arXiv research

The paper studies geometric properties of quasi-trees and tree approximations.

The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the $h-$principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working on…

Proves a generalized Whitehead cut vertex lemma for tree groups.

We present a practical algorithm which, given a non-archimedean local field $K$ and any two elements $A,B\in {\rm SL_2}(K)$, determines after finitely many steps whether or not the subgroup $\langle A, B \rangle\le {\rm SL_2}(K)$ is discrete and free of rank two. This makes use of the Ping Pong Lemma applied to the act…

Proves cobordism of CP^2 bundles generating oriented ring.

Two new proofs of Gromov's non-squeezing theorem using curve reparametrization and gradient bounds.

Method learns graphons from graphs via Gromov-Wasserstein barycenters.

There is a gap in the proof of the main theorem in the article [ShCh13a] on optimal bounds for the Morse lemma in Gromov-hyperbolic spaces. We correct this gap, showing that the main theorem of [ShCh13a] is correct. We also describe a computer certification of this result.

We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assum…

A labeled oriented tree is called injective if each generator occurs at most once as an edge label. We show that injective labeled oriented trees are aspherical. The proof relies on a new relative asphericity test based on a lemma of Stallings.

Differentiable optimization bridges arbitrary metrics to tree metrics.

We prove that for a relatively hyperbolic group G there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of G. Under natural assumptions, the same conclusion holds for the critical exponent of a cusp-uniform action of G on a hyperbolic metric space. As a c…

Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.

Study finds a linear lower bound on conformal dimension for random hyperbolic groups.

We define analogues of the graphs of free splittings, of cyclic splittings, and of maximally-cyclic splittings of $F_N$ for free products of groups, and show their hyperbolicity. Given a countable group $G$ which splits as $G=G_1\ast\dots\ast G_k\ast F$, where $F$ denotes a finitely generated free group, we identify th…

Geodesic currents on hyperbolic surfaces have dual spaces that are metric trees.

Geometric group theory explores groups through their geometric properties.

Verifying a conjecture of Gromov we establish a generalized Margulis Lemma for manifolds with lower Ricci curvature bound. Among the various applications are finiteness results for fundamental groups of compact $n$-manifolds with upper diameter and lower Ricci curvature bound modulo nilpotent normal subgroups.

Globalisation theorem for Lorentzian spaces with curvature bounds.

Solves weighted bi-colored plane tree enumeration and applies to geometric problems.

Transforms uniform learners to work under arbitrary distributions efficiently.

Researchers create a new compactification of character varieties using geometric and algebraic methods.

We show that the Gromov boundary of the free factor graph for the free group Fn with n>2 generators is the space of equivalence classes of minimal very small indecomposable projective Fn-trees without point stabilizer containing a free factor equipped with a quotient topology. Here two such trees are equivalent if the …

We prove that every visual Gromov hyperbolic space X whose boundary at infinity has the finite capacity dimension n admits a quasi-isometric embedding into (n+1)-fold product of metric trees.

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.

Explains visual metrics on hyperbolic space boundaries.

We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary dimension, obtained by the procedure of strict hyperbolization. We also recognize these …

Study shows saddle connection graph's geometry and quasi-isometry properties.

It was pointed out to us that the proof of a crucial lemma (Lemma 5.3) in the paper is incorrect. Thus the approximation theorem (Theorem 0.1) for L^2 torsion of an amenable covering of a finite simplicial complex remains unproved. However, results and proofs of the first four sections (in particular, the approximation…

When 1 -> H -> G -> Q -> 1 is a short exact sequence of three infinite, word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates t…

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show tha…

Proves theorem for Riemannian manifolds, extending previous work.

Given a closed Riemannian manifold of dimension $n$ and a Morse-Smale function, there are finitely many $n$-part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of $n$-part broken trajectories is always at least the hyperbolic volume. The proof…

We describe, under some additional technical assumptions, the Gromov boundary of the free product of several $G_i$'s amalgamated wrt. $H$, where $G_i$ are hyperbolic groups with boundary homeomorphic to a densely punctured $n$-sphere, and $H$ is their common subgroup corresponding to a peripheral sphere in each of the …

Defines super stable maps and proves quotient superorbifolds for genus zero.

In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we…

This paper surveys various methods for dimensionality reduction and nearest neighbor search.

The paper proves stability of the positive mass theorem using intrinsic flat convergence.

We study actions of finitely generated groups on $\bbR$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and a…

In [9] Kaimanovich introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown in [13] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in det…

The Random Projection Tree structures proposed in [Freund-Dasgupta STOC08] are space partitioning data structures that automatically adapt to various notions of intrinsic dimensionality of data. We prove new results for both the RPTreeMax and the RPTreeMean data structures. Our result for RPTreeMax gives a near-optimal…

A real valued function $\varphi$ of one variable is called a metric transform if for every metric space $(X,d)$ the composition $d_\varphi = \varphi\circ d$ is also a metric on $X$. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms $\varphi$ such that the trans…

The proof of Brouwer's fixed-point theorem based on Sperner's lemma is often presented as an elementary combinatorial alternative to advanced proofs based on algebraic topology. The goal of this note is to show that: (i) the combinatorial proof of Sperner's Lemma can be considered as a cochain-level version, written in…

Algorithm learns CNF formulas from random solutions under specific conditions.

In 1969 M. Gromov in his PhD thesis greatly generalized Smale-Hirsch-Phillips immersion-submersion theory by proving what is now called the h-principle for invariant open differential relations over open manifolds. Gromov extracted the original geometric idea of Smale and put it to work in the maximal possible generali…

Dense neural networks can't approximate all functions.

For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bou…

Paper studies ensemble probabilistic regression trees for smooth approximations.