scholarly journals The Furstenberg–Poisson boundary and CAT(0) cube complexes

2017 ◽  
Vol 38 (6) ◽  
pp. 2180-2223 ◽  
Author(s):  
TALIA FERNÓS
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Stationary Measure ◽  
Poisson Boundary ◽  
Proper Action ◽  
Cube Complex ◽  
Finite Dimensional ◽  
Cube Complexes

We show under weak hypotheses that $\unicode[STIX]{x2202}X$, the Roller boundary of a finite-dimensional CAT(0) cube complex $X$ is the Furstenberg–Poisson boundary of a sufficiently nice random walk on an acting group $\unicode[STIX]{x1D6E4}$. In particular, we show that if $\unicode[STIX]{x1D6E4}$ admits a non-elementary proper action on $X$, and $\unicode[STIX]{x1D707}$ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a $\unicode[STIX]{x1D707}$-stationary measure on $\unicode[STIX]{x2202}X$ making it the Furstenberg–Poisson boundary for the $\unicode[STIX]{x1D707}$-random walk on $\unicode[STIX]{x1D6E4}$. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.

scholarly journals An Abramov formula for stationary spaces of discrete groups

2013 ◽  
Vol 34 (3) ◽  
pp. 837-853 ◽  
Author(s):  
YAIR HARTMAN ◽  
YURI LIMA ◽  
OMER TAMUZ
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Finite Index ◽  
Discrete Group ◽  
Return Time ◽  
Discrete Groups ◽  
Poisson Boundary ◽  
Index Subgroup ◽  
Expected Return ◽  
Finite Index Subgroup

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

scholarly journals FROM WALL SPACES TO CAT(0) CUBE COMPLEXES

2005 ◽  
Vol 15 (05n06) ◽  
pp. 875-885 ◽  
Author(s):  
INDIRA CHATTERJI ◽  
GRAHAM NIBLO
Keyword(s):  
Proper Action ◽  
Cube Complex ◽  
Cube Complexes

We explain how to adapt a construction due to M. Sageev in order to construct a proper action of a group on a CAT(0) cube complex starting from a proper action of the group on a wall space.

scholarly journals Hyperbolic and cubical rigidities of Thompson’s group V

2019 ◽  
Vol 22 (2) ◽  
pp. 313-345 ◽  
Author(s):  
Anthony Genevois
Keyword(s):  
General Criterion ◽  
Isometric Action ◽  
Group V ◽  
Cube Complex ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Fixed Point Properties ◽  
Thompson’S Group ◽  
Cube Complexes ◽  
Finitely Presented

Abstract In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity, or stabilises a pair of points at infinity, or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson’s group V is hyperbolically elementary, and we deduce that it satisfies Property {({\rm FW}_{\infty})} , i.e., every isometric action of V on a finite-dimensional CAT(0) cube complex fixes a point. It provides the first example of a (finitely presented) group acting properly on an infinite-dimensional CAT(0) cube complex such that all its actions on finite-dimensional CAT(0) cube complexes have global fixed points.

Limiting diffusion for random walks with drift conditioned to stay positive

1978 ◽  
Vol 15 (02) ◽  
pp. 280-291 ◽  
Author(s):  
Peichuen Kao
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Function ◽  
Random Variables ◽  
Principal Result ◽  
Hitting Time ◽  
Brownian Excursion ◽  
Norming Constant ◽  
Finite Dimensional ◽  
Independent Identically Distributed

Let {ξ k : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ 1} = μ ≠ 0. Form the random walk {S n : n ≧ 0} by setting S 0, S n = ξ 1 + ξ 2 + ··· + ξ n , n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ 1) that the finite-dimensional distributions of Xn , conditioned on n < N < ∞ converge to those of the Brownian excursion process.

scholarly journals On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points

2019 ◽  
Vol 22 (6) ◽  
pp. 1089-1099
Author(s):  
Motoko Kato
Keyword(s):  
Fixed Points ◽  
Topological Dimension ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Thompson’S Group ◽  
Simple Action ◽  
Cube Complexes

Abstract We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson’s group T and various generalizations of Thompson’s group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces, while it is known that T and V act properly on infinite-dimensional CAT(0) cube complexes.

"MÜNCHHAUSEN TRICK" AND AMENABILITY OF SELF-SIMILAR GROUPS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 907-937 ◽  
Author(s):  
VADIM A. KAIMANOVICH
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Group Identity ◽  
Convex Combination ◽  
Similar Group ◽  
Grigorchuk Group ◽  
Intermediate Growth ◽  
Group A ◽  
Self Similar ◽  
Asymptotic Entropy

The structure of a self-similar group G naturally gives rise to a transformation which assigns to any probability measure μ on G and any vertex w in the action tree of the group a new probability measure μw. If the measure μ is self-similar in the sense that μw is a convex combination of μ and the δ-measure at the group identity, then the asymptotic entropy of the random walk (G, μ) vanishes; therefore, the random walk is Liouville and the group G is amenable. We construct self-similar measures on several classes of self-similar groups, including the Grigorchuk group of intermediate growth.

scholarly journals Non-hyperbolic automatic groups and groups acting on CAT(0) cube complexes

2014 ◽  
Vol 24 (06) ◽  
pp. 795-813
Author(s):  
Yoshiyuki Nakagawa ◽  
Makoto Tamura ◽  
Yasushi Yamashita
Keyword(s):  
Cube Complex ◽  
Automatic Group ◽  
Mild Conditions ◽  
Automatic Groups ◽  
Cube Complexes

We discuss a problem posed by Gersten: Is every automatic group which does not contain ℤ × ℤ subgroup, hyperbolic? To study this question, we define the notion of "n-track of length n", which is a structure like ℤ × ℤ, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts freely, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is "weakly special", then the above question is answered affirmatively.

scholarly journals A Consistent Markov Partition Process Generated from the Paintbox Process

2011 ◽  
Vol 48 (3) ◽  
pp. 778-791 ◽  
Author(s):  
Harry Crane
Keyword(s):  
Probability Measure ◽  
Point Process ◽  
Markov Processes ◽  
Poisson Point Process ◽  
Transition Probabilities ◽  
Stationary Measure ◽  
Product Measure ◽  
Natural Numbers ◽  
Reversible Processes ◽  
Partition Process

We study a family of Markov processes on P(k), the space of partitions of the natural numbers with at most k blocks. The process can be constructed from a Poisson point process on R+ x ∏i=1kP(k) with intensity dt ⊗ ϱν(k), where ϱν is the distribution of the paintbox based on the probability measure ν on Pm, the set of ranked-mass partitions of 1, and ϱν(k) is the product measure on ∏i=1kP(k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

A Comparison of Methods for Constructing Probability Measures on Infinite Product Spaces

1987 ◽  
Vol 30 (3) ◽  
pp. 282-285 ◽  
Author(s):  
Charles W. Lamb
Keyword(s):  
Probability Measure ◽  
Conditional Probability ◽  
Product Space ◽  
Infinite Product ◽  
Probability Measures ◽  
Classical Theorem ◽  
Comparison Of Methods ◽  
Product Spaces ◽  
Probability Functions ◽  
Finite Dimensional

AbstractThe construction, from a consistent family of finite dimensional probability measures, of a probability measure on a product space when the marginal measures are perfect is shown to follow from a classical theorem due to Ionescu Tulcea and known results on the existence of regular conditional probability functions.

scholarly journals Finiteness properties of cubulated groups

2014 ◽  
Vol 150 (3) ◽  
pp. 453-506 ◽  
Author(s):  
G. C. Hruska ◽  
Daniel T. Wise
Keyword(s):  
Hyperbolic Groups ◽  
Cube Complex ◽  
Relatively Hyperbolic Groups ◽  
Finiteness Properties ◽  
Cube Complexes ◽  
Relatively Hyperbolic

AbstractWe give a generalized and self-contained account of Haglund–Paulin’s wallspaces and Sageev’s construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let $H_1,\ldots, H_s$ be relatively quasiconvex codimension-1 subgroups of a group $G$ that is hyperbolic relative to $P_1, \ldots, P_r$. We prove that $G$ acts relatively cocompactly on the associated dual CAT(0) cube complex $C$. This generalizes Sageev’s result that $C$ is cocompact when $G$ is hyperbolic. When $P_1,\ldots, P_r$ are abelian, we show that the dual CAT(0) cube complex $C$ has a $G$-cocompact CAT(0) truncation.

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