arXiv research

We study functions whose truncations are convex or quasiconvex.

We provide a dual representation of quasiconvex maps between two lattices of random variables in terms of conditional expectations. This generalizes the dual representation of quasiconvex real valued functions and the dual representation of conditional convex maps.

Study dual representations for quasiconvex systemic risk measures.

In the conditional setting we provide a complete duality between quasiconvex risk measures defined on $L^{0}$ modules of the $L^{p}$ type and the appropriate class of dual functions. This is based on a general result which extends the usual Penot-Volle representation for quasiconvex real valued maps.

Paper shows non-convexity in solutions to Hessian equations.

Paper discusses natural quasiconvexity and its relation to decomposable sums in risk measures.

Exact sequence results show infinite index quasiconvex subgroups in certain groups.

New principles for collapsing law-invariant functionals to means, extending beyond convexity.

New method constructs non-quasiconvex subgroups in hyperbolic groups.

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1 \ast_{Q_1 \cap Q_2} Q_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, an…

Since the quasiconvex risk measures is a bigger class than the well known convex risk measures, the study of quasiconvex risk measures makes sense especially in the financial markets with volatility. In this paper, we will study the quasiconvex risk measures defined on a special space $L^{p(\cdot)}$ where the variable …

We study different notions of quasiconvexity for a subgroup $H$ of a relatively hyperbolic group $G.$ The first result establishes equivalent conditions for $H$ to be relatively quasiconvex. As a corollary we obtain that the relative quasiconvexity is equivalent to the dynamical quasiconvexity. This answers to a questi…

Deep neural networks (DNNs) have demonstrated success for many supervised learning tasks, ranging from voice recognition, object detection, to image classification. However, their increasing complexity might yield poor generalization error that make them hard to be deployed on edge devices. Quantization is an effective…

Proves existence of certain subgroups in hyperbolic groups.

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general…

No hyperbolic group can have an infinite chain of free subgroups of fixed rank.

We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain conditions. We then provide a criterion for the relative quasiconvexity of a subgroup H de…

In this paper, we state two combination theorems for relatively quasiconvex subgroups in a relatively hyperbolic group. Applications are given to the separability of double cosets of certain relatively quasiconvex subgroups and the existence of closed surface subgroups in relatively hyperbolic groups.

In this paper, we study strongly quasiconvex subgroups in a finitely generated $3$--manifold group $π_1(M)$. We prove that if $M$ is a compact, orientable $3$--manifold that does not have a summand supporting the Sol geometry in its sphere-disc decomposition then a finitely generated subgroup $H \le π_1(M)$ has finite …

Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3--manifold groups. We investigate strongly quasiconvex subsets in this class and characterize them in terms of their contracting pro…

A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to whic…

A Kleinian group $Γ< \mathrm{Isom}(\mathbb H^3)$ is called convex cocompact if any orbit of $Γ$ in $\mathbb H^3$ is quasiconvex or, equivalently, $Γ$ acts cocompactly on the convex hull of its limit set in $\partial \mathbb H^3$. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which…

Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric sp…

We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-end…

Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually co…

Classifies hyperbolic groups with surface-like boundaries.

Extends portfolio optimization with two quasiconvex risk measures.

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined stability and proved stability is equivalent to convex cocompactness in mapping class …

The paper provides conditions for amalgamation of certain subgroups and preserves convexity properties.

We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with a recent result of Cooper and Futer, we provide a new proof of the virtual fibering of non-compact finite-volume hy…

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p$ punctures. We show that nested train track sequences constitute $O((g,p)^{2})$-quasiconvex subsets of the curve graph, effectivizing a theorem of Masur and Minsky. As a consequence, the genus $g$ disk set is $O(g^{2})$-quasiconvex. We also show that splitti…

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the f…

Ridge regression CV loss may have multiple local optima.

The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and run…

We give a proof that there exists a universal constant $K$ such that the disc graph associated to a surface $S$ forming a boundary component of a compact, orientable 3-manifold $M$ is $K$-quasiconvex in the curve graph of $S$. Our proof does not require the use of train tracks.

We introduce the notions of geometric height and graded (geometric) relative hyperbolicity in this paper. We use these to characterize quasiconvexity in hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and convex cocompactness in mapping class groups and $Out(F_n)$. Corrigendum: there is an u…

We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with…

For any finitely generated, non-elementary, torsion-free group $G$ that is hyperbolic relative to $\mathbb P$, we show that there exists a group $G^*$ containing $G$ such that $G^*$ is hyperbolic relative to $\mathbb P$ and $G$ is not relatively quasiconvex in $G^*$. This generalizes a result of I. Kapovich for hyperbo…

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…

We propose a new PAC-Bayesian bound and a way of constructing a hypothesis space, so that the bound is convex in the posterior distribution and also convex in a trade-off parameter between empirical performance of the posterior distribution and its complexity. The complexity is measured by the Kullback-Leibler divergen…

We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompac…

A random group contains many quasiconvex surface subgroups.

This paper refines bounds on random walk speed in Teichmüller space.

Improved resource allocation method reduces procurement costs.

We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence $1\to H\to G \to Q \to 1$ of hyperbolic groups. These laminations arise in different contexts: existence of Cannon-Thurston maps; closed geodesics exiting ends of manifolds; dual to …

We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) $X$ is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\JJ$ an equivariant collection of ma…

The paper consists of two parts. In the first one we show that a relatively hyperbolic group $G$ splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups. As a corollary we obtain that every group which admits 3-discontinuous and 2-coc…

Let $1 \to H \to G \to Q \to 1$ be an exact sequence where $H= π_1(S)$ is the fundamental group of a closed surface $S$ of genus greater than one, $G$ is hyperbolic and $Q$ is finitely generated free. The aim of this paper is to provide sufficient conditions to prove that $G$ is cubulable and construct examples satis…