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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
An abramov formula for stationary spaces of discrete groups, arXiv preprint arXiv:1204.5414
, 2012
"... Abstract. Let (G, µ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A µ-random walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the rand ..."
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Cited by 6 (1 self)
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Abstract. Let (G, µ) be a discrete group equipped with a generating probability measure, and let Γ be a finite index subgroup of G. A µ-random walk on G, starting from the identity, returns to Γ with probability one. Let θ be the hitting measure, or the distribution of the position in which the random walk first hits Γ. We prove that the Furstenberg entropy of a (G, µ)-stationary space, with respect to the action of (Γ, θ), is equal to the Furstenberg entropy with respect to the action of (G, µ), times the index of Γ in G. The index is shown to be equal to the expected return time to Γ. As a corollary, when applied to the Furstenberg-Poisson boundary of (G, µ), we prove that the random walk entropy of (Γ, θ) is equal to the random walk entropy of (G, µ), times the index of Γ in G.
STABILIZER RIGIDITY IN IRREDUCIBLE GROUP ACTIONS
"... Abstract. We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and Nevo-Zimmer, we show that the action stabilizers, and hence all irreducible invariant random subgroups, are co-amenable i ..."
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Cited by 4 (0 self)
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Abstract. We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and Nevo-Zimmer, we show that the action stabilizers, and hence all irreducible invariant random subgroups, are co-amenable in some normal subgroup. As a consequence, we derive rigidity results on irreducible actions that provide generalizations, and new proofs,
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
Generic Stationary Measures and Actions
, 2015
"... Let G be a countably infinite group, and let µ be a generating probability measure on G. We study the space of µ-stationary Borel probability measures on a topological G space, and in particular on ZG, where Z is any perfect Polish space. We also study the space of µ-stationary, measurable G-actions ..."
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Let G be a countably infinite group, and let µ be a generating probability measure on G. We study the space of µ-stationary Borel probability measures on a topological G space, and in particular on ZG, where Z is any perfect Polish space. We also study the space of µ-stationary, measurable G-actions on a standard, nonatomic probability space. Equip the space of stationary measures with the weak * topology. When µ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of (G,µ). When Z is compact, this implies that the simplex of µ-stationary measures on ZG is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on {0, 1}G. We furthermore show that if the action of G on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary. Next, we consider the space of stationary actions, equipped with a standard topol-ogy known as the weak topology. Here we show that when G has property (T), the
FURSTENBERG ENTROPY REALIZATIONS FOR VIRTUALLY FREE GROUPS AND LAMPLIGHTER GROUPS
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