Finite height in relatively hyperbolic groups ensures relative quasiconvexity.
problem Conditions for relative quasiconvexity in relatively hyperbolic groups.
method Graph of groups structure and quasi-isometric embeddings.
result Finite height in vertex groups ensures relative quasiconvexity.
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…
For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups Q1 and Q2 is relatively quasiconvex and isomorphic to Q1∗Q1∩Q2Q2. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, an…
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…
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.
New condition controls quasiconvex subgroups in hyperbolic 3-manifolds.
problem Controlling behavior of quasiconvex subgroups in hyperbolic 3-manifolds.
method New condition on relatively hyperbolic Dehn filling.
result Provides new proof of virtual fibering of hyperbolic 3-manifolds.
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…
The paper introduces new geometric concepts to study quasiconvexity in hyperbolic groups.
problem Characterizing quasiconvexity in hyperbolic and relatively hyperbolic groups.
method Introducing geometric height and graded relative hyperbolicity, using these to characterize quasiconvexity.
result Characterization of quasiconvexity in hyperbolic and relatively hyperbolic groups.
New hierarchy for a special group type.
problem Classifying relatively hyperbolic virtually special groups.
method Constructing a new virtual quasiconvex hierarchy.
result Generalized Malnormal Special Quotient Theorem.
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…
For any finitely generated, non-elementary, torsion-free group G that is hyperbolic relative to P, we show that there exists a group G∗ containing G such that G∗ is hyperbolic relative to P and G is not relatively quasiconvex in G∗. This generalizes a result of I. Kapovich for hyperbo…
Exact sequence results show infinite index quasiconvex subgroups in certain groups.
problem Understanding quasiconvex subgroups in specific group structures.
method Analyzing exact sequences of hyperbolic groups and their subgroups.
result Infinite index quasiconvex subgroups in G1 are quasiconvex in G under mild conditions. 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 study proves stabilizing of ascending chains in specific groups.
problem Stabilization of ascending chains in bounded rank subgroups of 3-manifold groups.
method Reduction to hyperbolic 3-manifolds and use of geometrization.
result Ascending chains in toral relatively hyperbolic groups stabilize.
Groups grow faster than their subgroups, proven using special tools.
problem Growth rates of subgroups within hyperbolic groups.
method Automatic structures, Perron-Frobenius theory, growth tightness, rotating families.
result Non-elementary hyperbolic groups grow exponentially faster than their quasiconvex subgroups.
Characterizes strongly quasiconvex subsets in hierarchically hyperbolic spaces.
problem Understanding the structure of strongly quasiconvex subsets in HHSs.
method Characterization through contracting properties, relative divergence, and hierarchical structure.
result Proves characterization of hyperbolically embedded subgroups in hierarchically hyperbolic groups.
Proves stability pulls back under proper actions, with applications to mapping class groups and free groups.
problem Stability of subgroups in mapping class groups and free groups.
method Proves stability pulls back under proper actions on metric spaces.
result Stability of convex cocompact subgroups in mapping class groups and free groups.
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…
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…
The paper develops algorithms to detect stability and Morse properties in various groups.
problem Detecting stability and Morse properties in finitely generated groups.
method Various detection and decidability algorithms for stability and Morse properties in specific types of groups.
result The algorithms provide a way to determine if a subgroup is stable or Morse in specific group types.
Study quasiconvex risk measures in volatile financial markets.
problem Financial risk measurement in markets with variable volatility.
method Defined quasiconvex risk measures on Lp(⋅) space with p(⋅) as a random variable. result Deduced dual representation for the defined quasiconvex risk measures.
Paper discusses natural quasiconvexity and its relation to decomposable sums in risk measures.
problem Understanding natural quasiconvexity and its implications in risk measures.
method Relates natural quasiconvexity to decomposable sums, proposes a general treatment of convexity index, and proves equivalence for certain spaces.
result Natural quasiconvexity and convexity are equivalent for conditional risk measures on Lp spaces under mild conditions. New method constructs non-quasiconvex subgroups in hyperbolic groups.
problem Creating non-quasiconvex subgroups in hyperbolic groups.
method Using Stallings-like techniques on right-angled Coxeter groups (RACGs).
result Explicit examples of non-quasiconvex subgroups constructed.
Proves existence of certain subgroups in hyperbolic groups.
problem Existence of weakly malnormal quasiconvex subgroups in hyperbolic groups.
method Proof of existence in nonelementary hyperbolic groups.
result Existence of weakly malnormal, virtually free, quasiconvex subgroups in hyperbolic groups.
Study of quasiconvex subgroups in 3-manifold groups.
problem Characterize quasiconvex subgroups in 3-manifold groups.
method Analyzes strongly quasiconvex subgroups in finitely generated 3-manifold groups.
result Characterizes quasiconvex subgroups in graph manifold groups and 3-manifold groups.
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.
Disc graphs are uniformly quasiconvex in curve graphs of surfaces.
problem Characterizing quasiconvexity in curve graphs of surfaces.
method Proof using a universal constant K without train tracks.
result Disc graphs are K-quasiconvex in curve graphs.
No hyperbolic group can have an infinite chain of free subgroups of fixed rank.
problem Infinite ascending chains of free subgroups in hyperbolic groups.
method Proof by contradiction and properties of hyperbolic groups.
result Hyperbolic groups do not contain strictly ascending chains of free quasiconvex subgroups of constant rank.
New insights into the structure of blown-up corona of hyperbolic groups.
problem Understanding the structure of blown-up corona of relatively hyperbolic groups.
method Equivariant compactification and cohomological dimension analysis.
result Blown-up corona of a relatively hyperbolic group is contractible and homeomorphic to the Gromov boundary.
New techniques reveal subgroup properties in Coxeter groups.
problem Characterizing and understanding subgroups of right-angled Coxeter groups.
method Using cube complexes and Stallings-like techniques to study subgroups.
result Reflection and one-ended subgroups are quasiconvex.
In the conditional setting we provide a complete duality between quasiconvex risk measures defined on L0 modules of the Lp 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.
Study dual representations for quasiconvex systemic risk measures.
problem Finding dual representations for quasiconvex systemic risk measures.
method Abstract infinite-dimensional setting, explicit formula for penalty function, nonstandard minimax inequality.
result Explicit formula for the penalty function of quasiconvex compositions.
The study characterizes subgroup stability in finitely generated groups using the Morse boundary.
problem Stability of subgroups in finitely generated groups.
method Using the Morse boundary, an equivalent characterization of subgroup stability is developed.
result Subgroup stability coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups.
We study functions whose truncations are convex or quasiconvex.
problem Understanding functions with specific truncation properties.
method Analyzing C2-smooth functions with positive definite Hessians. result Injectivity of restricted gradient in positive definite region.
The study proves quasiconvexity for subgroups in hyperbolic group sequences.
problem Proving quasiconvexity for subgroups in hyperbolic group sequences.
method Using relationships between different types of algebraic laminations.
result Proves quasiconvexity for finitely generated infinite index subgroups of H. Classifies hyperbolic groups with surface-like boundaries.
problem Classifying hyperbolic groups with specific surface-like boundaries.
method Analyzing quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.
result Identifies hyperbolic groups with surface-like boundaries.
Proves small cancellation free products have geometric actions on CAT(0) cube complexes.
problem Proving small cancellation free products have geometric actions on CAT(0) cube complexes.
method Using a blown-up complex of groups and a boundary separation criterion, proving wall stabilizers form a rich family of subgroups.
result Proves $C'( rac16)$--small cancellation free products of residually finite groups are residually finite.
The paper provides conditions for amalgamation of certain subgroups and preserves convexity properties.
problem Conditions for amalgamation of subgroups in hierarchically hyperbolic groups.
method Study of amalgamation conditions and preservation of convexity properties.
result Conditions under which amalgamation preserves hierarchical quasiconvexity and strong quasiconvexity.
Paper shows non-convexity in solutions to Hessian equations.
problem Non-convexity of solutions to k-Hessian equations in exterior domains. method Examples and new proof for quasiconvexity of harmonic functions.
result Solutions to k-Hessian equations are not quasiconvex in exterior domains. Let Sg,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(g2)-quasiconvex. We also show that splitti…
Characterizes geometric actions on graphs with flexible stabilizers.
problem Understanding geometric actions on flexible stabilizers.
method Defining generalized fine actions and proving relative quasi-convexity criteria.
result Characterizes Bowditch boundary points in relatively geometric actions.
Stable and Morse subgroups coincide in mapping class groups.
problem Understanding subgroup properties in mapping class groups.
method Analyzing stability and Morse properties in mapping class groups.
result Stability and Morse properties coincide for subgroups of infinite index in mapping class groups.
We give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to pro…
Ridge regression CV loss may have multiple local optima.
problem Can we globally optimize cross-validation loss in ridge regression?
method Analyzing quasiconvexity of CV loss in ridge regression.
result CV loss may fail to be quasiconvex and have multiple local optima.
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…
The local-to-global property is proven for Morse quasi-geodesics in various groups.
problem Proving local-to-global properties for Morse quasi-geodesics in different groups.
method Developing a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces.
result Generalization of combination theorems for stable subgroups of various groups.
This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of the simplist extrinsic notions. Results and examples concerning distorted subgrou…
The paper proves conditions for cubulating surface-by-free groups.
problem Proving conditions for cubulating surface-by-free groups.
method Combination theorem for virtually special hyperbolic groups, tracks theory, quasiconvex hierarchy theorem, distance estimates, and model geometry.
result Main result: sufficient conditions for cubulability of G.