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## FURSTENBERG ENTROPY REALIZATIONS FOR VIRTUALLY FREE GROUPS AND LAMPLIGHTER GROUPS

### Citations

1330 | Probability: theory and examples - Durrett - 1996 |

189 |
Random walks on discrete groups: boundary and entropy
- Kaimanovich, Vershik
- 1983
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Citation Context ...onary space with Furstenberg entropy h. 1.1. Related results. The Furstenberg entropy of any stationary space is bounded from above by the entropy of the Poisson boundary [9]. Kaimanovich and Vershik =-=[14]-=- show that when H (µ) is finite then the Furstenberg entropy of the Poisson boundary is equal to hRW (G,µ), the random walk entropy of µ, defined by hRW (G,µ) = lim n→∞ 1 n H (µn) . Little is known ab... |

147 |
Noncommuting random products.
- Furstenberg
- 1963
(Show Context)
Citation Context ... Hence a stationary measure ν is not in general G-invariant, but it is invariant “on average”, when the average is taken over µ. An important invariant of stationary spaces is the Furstenberg entropy =-=[8]-=-, given by hµ(X, ν) = ∑ g∈G µ(g) ∫ X − log dν dgν (x)dgν(x). Despite the fact that stationary spaces have been studied for several decades now, few examples are known, and the theory of their structur... |

132 | Random Walks on Infinite Graphs and Groups
- Woess
- 2000
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Citation Context ...onary space is also (G,µ)-stationary, and hµ(X, ν) = hϕ∗µ(X, ν). Consider the canonical lamplighter (Z/2Z) ≀ Z3. Since any generating random walk on Z3 is transient (see., e.g., Proposition 3.20 in =-=[20]-=-), by Erschler [6], for any finite first moment µ, it holds that hℓ((Z/2Z) ≀ Z3, µ) ≥ hconf((Z/2Z) ≀ Z3, µ) > 0. Therefore, Claim 4.1 and Lemma 4.2, together with Proposition 3.1, yield the following ... |

68 |
Amenable ergodic group actions and an application to Poisson boundaries of random walks,
- Zimmer
- 1978
(Show Context)
Citation Context ...)-stationary space is Π(G,µ), the Poisson boundary of (G,µ). The Poisson boundary can be defined as the Mackey realization [16] of the shift invariant sigma-algebra of the space of random walks (Ω,P) =-=[14, 21]-=-, also known as the space of shift ergodic components of Ω. Furstenberg’s original definition [9] used the Gelfand representation of the algebra of bounded µ-harmonic functions on G. For formal defini... |

53 |
Property T and asymptotically invariant sequences
- Connes, Weiss
- 1980
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Citation Context ...holds that λp,m(γ is closed) = p (3) For any finite S ⊂ Γ and all p ∈ (0, 1) it holds that lim m→∞ λp,m(S is open ∪ S is closed) = 1. We prove this lemma in Appendix A below. For a related result see =-=[4]-=-. The limit limm λp,m is the non-ergodic percolation λp = pδ∅+(1−p)δΓ. Clearly, the Furstenberg entropy of the associated Bowen space is p ·hRW (G,µ). This is the basic intuition behind Proposition 3.... |

48 |
Factor and normal subgroup theorems for lattices in products of groups, Invent
- Bader, Shalom
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Citation Context ...X : Ω → X that assigns to (Z1, Z2, . . .) the point x is called the boundary map of (X, ν). For further 5discussion and a definition of boundaries that is independent of topology see Bader and Shalom =-=[1]-=-. We shall also consider the Poisson boundary of general Markov chains, defined again as the space of ergodic components of the shift invariant sigma-algebra [12]. 2.3. Furstenberg entropy. The Furste... |

41 |
Random walks on groups and random transformations,
- Furman
- 2002
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Citation Context ...berg’s original definition [9] used the Gelfand representation of the algebra of bounded µ-harmonic functions on G. For formal definitions see also Furstenberg and Glasner [10], or a survey by Furman =-=[7]-=-. G-factors of the Poisson boundary are stationary spaces called (G,µ)-boundaries; the Mackey realization of each G-invariant, shift invariant sigma-algebra is a (G,µ)- boundary. A different perspecti... |

41 |
Stabilizers for ergodic actions of higher rank semisimple groups
- Stuck, Zimmer
- 1994
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Citation Context ...py gap result for (G,µ). 4.3. Lifting Bowen spaces from lattices. The following construction applies to a more general settings, where G is a locally compact group and Γ is a lattice in G (see, e.g., =-=[19]-=-). That is, there exists a G-invariant measure η ∈ P(G/Γ). Let λ ∈ IRS (Γ). Then λ is Γ-invariant but not, in general, G-invariant. Note, however, that if g1Γ = g2Γ then there exists a γ ∈ Γ such that... |

36 | Measure-theoretic boundaries of Markov chains, 0-2 laws and entropy. In: Harmonic analysis and discrete potential theory
- Kaimanovich
- 1991
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Citation Context ...ndent of topology see Bader and Shalom [1]. We shall also consider the Poisson boundary of general Markov chains, defined again as the space of ergodic components of the shift invariant sigma-algebra =-=[12]-=-. 2.3. Furstenberg entropy. The Furstenberg entropy of a (G,µ)-stationary space (X, ν) is given by hµ(X, ν) = ∑ g∈G µ(g) ∫ X − log dν dgν (x)dgν(x). Alternatively, it can be written as hµ(X, ν) = E [D... |

24 |
Point realizations of transformation groups
- Mackey
- 1962
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Citation Context ...hism of the probability spaces, then pi is a G-isomorphism. An important (G,µ)-stationary space is Π(G,µ), the Poisson boundary of (G,µ). The Poisson boundary can be defined as the Mackey realization =-=[16]-=- of the shift invariant sigma-algebra of the space of random walks (Ω,P) [14, 21], also known as the space of shift ergodic components of Ω. Furstenberg’s original definition [9] used the Gelfand repr... |

17 |
Random walks on coset spaces with applications to Furstenberg entropy
- Bowen
- 1008
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Citation Context ... properties is still far from complete [10]. For example, it is in general not known which Furstenberg entropy values they may take; this problem is called the Furstenberg entropy realization problem =-=[2, 18]-=-. More specifically, it is not known which groups have an entropy gap: Definition. (G,µ) has an entropy gap if it admits stationary spaces of positive Furstenberg entropy, and if there exists an ε > 0... |

13 |
Stationary dynamical systems, Dynamical Numbers—Interplay Between Dynamical Systems and
- Furstenberg, Glasner
(Show Context)
Citation Context ...ν dgν (x)dgν(x). Despite the fact that stationary spaces have been studied for several decades now, few examples are known, and the theory of their structure and properties is still far from complete =-=[10]-=-. For example, it is in general not known which Furstenberg entropy values they may take; this problem is called the Furstenberg entropy realization problem [2, 18]. More specifically, it is not known... |

11 |
The spectral theory of amenable actions and invariants of discrete groups, Geometriae Dedicata 100
- Nevo
(Show Context)
Citation Context ...such that the Furstenberg entropy of any ergodic (G,µ)-stationary space is either zero or greater than ε. A group G has an entropy gap if (G,µ) has an entropy gap for every generating measure µ. Nevo =-=[17]-=- shows that any group with Kazhdan’s property (T) has an entropy gap. A natural conjecture is that groups without property (T) do not have an entropy gap, so that having an entropy gap is a characteri... |

10 |
walks and discrete subgroups of Lie groups
- Random
- 1971
(Show Context)
Citation Context ...exists an ergodic (G,µ)-stationary space with Furstenberg entropy h. 1.1. Related results. The Furstenberg entropy of any stationary space is bounded from above by the entropy of the Poisson boundary =-=[9]-=-. Kaimanovich and Vershik [14] show that when H (µ) is finite then the Furstenberg entropy of the Poisson boundary is equal to hRW (G,µ), the random walk entropy of µ, defined by hRW (G,µ) = lim n→∞ 1... |

10 |
Rigidity of Furstenberg entropy for semisimple Lie group actions
- Nevo, Zimmer
- 2000
(Show Context)
Citation Context ... properties is still far from complete [10]. For example, it is in general not known which Furstenberg entropy values they may take; this problem is called the Furstenberg entropy realization problem =-=[2, 18]-=-. More specifically, it is not known which groups have an entropy gap: Definition. (G,µ) has an entropy gap if it admits stationary spaces of positive Furstenberg entropy, and if there exists an ε > 0... |

8 | Invariant random subgroups of the lamplighter group
- Bowen, Grigorchuk, et al.
- 2012
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Citation Context ...are not able to show this, and instead resort to a more elaborate construction which only realizes a dense set of entropies (see Section 3). For more on invariant random subgroups of lamplighters see =-=[3]-=-. 2.7. Digression: the Radon-Nikodym compact is not necessarily a boundary. The Radon-Nikodym factor rn : X → RG assigns to almost every point x in a (G,µ)-stationary space (X, ν) the function fx(g) =... |

8 | The Poisson boundary of lamplighter random walks on trees
- Karlsson, Woess
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Citation Context ...[6] shows that these boundaries are equal for ((Z/2Z) ≀ Zd, µ), when d ≥ 5 and µ has with finite third moment. An additional equality result on non-amenable base groups is given by Karlsson and Woess =-=[15]-=-. 4. No entropy gap for virtually free groups In this section we prove our main result, Theorem 1, which states that when G is virtually free and µ has finite first moment then (G,µ) does not have an ... |

6 | An abramov formula for stationary spaces of discrete groups, arXiv preprint arXiv:1204.5414
- Hartman, Lima, et al.
- 2012
(Show Context)
Citation Context ...lized. We also realize entropies using Bowen spaces: to prove Theorem 1, we construct Bowen spaces of lamplighter groups, and lift them to Bowen spaces of virtually free groups. Using a recent result =-=[11]-=- that relates the entropies of the actions of groups and their finite index subgroups, we control the entropies of the lifted spaces, and show that they can be made arbitrarily small. A natural statio... |

5 |
Poisson–Furstenberg boundary of random walks on wreath products and free metabelian groups
- Erschler
- 2011
(Show Context)
Citation Context ...tural stationary space of lamplighter groups is the limit configuration boundary (see Section 2.6.1), which, in some classical lamplighters, has been shown to 3coincide with the Poisson boundary (see =-=[6, 13, 15]-=-). We denote by hconf(G,µ) its Furstenberg entropy. Our construction of Bowen spaces for lamplighter groups yields the following realization result. Theorem 2. Let G = L ≀ Γ be a finitely generated di... |