Quasi-isometric Embedding Research Articles

Random walks and contracting elements II: Translation length and quasi-isometric embedding

Continuing from Choi (2022), we study random walks on metric spaces with contracting elements. We prove that random subgroups of the isometry group of a metric space are quasi-isometrically embedded into the space. We discuss this problem in two ways, namely in the sense of random walks and counting problem. We also establish the genericity of contracting elements and the central limit theorem and its converse for translation length.

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

Let N be a connected nonorientable surface with or without boundary and punctures, and j\colon S\rightarrow N be the orientation double covering. It has previously been proved that j induces an embedding \iota\colon\mathrm{Mod}(N)\hookrightarrow\mathrm{Mod}(S) with one exception. In this paper, we prove that the injective homomorphism \iota is a quasi-isometric embedding. The proof is based on the semihyperbolicity of \mathrm{Mod}(S) , which has already been established. We also prove that the embedding \mathrm{Mod}(F') \hookrightarrow \mathrm{Mod}(F) induced by an inclusion of a pair of possibly nonorientable surfaces F' \subset F is a quasi-isometric embedding.

Maps between relatively hyperbolic spaces and between their boundaries

We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups.

On the quasi-Figiel problem and extension of ε-isometry on unit sphere of [formula omitted] space

This paper aims to study the extension of a bijective ε-isometry and the existence of the Figiel operator of an isometric embedding between two unit spheres of Banach spaces. In the first part, we introduce and study the quasi-Figiel problem about a quasi-isometric embedding between unit spheres of two Banach spaces. Consequently, a quasi-anti-Lipschitz type inequality is obtained when the domain space is l∞n. Based on this quasi-anti-Lipschitz type inequality, the corresponding quasi-anti-Lipschitz type inequality is also obtained for an ε-isometric embedding defined on the unit sphere of an L∞,1+ space, i.e., a Banach space whose finite-dimensional subspaces can be any close to the finite-dimensional subspaces of l∞ in the sense of the Banach-Mazur distance. As an application of the quasi-anti-Lipschitz type inequality, we show that every bijective ε-isometry between the unit spheres of an L∞,1+ space and another Banach space can be extended to a bijective 5ε-isometry between their corresponding unit balls. In particular, this implies that every L∞,1+ space admits the Mazur-Ulam property. Furthermore, in this paper, some attempts are also made to generalize the classical Figiel's theorem to the local case with respect to an isometric embedding between unit spheres.

Embeddings of trees, cantor sets and solvable Baumslag–Solitar groups

We characterise when there exists a quasiisometric embedding between two solvable Baumslag–Solitar groups. This extends the work of Farb and Mosher on quasiisometries between the same groups. More generally, we characterise when there can exist a quasiisometric embedding between two treebolic spaces. This allows us to determine when two treebolic spaces are quasiisometric, confirming a conjecture of Woess. The question of whether there exists a quasiisometric embedding between two treebolic spaces turns out to be equivalent to the question of whether there exists a bilipschitz embedding between two symbolic Cantor sets, which in turn is equivalent to the question of whether there exists a rough isometric embedding between two regular rooted trees. Hence we answer all three of these questions simultaneously. It turns out that the existence of such embeddings is completely determined by the boundedness of an intriguing family of integer sequences.

Quasi-isometric Embeddings from Generalised Thompson’s Groups to Thompson’s Group T

Brown has defined the generalised Thompson's group $F_n$, $T_n$, where $n$ is an integer at least $2$ and Thompson's groups $F= F_2$ and $T =T_2$ in the 80's. Burillo, Cleary and Stein have found that there is a quasi-isometric embedding from $F_n$ to $F_m$ where $n$ and $m$ are positive integers at least 2. We show that there is a quasi-isometric embedding from $T_n$ to $T_2$ for any $n \geq 2$ and no embeddings from $T_2$ to $T_n$ for $n \geq 3$.

Poincaré profiles of Lie groups and a coarse geometric dichotomy

Poincaré profiles are analytically defined invariants, which provide obstructions to the existence of coarse embeddings between metric spaces. We calculate them for all connected unimodular Lie groups, Baumslag–Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. For Lie groups, our dichotomy extends both the rank one versus higher rank dichotomy for semisimple Lie groups and the polynomial versus exponential growth dichotomy for solvable unimodular Lie groups. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. As a consequence, we deduce that for groups of the form Ntimes S, where N is a connected nilpotent Lie group, and S is a rank one simple Lie group, both the growth exponent of N, and the conformal dimension of S are non-decreasing under coarse embeddings. These results are new even for quasi-isometric embeddings and give obstructions which in many cases improve those previously obtained by Buyalo–Schroeder.

The Schwarz-Milnor lemma for braids and area-preserving diffeomorphisms

We prove a number of new results on the large-scale geometry of the \(L^p\)-metrics on the group of area-preserving diffeomorphisms of each orientable surface. Our proofs use in a key way the Fulton-MacPherson type compactification of the configuration space of n points on the surface due to Axelrod-Singer and Kontsevich. This allows us to apply the Schwarz-Milnor lemma to configuration spaces, a natural approach which we carry out successfully for the first time. As sample results, we prove that all right-angled Artin groups admit quasi-isometric embeddings into the group of area-preserving diffeomorphisms endowed with the \(L^p\)-metric, and that all Gambaudo-Ghys quasi-morphisms on this metric group coming from the braid group on n strands are Lipschitz. This was conjectured to hold, yet proven only for small values of n and g, where g is the genus of the surface.

Stable subgroups of the genus 2 handlebody group

We show that a finitely generated subgroup of the genus two handlebody group is stable if and only if the orbit map to the disk graph is a quasi-isometric embedding. To this end, we prove that the genus two handlebody group is a hierarchically hyperbolic group, and that the maximal hyperbolic space in the hierarchy is quasi-isometric to the disk graph of a genus two handlebody by appealing to a construction of Hamenst\"adt-Hensel. We then utilize the characterization of stable subgroups of hierarchically hyperbolic groups provided by Abbott-Behrstock-Berlyne-Durham-Russell. We also present several applications of the main theorems, and show that the higher genus analogues of the genus two results do not hold.

QUASI-ISOMETRIC EMBEDDINGS INAPPROXIMABLE BY ANOSOV REPRESENTATIONS

AbstractWe construct examples of quasi-isometric embeddings of word hyperbolic groups into $\mathsf {SL}(d,\mathbb {R})$ for $d \geq 4$ which are not limits of Anosov representations into $\mathsf {SL}(d,\mathbb {R})$ . As a consequence, we conclude that an analogue of the density theorem for $\mathsf {PSL}(2,\mathbb {C})$ does not hold for $\mathsf {SL}(d,\mathbb {R})$ when $d \geq 4$ .

Quasi-isometric embedding of Kerr poloidal submanifolds

We propose two approaches to obtain an isometric embedding of the poloidal Kerr submanifold. The first one relies on the convex integration process using the corrugation from a primitive embedding. This allows us to obtain one parameter family of embeddings reaching the limits of an isometric embedding. The second one consists in consecutive numerical resolutions of the Gauss–Codazzi–Mainardi and frame equations. This method requires geometric assumptions near the equatorial axis of the poloidal submanifold to get initial and boundary conditions. The second approach allows to understand some physical properties in the vicinity of a Kerr black hole, in particular the fast increasing ergoregion extent with angular momentum.

Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by [Formula: see text]. The relevant persistence modules come from filtered symplectic homology and are stable with respect to [Formula: see text]. Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped with Riemannian metrics. We consider some questions about large-scale geometry of this space and in particular we give a construction of a quasi-isometric embedding of [Formula: see text] into this space for all [Formula: see text]. On the other hand, in the case of domains in [Formula: see text], we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.

Affine diffeomorphism groups are undistorted

The affine diffeomorphism group $\mathrm{Aff}(S,q)$ of a half-translation surface $(S,q)$ comprise the self-diffeomorphisms with constant differential away from the singularities. This group coincides with the stabiliser of the associated Teichm\"uller disc under the action of the mapping class group on Teichm\"uller space. We prove that any finitely generated subgroup of $\mathrm{Aff}(S,q)$ is undistorted in the mapping class group. We also show that the systole map restricted to the associated electrified Nielsen core in the Teichm\"uller disc is a quasi-isometric embedding into the curve complex.

Quasi-isometric embeddings of symmetric spaces and lattices: reducible case

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semi-simple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semi-simple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.

Symplectic Banach–Mazur distances between subsets of ℂn

Following proposals of Ostrover and Polterovich, we introduce and study “coarse” and “fine” versions of a symplectic Banach–Mazur distance on certain open subsets of [Formula: see text] and other open Liouville domains. The coarse version declares two such domains to be close to each other if each domain admits a Liouville embedding into a slight dilate of the other; the fine version, which is similar to the distance on subsets of cotangent bundles of surfaces recently studied by Stojisavljević and Zhang, imposes an additional requirement on the images of these embeddings that is motivated by the definition of the classical Banach–Mazur distance on convex bodies. Our first main result is that the coarse and fine distances are quite different from each other, in that there are sequences that converge coarsely to an ellipsoid but diverge to infinity with respect to the fine distance. Our other main result is that, with respect to the fine distance, the space of star-shaped domains in [Formula: see text] admits quasi-isometric embeddings of [Formula: see text] for every finite dimension D. Our constructions are obtained from a general method of constructing [Formula: see text]-dimensional Liouville domains whose boundaries have Reeb dynamics determined by certain autonomous Hamiltonian flows on a given [Formula: see text]-dimensional Liouville domain. The bounds underlying our main results are proven using filtered equivariant symplectic homology via methods from [J. Gutt and M. Usher, Symplectically knotted codimension-zero embeddings between domains in [Formula: see text], Duke Math. J. 168 (2019) 2299–2363].

Quasi-isometric embeddings of non-uniform lattices

Let G and G' be simple Lie groups of equal real rank and real rank at least 2 . Let \Gamma <G and \Lambda < G' be non-uniform lattices. We prove a theorem that often implies that any quasi-isometric embedding of \Gamma into \Lambda is at bounded distance from a homomorphism. For example, any quasi-isometric embedding of SL(n,\mathbb Z) into SL(n, \mathbb Z[i]) is at bounded distance from a homomorphism. We also include a discussion of some cases when this result is not true for what turn out to be purely algebraic reasons.

Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the RCD(0,N), with N > 1. In fact, we can prove that a subRiemannian manifold whose generic degree of nonholonomy is not smaller than 2 can not be biLipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the CD(K,N) with N > 1. We also prove that the subRiemannian manifold is infinitesimally Hilbert space.

Quasi-hyperbolic planes in relatively hyperbolic groups

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric properties of the boundaries of relatively hyperbolic groups, such as linear connectedness. In particular, we prove a new existence result for quasi-arcs that avoid obstacles.

Symplectic structure perturbations and continuity of symplectic invariants

This paper studies how symplectic invariants created from Hamiltonian Floer theory change under the perturbations of symplectic structures, not necessarily in the same cohomology class. These symplectic invariants include spectral invariants, boundary depth, and (partial) symplectic quasi-states. This paper can split into two parts. In the first part, we prove some energy estimations that control the shifts of symplectic action functionals. These directly imply positive conclusions on the continuity of spectral invariants and boundary depth, in some important cases including any symplectic surface $\Sigma_{g \geq1}$ and any closed symplectic manifold $M$ with $\dim_{\mathcal K} H^2(M; \mathcal K) = 1$. This follows by applications on some rigidity of the subsets of a symplectic manifold in terms of heaviness and superheaviness, as well as on the continuity property of some symplectic capacities. In the second part, we generalize the construction in the first part to any closed symplectic manifold. In particular, to deal with the change of Novikov rings from symplectic structure perturbations, we construct a family of variant Floer chain complexes over a common Novikov-type ring. In this set-up, we define a new family of spectral invariants called $t$-spectral invariants, and prove that they are upper semicontinuous under the symplectic structure perturbations. This implies a quasi-isometric embedding from $(\mathbb R^{\infty}, |-|_{\infty})$ to $({\widetilde {\rm Ham}}(M, \omega), |-|_{\rm Hofer})$ under some dynamical assumption, imitating a result from Usher.

The geometry of the knot concordance space

Most of the 50-year history of the study of the set of knot concordance classes, C, has focused on its structure as an abelian group. Here we take a different approach, namely we study C as a metric space admitting many natural geometric operators, especially satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi-n-flats for every n, implying that C admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasi-homomorphism P: C --- C with respect to the metric given by the slice genus. We show that winding number one satellite operators induce quasi-isometries. We prove that strong winding number one satellite operators induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasi-contractions. These results contribute to the conjecture that C is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators.