Asymptotic geometry in products of Hadamard spaces with rank one isometries

Link, Guido

doi:10.2140/gt.2010.14.1063

Cited by 12 publications

(31 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As mentioned above, when we combine our theorem with the results of [11], we get the following corollary. In any case, all boundaries of the Croke-Kleiner group are now seen to be equivariantly cell-like equivalent.…”

Section: The Main Resultsmentioning

confidence: 64%

See 1 more Smart Citation

Boundaries of Croke-Kleiner-admissible groups and equivariant cell-like equivalence

Guilbault

Mooney

2014

Journal of Topology

View full text Add to dashboard Cite

Abstract. Question 2.6 of Bestvina's Questions in Geometric Group Theory asks whether every pair of boundaries of a given CAT(0) group G is cell-like equivalent [1]. The question was posed by Bestvina shortly after the discovery, by Croke and Kleiner [5], of a CAT(0) group Γ that admits multiple boundaries. Previously, it had been observed by Bestvina and Geoghegan that all boundaries of a torsion free CAT(0) G would necessarily have the same shape. Since "cell-like equivalence" is weaker than topological equivalence, but in most circumstances, stronger (and more intuitive) than shape equivalence, this question is a natural one when working with the pathological types of spaces that occur as group boundaries. Furthermore, the definition of cell-like equivalence allows for a obvious G-equivariant extension. In private conversations, Bestvina has indicated a preference for the G-equivariant formulation of Q2.6.In this paper we provide a positive answer to Bestvina's G-equivariant Cell-like Equivalence Question for the class of admissible groups studied by Croke and Kleiner in [5]. Since that collection includes the original Croke-Kleiner group Γ, our result provides a strong solution to Q2.6, for the group that originally motivated the question.In [8], we described a general strategy for attacking the following:Bestvina's Equivariant Cell-like Equivalence Question. For a CAT(0) group G, are all boundaries G-equivariantly cell-like equivalent?This question is a strong version of Q2.6 from Bestvina's Questions in Geometric Group Theory [1], where G-equivariance is not required. Both versions of the question were motivated by a desire to understand the now-famous example, due to Croke and Kleiner [5], of a CAT(0) group Γ that admits boundaries that are not topologically equivalent. Prior to the emergence of that example, it had been observed by both Bestvina and Geoghegan that all boundaries of a given CAT(0) group G would necessarily be shape equivalent. The point then is that, in the proper context, cell-like equivalence is a relationship more flexible than topological equivalence, but stronger than mere shape equivalence. In addition, by the nature of its definition, a celllike equivalence allows for a more direct comparison of the spaces involved. One of those ways is that an equivariance requirement can easily be added-a requirement that is obviously desireable in the context of group boundaries. (Definitions of all terms used in this paragraph will be discussed shortly.)In [6], Croke and Kleiner built upon their work in [5] by analyzing a collection of CAT(0) groups which they called admissible, and which we term Croke-Kleiner admissible (or CKA groups). The main result of this paper is the following.Theorem A. If G is a Croke-Kleiner admissible group, then all CAT(0) boundaries of G are G-equivariantly cell-like equivalent.Since the the original Croke-Kleiner group Γ is a CKA group, our most striking application is the corresponding:Corollary. All boundaries of the Croke-Kleiner group Γ are Γ-equivariantly cel...

show abstract

Section: The Main Resultsmentioning

confidence: 64%

“…As an application, we used this to prove the following theorem. Note that since those groups have higher rank, the work of [11] does not apply to them. We can now state the main theorem of this paper; it immediately implies Theorem A.…”

Section: The Main Resultsmentioning

confidence: 99%

Boundaries of Croke-Kleiner-admissible groups and equivariant cell-like equivalence

Guilbault

Mooney

2014

Journal of Topology

View full text Add to dashboard Cite

show abstract

“…Recall that the support of a Borel measure ν on a topological space Y is defined as the set (16) supp…”

Section: Geodesic Currents and Ricks' Measurementioning

confidence: 99%

Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces

Link

2020

Tunisian J. Math.

Self Cite

View full text Add to dashboard Cite

Let X be a proper, geodesically complete Hadamard space, and Γ < Is(X) a discrete subgroup of isometries of X with the fixed point of a rank one isometry of X in its infinite limit set. In this paper we prove that if Γ has non-arithmetic length spectrum, then the Ricks' Bowen-Margulis measure -which generalizes the well-known Bowen-Margulis measure in the CAT(−1) setting -is mixing. If in addition the Ricks' Bowen-Margulis measure is finite, then we also have equidistribution of Γ-orbit points in X, which in particular yields an asymptotic estimate for the orbit counting function of Γ. This generalizes well-known facts for non-elementary discrete isometry groups of Hadamard manifolds with pinched negative curvature and proper CAT(−1)spaces.1991 Mathematics Subject Classification. Primary: 22D40, 20F69; Secondary: 37D40, 20F67, 37D25.1 Using the Poincaré series from above, a remarkable Γ-equivariant family of measures (µ x ) x∈X supported on the geometric boundary ∂X of X -a so-called conformal density -can be constructed in our very general setting (see [19] and [23] for the original constructions in hyperbolic n-space).Let G denote the set of parametrized geodesic lines in X endowed with the compact-open topology (which can be identified with the unit tangent bundle SX if X is a Riemannian manifold) and consider the action of R on G by reparametrization. This action induces a flow g Γ on the quotient space Γ G. If X is geodesically complete, then thanks to the construction due to R. Ricks ([21, Section 7]) -which uses the conformal density (µ x ) x∈X described above -we obtain a g Γ -invariant Radon measure m Γ on Γ G. This possibly infinite measure will be called the Ricks' Bowen-Margulis measure, since it generalizes the classical Bowen-Margulis measure in the CAT(−1)-setting.If Γ is divergent, then according to Theorem 10.2 in [17] the dynamical system (Γ G, g Γ , m Γ ) is conservative and ergodic. We also want to mention here that if X is a Hadamard manifold, then Ricks' Bowen-Margulis measure m Γ is equal to Knieper's measure first introduced in Section 2 of [14] for cocompact groups Γ (and which was used in [18] for arbitrary rank one groups). In the cocompact case Knieper's work further implies that the Ricks' Bowen-Margulis measure is the unique measure of maximal entropy on the unit tangent bundle of the compact quotient Γ X (see again Section 2 in [14]).In this article we will first address the question under which hypotheses the dynamical system (Γ G, g Γ , m Γ ) is mixing. We remark that in our very general setting we cannot hope to get mixing without further restrictions on the group Γ: F. Dal'Bo ([10, Theorem A]) showed that even in the special case of a CAT(−1)-Hadamard manifold X, the dynamical system (Γ SX, g Γ , m Γ ) with the classical Bowen-Margulis measure m Γ is not mixing, if the length spectrum of Γ is arithmetic (that is if the set of lengths of closed geodesics in Γ X is a discrete subgroup of R). However, we obtain the best possible result:Theorem A. Let X be a proper, geodesically co...

show abstract

“…In order to prove that the radial limit set of Γ has full measure with respect to µ o we follow as in [29, Section 6] Roblin's exposition. As we want to apply the generalization of the second Borel-Cantelli lemma Lemma 2 in [2], we need to work with a weak Ricks' measure m Γ on Γ [G] and find an appropriate Borel set K ⊆ [G] whose projection to Γ [G] has finite m Γ -measure and which satisfies the two Renyi inequalities (27) and (28) below. Notice that in order to get a better control -and a proof even without the presence of a zero width rank one element -apart from using the weak Ricks' measure we need to choose the set K more carefully than in [29,Section 6].…”

Section: Conservativity In the Case Of Divergent Groupsmentioning

confidence: 99%

“…But by Lemma 6.3 (b) this is a contradiction to µ o (L rad Γ ) = 0. We begin with the proof of (27): From the definition of the weak Ricks' measure and the estimates (25) and (26) it follows that for all γ, ϕ ∈ Γ is uniformly bounded in T as a direct consequence of Corollary 3.8 in [26], we have established (27) with a constant C > 0 depending only on c.…”

Section: Conservativity In the Case Of Divergent Groupsmentioning

confidence: 99%

Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry

Link¹

2018

Discrete &Amp; Continuous Dynamical Systems - A

Self Cite

View full text Add to dashboard Cite

Let X be a proper Hadamard space and Γ < Is(X) a nonelementary discrete group of isometries with a rank one isometry. We discuss and prove Hopf-Tsuji-Sullivan dichotomy for the geodesic flow on the set of parametrized geodesics of the quotient Γ X and with respect to Ricks' measure introduced in [35]. This generalizes previous work of the author and J. C. Picaud on Hopf-Tsuji-Sullivan dichotomy in the analogous manifold setting and with respect to Knieper's measure.1991 Mathematics Subject Classification. Primary: 37D40, 20F67; Secondary: 37D25.

show abstract

Asymptotic geometry in products of Hadamard spaces with rank one isometries

Cited by 12 publications

References 18 publications

Boundaries of Croke-Kleiner-admissible groups and equivariant cell-like equivalence

Boundaries of Croke-Kleiner-admissible groups and equivariant cell-like equivalence

Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces

Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry

Contact Info

Product

Resources

About