arXiv research

Groups of contact transformations are uniformly simple if supporting open books are flexible.

New framework shows $C^*$-simplicity for groups without certain subalgebras.

The study examines arithmetic orbifolds and their length spectra, proving uniform discreteness and linear dependence of geodesic lengths.

Uniformly perfect Morse boundaries characterize geometric properties of groups.

Study uniformly differentiable graphs in Carnot groups, proving area formulas.

Groups with specific subgroup actions have proper actions on uniformly convex spaces.

Characterizes a specific homology group for certain graphs.

We develop a theory of `non-uniformly local' tent spaces on metric measure spaces. As our main result, we give a remarkably simple proof of the atomic decomposition.

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, the group of homeomorphisms of M …

We show that the mapping class group of an orientable finite type surface has uniformly exponential growth, as well as various closely related groups. This provides further evidence that mapping class groups may be linear.

The paper examines boundedness of diffeomorphism groups of manifold pairs, focusing on the circle case.

Study calorons and their constituent monopoles under collapse.

Groups can embed uniformly but not act properly on contractible manifolds.

Study mapping class groups of infinite type surfaces with noncompact boundaries.

New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.

It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an application, we give solutions to some group-theoretic problems.

Triangulates permutahedra for Coxeter groups, revealing braid group connections.

Uniform undistortion in cyclic subgroups of certain groups.

Let $Γ$ be a finitely generated group which is hyperbolic relative to a finite family $\{H_1,...,H_n\}$ of subgroups. We prove that $Γ$ is uniformly embeddable in a Hilbert space if and only if each subgroup $H_i$ is uniformly embeddable in a Hilbert space.

Uniform drift estimates found for random walks on graph products.

Aspherical manifolds with bounded curvature have non-trivial abelian subgroups in their fundamental groups.

Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.

Diffeomorphisms of surfaces have many unbounded quasi-morphisms.

Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie algebras and we classify uniform Lie algebras with five or fewer generators. We defi…

Study of cable links of uniformly thick knots, revealing new isotopy phenomena.

Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.

This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.

Study automorphic and Lattès-type maps on manifolds, finding restrictions on their groups.

Given a non-compact Riemannian manifold M and a submanifold N of codimension q, we will construct under certain assumptions on both M and N a wrong way map in uniformly finite homology. Using an equivariant version of the construction and applying it to universal covers, we will construct wrong way maps in homology of …

In this paper, we study the Hausdorff dimension of the Floyd and Bowditch boundaries of a relatively hyperbolic group, and show that for the Floyd metric and shortcut metrics respectively, they are are both equal to a constant times the growth rate of the group. In the proof, we study a special class of conical points …

We show, finitely generated rational $\mathsf{VIC}_{\mathbb Q}$-modules and $\mathsf{SI}_{\mathbb Q}$-modules are uniformly representation stable and all their submodules are finitely generated. We use this to prove two conjectures of Church and Farb, which state that the quotients of the lower central series of the To…

We show that for acylindrically hyperbolic groups $Γ$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $ρ$ of $Γ$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(Γ;ρ)$ is infinite dimensional. The result was known for the regular representations on $\ell^p(Γ)$ with …

Uniformly random permutations converge to regular representation on surface groups.

Let $Γ$ be a finite index subgroup of the mapping class group $MCG(Σ)$ of a closed orientable surface $Σ$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element $g\inΓ$ has positive stable commutator length. In addition, we show that in these situations th…

Compactness proven for manifolds with nonnegative Ricci curvature and uniformly convex boundary.

Based on C. Li and Y. Rubinstein's upper bisectional curvature bound estimate for the conic Kähler metric, we can construct a smoothing sequence for the conic metric with uniformly upper bisectional curvature bound. For the conic metric along a simple normal crossing divisor with triple or higher multiple points we may…

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…

Deep neural nets approximate random dynamical system trajectories uniformly in time.

We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk …

Let S be a compact surface, and M be the double of a handlebody. Given a homotopy class of maps from S to M inducing an isomorphism of fundamental groups, we describe a canonical uniformly lipschitz retraction of the sphere graph of M to the arc graph of S. We also show that this retraction is a uniformly bounded dista…

New insights into matrix factorization show strict saddles have bounded eigenvalues.

In this note, we provide a description of the structure of homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jorgensen-Thurston Theorem in hyperbolic geometry.

Uniform Jordan property proven for Lie groups and complex space transformations.

We analyze the obstruction to metrics of positive scalar curvature within a given bounded distortion class of metrics. This obstruction lives in a non-Hausdorff cohomology group Poincare dual to the uniformly finite homology studied by Block and Weinberger. One of the applications is a converse to their theorem on infi…

We prove injectivity and a support theorem for the X-ray transform on $2$-step nilpotent Lie groups with many totally geodesic $2$-dimensional flats. The result follows from a general reduction principle for manifolds with uniformly escaping geodesics.

We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we show that the algebraic entropy of discrete subgroups of a given Lie group is unif…

Study of quasimorphisms and bounded cohomology in braided Thompson groups.

We prove a finiteness result for the systolic area of groups, answering a question of M. Gromov. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of~2-complexes whose systolic area is uniformly bounded. Furthermore, we prove a uniform systolic inequality for all 2-complexe…