIGHTER GROUPS
, 1210
"... Abstract. Let (G,µ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε> 0 such that the Furstenberg entropy of any (G,µ)stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have proper ..."
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Abstract. Let (G,µ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε> 0 such that the Furstenberg entropy of any (G,µ)stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have property (T). For these groups, we construct stationary actions with arbitrarilysmall, positive entropy. This construction involves building and lifting spaces of lamplighter groups. For some classical lamplighters, these spaces realize a dense set of entropies
The Poisson formula for groups .. .
, 1998
"... The Poisson boundary of a group G with a probability measure µ on it is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded µharmonic functions on G. ..."
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The Poisson boundary of a group G with a probability measure µ on it is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded µharmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with infinitely many ends, cocompact lattices in Cartan–Hadamard manifolds, discrete subgroups of semisimple Lie groups, polycyclic groups, some wreath and semidirect products including Baumslag–Solitar groups.