scholarly journals Rank-one isometries of CAT(0) cube complexes and their centralisers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Anthony Genevois
Keyword(s):  
Infinite Order ◽  
Cube Complex ◽  
Rank One ◽  
Cube Complexes

Abstract If G is a group acting geometrically on a CAT(0) cube complex X and if g ∈ G has infinite order, we show that exactly one of the following situations occurs: (i) g defines a rank-one isometry of X; (ii) the stable centraliser SCG (g) = {h ∈ G ∣ ∃ n ≥ 1, [h, gn ] = 1} of g is not virtually cyclic; (iii) Fix Y (gn ) is finite for every n ≥ 1 and the sequence (Fix Y (gn )) takes infinitely many values, where Y is a cubical component of the Roller boundary of X which contains an endpoint of an axis of g. We also show that (iii) cannot occur in several cases, providing a purely algebraic characterisation of rank-one isometries.

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 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 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.

scholarly journals Cube complexes and abelian subgroups of automorphism groups of RAAGs

2019 ◽  
pp. 1-25
Author(s):  
BENJAMIN MILLARD ◽  
KAREN VOGTMANN
Keyword(s):  
Lower Bounds ◽  
Automorphism Groups ◽  
Cohomological Dimension ◽  
Upper And Lower Bounds ◽  
Maximal Dimension ◽  
Cube Complex ◽  
Outer Automorphisms ◽  
Abelian Subgroups ◽  
Cube Complexes ◽  
Virtual Cohomological Dimension

Abstract We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

Multiple Mixing and Rank One Group Actions

2000 ◽  
Vol 52 (2) ◽  
pp. 332-347 ◽  
Author(s):  
Andrés del Junco ◽  
Reem Yassawi
Keyword(s):  
Abelian Group ◽  
Large Class ◽  
Probability Space ◽  
Infinite Order ◽  
Group Actions ◽  
Intersection Property ◽  
Product Measure ◽  
Multiple Mixing ◽  
Rank One ◽  
Countable Abelian Group

AbstractSuppose G is a countable, Abelian group with an element of infinite order and let be amixing rank one action of G on a probability space. Suppose further that the Følner sequence {Fn} indexing the towers of satisfies a “bounded intersection property”: there is a constant p such that each {Fn} can intersect no more than p disjoint translates of {Fn}. Then is mixing of all orders. When G = Z, this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of k copies of is necessarily product measure. This method generalizes Ryzhikov’s technique.

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.

scholarly journals Relative cubulations and groups with a 2-sphere boundary

2020 ◽  
Vol 156 (4) ◽  
pp. 862-867
Author(s):  
Eduard Einstein ◽  
Daniel Groves
Keyword(s):  
Finite Volume ◽  
Hyperbolic Group ◽  
Kleinian Groups ◽  
Cube Complex ◽  
Relatively Hyperbolic Group ◽  
Cube Complexes ◽  
Relatively Hyperbolic ◽  
Geometric Action

We introduce a new kind of action of a relatively hyperbolic group on a $\text{CAT}(0)$ cube complex, called a relatively geometric action. We provide an application to characterize finite-volume Kleinian groups in terms of actions on cube complexes, analogous to the results of Markovic and Haïssinsky in the closed case.

scholarly journals Tubular Free by Cyclic Groups Act Freely on CAT(0) Cube Complexes

2017 ◽  
Vol 60 (1) ◽  
pp. 54-62 ◽  
Author(s):  
Jack Button
Keyword(s):  
Fundamental Group ◽  
Cyclic Group ◽  
Graph Of Groups ◽  
Cube Complex ◽  
Cyclic Groups ◽  
Cube Complexes

AbstractWe identify when a tubular group (the fundamental group of a ûnite graph of groups with ℤ2 vertex and ℤ edge groups) is free by cyclic and show, using Wise’s equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex.

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.

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