arXiv research

In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion of analytic finiteness in dimension two. Then we extend the argument in the paper…

It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each …

In this note, we show that the asymptotic dimension of any building is finite and equal to the asymptotic dimension of an apartment in that building.

We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, …

After results by the author (1980, 1981), and by Vinberg (1981), finiteness of the number of maximal arithmetic reflection groups in Lobachevsky spaces was not known in dimensions $2\le n\le 9$ only. Recently (2005), the finiteness was proved in dimension 2 by Long, Maclachlan and Reid, and in dimension 3 by Agol. Here…

It is proven that if a finitely presented group is one ended it has asymptotic dimension bigger than one. It follows that finitely presented groups with asdim 1 are virtually free. A counterexample is given for the finitely generated case.

The paper proves scalar curvature decay for uniformly contractible manifolds with finite asymptotic dimension.

New findings on metric spaces with finite Nagata dimension.

Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.

The study establishes equivalence of conditions on metric manifolds with finite volume.

Study shows Roller compactification's median graph has limited asymptotic dimension.

This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Suc…

The aim of this paper is to investigate properties preserved and co-preserved by coarsely $n$-to-1 functions, in particular by the quotient maps $X\to X/\sim$ induced by a finite group $G$ acting by isometries on a metric space $X$. The coarse properties we are mainly interested in are related to asymptotic dimension a…

Compact manifolds with boundary have finite type mapping class groups.

New bounds found for vertices of hyperbolic polyhedra in dimensions 5 to 12.

Learnable multiclass hypothesis classes don't always have a sample compression scheme of fixed size.

By recognizing them as fundamental groups of developable complexes of groups we prove that mapping class groups of compact orientable surfaces have finite asymptotic dimension.

We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An amalgamated product of asymptotically finite dimensional groups has finite asympt…

We compute many dimensions of spaces of finite type invariants of virtual knots (of several kinds) and the dimensions of the corresponding spaces of "weight systems", finding everything to be in agreement with the conjecture that "every weight system integrates".

We introduce the notion of regular finite decomposition complexity of a metric family. This generalizes Gromov's finite asymptotic dimension and is motivated by the concept of finite decomposition complexity (FDC) due to Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC and has all the perma…

New hyperbolic manifolds discovered that fiber algebraically up to dimension 8.

A famous open problem asks whether the asymptotic dimension of a CAT(0) group is necessarily finite. For hyperbolic groups, it is known that asymptotic dimension of the group is bounded above by the dimension of the boundary plus one, which is known to be finite. For CAT(0) groups, the latter quantity is also known to …

We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.

The virtually cyclic dimension of Out(F_N) is finite and related properties are established.

We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their…

In this paper, we define the curvature dimension inequalities CD(m, K) on finite directed graphs modifying the case of undirected graphs. As a main result, we evaluate m and K on finite directed graphs.

In this work we study two problems about Assouad-Nagata dimension: 1) Is there a metric space of non zero Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes) 2) Suppose $G$ is a locally finite group with a proper left invariant metric $d_G$. If $\dim_{A…

Our main result asserts that for any given numbers C and D the class of simply connected closed smooth manifolds of dimension m<7 which admit a Riemannian metric with sectional curvature bounded in absolute value by C and diameter uniformly bounded from above by D contains only finitely many diffeomorphism types. Thus …

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension $\as_{\Z} X$ of metric spaces. We show that it agrees with the asymptotic dimension $\as X$ when the later is finite. Then we use this fact to construct an example of a metric space $X$ of boun…

We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for …

Paper studies Hausdorff dimensions of specific limit sets for groups on curved spaces.

Entropy rigidity proven for 3D and higher convex projective manifolds.

We prove that for geometrically finite groups cohomological dimension of the direct product of a group with itself equals 2 times the cohomological dimension dimension of the group.

Estimates intrinsic dimension of data for GANs.

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Lück, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invar…

Extends Dirac structures to infinite dimensions, focusing on convenient Lie algebroids and manifolds.

Study computable multiclass learning within PAC framework.

Let L be a countable and locally finite CW complex. Suppose that the class of all metrizable compacta of extension dimension not greater than L contains a universal element which is an absolute extensor in dimension L. Our main result shows that L is quasi-finite.

It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus g > 1 is bounded by the linear polynomial 12(g-1), and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group…

This work generalizes a geometric Laplacian determinant description to higher dimensions.

Proves homological inequality for cycles in Hadamard spaces of asymptotic rank 2.

Finite index solutions to Bernoulli problem are always axially symmetric.

Given a metric space $X$ of finite asymptotic dimension, we consider a quasi-isometric invariant of the space called dimension function. The space is said to have asymptotic Assouad-Nagata dimension less or equal $n$ if there is a linear dimension function in this dimension. We prove that if $X$ is a tree-graded space …

We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in $[0,\infty]$ which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits all its useful properties such as additivity, cofinality and continuity. This all…

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that…

We find an upper bound for the asymptotic dimension of a hyperbolic metric space with a set of geodesics satisfying a certain boundedness condition studied by Bowditch. The primary example is a collection of tight geodesics on the curve graph of a compact orientable surface. We use this to conclude that a curve graph h…

Investigates the impact of finite VC dimension on neural network approximation and learning.

In this paper we extend our previous work on singularities of Monge-Ampère foliations to the case of pseudoconvex finite type domains. We are able to answer the questin of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge-Ampère equation completely in dimension 2 . We are also able to general…