Superexpanders from group actions on compact manifolds

Laat, Tim de; Vigolo, Federico

doi:10.1007/s10711-018-0371-0

Cited by 8 publications

(13 citation statements)

References 27 publications

(37 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nonetheless, quasi-isometry of warped cones implies the following 'stable' quasi-isometry (Theorem 3.1), which was very recently obtained independently by de Laat and Vigolo [19].…”

Section: Resultsmentioning

confidence: 79%

“…Nonetheless, quasi‐isometry of warped cones implies the following ‘stable’ quasi‐isometry (Theorem ), which was very recently obtained independently by de Laat and Vigolo . Theorem Quasi‐isometry of warped cones

{scriptO}_{normalΓ} M ≃ {scriptO}_{normalΔ} N

implies the following quasi‐isometry of groups

Γ \times Z^{m} ≃ Δ \times Z^{n}

, for manifolds

M, N

of dimension

m

and

n

, respectively.…”

Section: Introductionmentioning

confidence: 90%

“…Tim de Laat and Federico Vigolo very recently obtained independently a different strengthening of Theorem in which they assume only that the actions

Γ ↷ M

and

Λ ↷ N

are essentially free.…”

Section: Quasi‐isometry Of Cones Implies Stable Quasi‐isometry Of Groupsmentioning

confidence: 99%

See 2 more Smart Citations

Warped cones, (non‐)rigidity, and piecewise properties

Sawicki

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

We prove that if a quasi‐isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi‐isometry of the respective warped cones. For a general quasi‐isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi‐isometric after taking Cartesian products with suitable powers of the integers. Second, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone, improve bounds by Szabó, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups, and give a partial answer to a question of Willett about dynamic asymptotic dimension. In the Appendix, we justify optimality of the aforementioned result on general quasi‐isometries by showing that quasi‐isometric warped cones need not come from quasi‐isometric groups, contrary to the case of box spaces.

show abstract

“…Nonetheless, quasi-isometry of warped cones implies the following 'stable' quasi-isometry (Theorem 3.1), which was very recently obtained independently by de Laat and Vigolo [19].…”

Section: Resultsmentioning

confidence: 79%

{scriptO}_{normalΓ} M ≃ {scriptO}_{normalΔ} N

implies the following quasi‐isometry of groups

Γ \times Z^{m} ≃ Δ \times Z^{n}

, for manifolds

M, N

of dimension

m

and

n

, respectively.…”

Section: Introductionmentioning

confidence: 90%

See 1 more Smart Citation

Warped cones, (non‐)rigidity, and piecewise properties

Sawicki

2018

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The estimate (39) corresponds to the original definition of metric cotype in [121], while here we considered the variant (40), in which the averaging in the right hand side is over the 2 n possible sign vectors in {−1, 1} n rather than over the 3 n possible ℓ ∞ -edges in {−1, 0, 1} n . We will now show that these definitions are essentially equivalent, up to universal constant factors, by establishing the two estimates in (41) below, which hold for every metric space (X , d X ), every q ∈ [1, ∞) and every m, n ∈ N. This confirms a prediction of [135,Section 5.2], where a special case was treated.…”

Section: The Definitions Of Metric Cotype With ℓ ∞ Edges and Sign Edgmentioning

confidence: 99%

Nonpositive curvature is not coarsely universal

Eskenazis

Mendel

2019

Invent. math.

View full text Add to dashboard Cite

We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni, Naor and Neiman (2015). We establish this statement by proving that a metric space which is q-barycentric for some q ∈ [1,∞) has metric cotype q with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the ℓ ∞ grid [m] n ∞ = ({1,... ,m} n , · ∞ ) into ℓ q for all q ∈ (2,∞), from which we deduce the following discrete converse to the fact that ℓ n ∞ embeds with distortion O(1) into ℓ q for q = O(logn). A rigidity theorem of Ribe (1976) implies that for every n ∈ N there exists m ∈ N such that if [m] n ∞ embeds into ℓ q with distortion O(1), then q is necessarily at least a universal constant multiple of log n. Ribe's theorem does not give an explicit upper bound on this m, but by the work of Bourgain (1987) it suffices to take m = n, and this was the previously best-known estimate for m. We show that the above discretization statement actually holds when m is a universal constant.

show abstract

“…More recently, attention has focused on constructions of (super)expanders with a wide range of geometric behavior. This began in [MN1,Theorem 9.1] and [Ost,Theorem 5.76] and has led to a rush of recent results by Hume,Hum,DK,dLV,Saw1]. Most of these papers study the problem from the point of view of its intrinsic interest, but there are also intriguing connections to the theory of computation in [MN2].…”

mentioning

confidence: 99%

Rigidity of warped cones and coarse geometry of expanders

Fisher¹,

Nguyen²,

Limbeek³

2019

Advances in Mathematics

View full text Add to dashboard Cite

We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian factors, then two such warped cones are quasi-isometric if and only if the actions are finite covers of conjugate actions. As a consequence, we produce continuous families of non-quasi-isometric expanders and superexpanders. The proof relies on the use of coarse topology for warped cones, such as a computation of their coarse fundamental groups.From the above expression, it is clear the large-scale geometry of these level sets is invariant under taking finite quotients or extensions of the action: Lemma 2.3. Let F ⊆ Γ be a finite normal subgroup, and write Γ := Γ/F and M := M/F . Write M t (resp. M t ) for the level set at time t of the cone over Γ M (resp. Γ M ). Then M t is (1, B)-quasi-isometric to M t , where B is the maximal word length in Γ of an element of F .Proof. Write π : M t → M t for the natural quotient map. It is easy to see that π is distance non-increasing, i.e. 1-Lipschitz. For the lower bound, let C be the maximal word length of an element of the finite group F . Further

show abstract

Superexpanders from group actions on compact manifolds

Cited by 8 publications

References 27 publications

Warped cones, (non‐)rigidity, and piecewise properties

Warped cones, (non‐)rigidity, and piecewise properties

Nonpositive curvature is not coarsely universal

Rigidity of warped cones and coarse geometry of expanders

Contact Info

Product

Resources

About