arXiv research

Sharp growth tightness proven for group quotients.

We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting eleme…

Word metrics from large balls in injective spaces are exponentially generic and growth tight.

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions us…

Among (isotopy classes of) automorphisms of handlebodies those called irreducible (or generic) are the most interesting, analogues of pseudo-Anosov automorphisms of surfaces. We consider the problem of isotoping an irreducible automorphism so that it is most efficient (has minimal growth rate) in its isotopy class. We …

In a financial market model, we consider variations of the problem of minimizing the expected time to upcross a certain wealth level. For exponential Levy markets, we show the asymptotic optimality of the growth-optimal portfolio for the above problem and obtain tight bounds for the value function for any wealth level.…

This paper presents a study of the asymptotic geometry of groups with contracting elements, with emphasis on a subclass of statistically convex-cocompact (SCC) actions. The class of SCC actions includes relatively hyperbolic groups, CAT(0) groups with rank-1 elements and mapping class groups, among others. We exploit a…

Enhances the Bishop-Gromov theorem for curved spaces, especially at late times.

Private optimization faster on interpolation problems with quadratic growth.

A quasi-geodesic is Morse if and only if it is strongly contracting in injective spaces.

New bounds for neural networks on curved manifolds improve generalization.

Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight t…

Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichmüller spaces, and hyperbolic 3-manifolds. In the same paper, they showed that there are at least one and at most finitely many tight geodesics betw…

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…

The study identifies conditions for algorithms to have tight generalization bounds.

Surgery on knots always admits a tight contact structure.

Tight maps was introduced along tight homomorphisms by Burger, Iozzi and Wienhard with aims towards maximal representations. In this paper we classify tight maps into classical Hermitian symmetric spaces and give a partial result for the exceptional spaces.

3-manifold triangulations are Golod and tight, proven through a topological characterization.

Simply connected spaces of tight frames identified.

The study finds many tight contact structures on hyperbolic 3-spheres.

Study tight contact structures on specific 3-manifolds.

Classifies tight contact structures on surgeries of the Whitehead link.

In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces. We also prove that any lens space admits only finitely many tight contact structures.

In \cite{confol} Y. Eliashberg and W. Thurston gave a definition of tight confoliations. We give an example of a tight confoliation $ξ$ on $T^3$ violating the Thurston-Bennequin inequalities. This answers a question from \cite{confol} negatively. Although the tightness of a confoliation does not imply the Thurston-Benn…

Classifies real tight contact structures on lens spaces and solid tori.

The study finds algebraically overtwisted tight 3-manifolds via contact surgeries.

New proof of Giroux Correspondence for tight contact 3-manifolds.

We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yie…

We determine all tight Lagrangian surfaces in $S^2 \times S^2$. In particular, globally tight Lagrangian surfaces in $S^2 \times S^2$ are nothing but real forms.

Study non-fibered links' relation to tight contact structures.

New 3D shapes found without certain special flows.

We prove gluing theorems for tight contact structures. In particular, we rederive (as special cases) gluing theorems due to Colin and Makar-Limanov, and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus 4 h…

In three-dimensional computational topology, the theory of normal surfaces is a tool of great theoretical and practical significance. Although this theory typically leads to exponential time algorithms, very little is known about how these algorithms perform in "typical" scenarios, or how far the best known theoretical…

We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.

Classifies tight contact structures on specific Seifert fibered manifolds.

This paper describes a characterization of tightness of closed contact 3-manifolds in terms of supporting open book decompositions. The main result is that tightness of a closed contact 3-manifold is preserved under Legendrian surgery.

The study finds tight contact structures without fillings in high dimensions.

New evidence supports the Euler class one conjecture for tight contact structures.

Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the …

We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched cover…

Discusses the tight versus overtwisted dichotomy in 3D contact geometry.

The paper extends conformal risk control to be valid with high probability over a growing calibration dataset.

The paper conjectures a 4D characterization of tight contact structures and proves it for certain cases.

Whether every hyperbolic 3-manifold admits a tight contact structure or not is an open question. Many hyperbolic 3-manifolds contain taut foliations and taut foliations can be perturbed to tight contact structures. The first examples of hyperbolic 3-manifolds without taut foliations were constructed by Roberts, Sharesh…

Contractibility of tight contact structures on $S^3$ proven.

Study on random hyperbolic surfaces with many cusps, focusing on tight geodesics.

Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying manifold. Tight triangulations are conjectured to be strongly minimal, and proven to be…

The paper proves tight fibered knots are minimal in a specific knot order.