Study group actions in metric spaces, proving convergence of lens spaces.
problem Understanding convergence in metric measure spaces with group actions.
method Generalized box and observable distances, applied mass-transport theory.
result Sequence of lens spaces converging to infinite-dimensional complex projective space.
Study equidistribution for flows on geometrically finite convergence group actions.
problem Counting, mixing and equidistribution for flows on geometrically finite convergence group actions.
method Establishing results for finite BMS measures on flow spaces associated to geometrically finite convergence group actions.
result Results apply to flow spaces associated to relatively Anosov groups.
The paper characterizes geometric infiniteness for convergence group actions using orbit uniform metrics.
problem Characterizing geometric infiniteness for convergence group actions.
method Introducing orbit uniform metrics and proving properties of discrete orbits.
result Characterization of geometric infiniteness in terms of uncountability of non-conical limit points and existence of escaping sequences.
Proves convergence of normal forms for infinite-dimensional Lie pseudo-group actions.
problem Analyzing convergence of normal forms for complex manifolds.
method Equivariant moving frame method and Cartan-Kähler Theorem.
result Proves convergence of normal form power series for infinite-dimensional Lie pseudo-group actions.
We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.
We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group H which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set Lambda H is the closure of a `tree of circles' (adjacent circles meeting in pairs…
Study of group boundaries and subgroup properties.
problem Characterizing boundaries of relatively hyperbolic group pairs.
method Analyzing Bowditch boundaries and convergence group actions.
result Rigidity of group pairs leads to specific subgroup properties.
Scl in groups acting on trees is rational and converges to limits.
problem Understanding stable commutator length in group actions on trees.
method Analyzing groups acting on trees with cyclic stabilizers, focusing on stable commutator length and its limits.
result Stable commutator length is rational and converges to limits in surgery families.
Study of geometric actions on CAT(0) spaces and their limits.
problem Understanding limits of geometric actions on CAT(0) spaces.
method Analysis of convergence and splitting/collapsing phenomena in CAT(0) lattices and orbispaces.
result Proof of compactness theorem for CAT(0) homology orbifolds.
Constructs equivariant cohomology models for differentiable stacks.
problem Developing cohomology theory for stacks with group actions.
method Extends classical results for smooth manifolds to differentiable stacks.
result Derives spectral sequences generalizing Bott's spectral sequence.
We prove that the space of actions of Z^d by C^1 (orientation-preserving) diffeomorphisms of either the interval or the circle is connected by arcs. This is proved by showing that all such actions can be C^0 conjugated via a 1-parameter family into diffeomorphisms that converge to either the trivial action or an action…
Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the co…
Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups-so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then p…
Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.
problem Positive entropy actions by higher-rank lattices in Lie groups.
method Analysis of sub-actions, fiber entropy upper semicontinuity, and conjugacy arguments.
result Actions by higher-rank lattices in SL(n,R) are conjugate to affine actions on (infra-)tori. If a sequence of Riemannian manifolds, Xi, converges in the pointed Gromov-Hausdorff sense to a limit space, X∞, and if Ei are vector bundles over Xi endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the Ei converges in the pointed Gromov-Hausdorff sense t…
Consider a manifold endowed with the action of a Lie group. We study the relation between the cohomology of the Cartan complex and the equivariant cohomology by using the equivariant De Rham complex developed by Getzler, and we show that the cohomology of the Cartan complex lies on the 0-th row of the second page of a …
Study on simply connected manifolds with discrete isometric actions and bounded quotient diameter.
problem Characterizing simply connected manifolds with discrete isometric cocompact group actions.
method Analyzing sequences of manifolds with bounded diameter and Ricci curvature lower bound, using Gromov-Hausdorff convergence and Lie group theory.
result The quotient space of the limit manifold is simply connected, and the fundamental group is generated by loops in the maximal torus orbit.
Constructs equivariant analytic torsion for proper actions on manifolds.
problem Analytic torsion for proper actions on Riemannian manifolds.
method Equivariant construction of Ray-Singer analytic torsion for proper, isometric actions.
result Generalizes earlier results to noncompact conjugacy classes under suitable conditions.
New method approximates hyperbolic lattices using cube complexes.
problem Metric approximation of hyperbolic lattices by cubulations.
method Study of co-geodesic currents and their intersection number.
result Isometric actions of hyperbolic lattices can be approximated by geometric actions on CAT(0) cube complexes.
Study on regularity of exceptional actions and moduli of continuity for circle diffeomorphisms.
problem Regularity of exceptional actions of groups on the circle.
method Analysis of C1,α diffeomorphisms with free orbits and bounded derivatives. result Existence of exceptional C1,α diffeomorphisms under certain conditions on α. Renormalized circle diffeos with breaks converge to Moebius maps with a symplectic structure.
problem Analyzing the renormalization of circle diffeomorphisms with breaks.
method Proving convergence to invariant piecewise Moebius maps and identifying the renormalization operator with a sub-action of the mapping class group.
result Renormalization identifies with a symplectic form preserved by the mapping class group.
Donaldson defined a parabolic flow on Kahler manifolds which arises from considering the action of a group of symplectomorphisms on the space of smooth maps between manifolds. One can define a moment map for this action, and then consider the gradient flow of the square of its norm. Chen discovered the same flow from a…
Researchers classify invariant translating solitons in the Heisenberg group.
problem Classifying invariant translating solitons in the Heisenberg group.
method Considering canonical deformations of the standard Riemannian metric, analyzing similarities and differences with Euclidean translators.
result Described analogous of Euclidean translators like grim reapers, bowl solutions, and catenoids, but some are not convex.
Mapping class group dynamics tracked through Teichmüller space.
problem Tracking mapping class group actions on Teichmüller space.
method Action on Teichmüller space and geometric intersection numbers.
result Effective estimate of mapping class group actions on Teichmüller space.
New distance function proves rigidity in geodesic lamination space.
problem Proving rigidity in geodesic lamination space.
method Introduced left Hausdorff distance function and proved rigidity result.
result Extended mapping class group is isomorphic to bijections preserving left Hausdorff convergence.
Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.
problem Behavior of harmonic 2-forms on K3 surfaces with Ricci-flat metrics.
method Analysis of convergence of harmonic forms to flat 4D orbifold.
result Decomposition of harmonic 2-forms into converging subspaces.
Given a free group Fn, a fully irreducible automorphism $f \in \aut$, and a generic element x∈Fn, the elements fk(x) converge in the appropriate sense to an object called an attracting lamination of f. When the action of f on [Fn,Fn]Fn has finite order, we introduce a homological version…
This paper studies symplectic vortices on compact manifolds, proving their asymptotic behavior.
problem Analyzing the asymptotic behavior of finite energy symplectic vortices on compact manifolds.
method Proves the convergence of symplectic vortices to sectors of symplectic reduction at cylinder ends.
result Finite energy symplectic vortices exponentially converge to un/untwisted sectors of the symplectic reduction.
The paper proves the existence of minimal surfaces on certain manifolds.
problem Existence of minimal surfaces on manifolds with specific symmetries.
method Developed a regularity theory for equivariant Allen--Cahn solutions, showing convergence to minimal hypersurfaces.
result Closed Riemannian manifolds with cohomogeneity 2 and no exceptional orbits admit minimal hypersurfaces with optimal regularity.
Researchers calculate the action dimension of various group structures.
problem Determining the minimum dimension of contractible manifolds with group actions.
method Computing the action dimension of specific group structures.
result Action dimensions computed for Artin groups, graph products, and aspherical complements.
Paper proves finite BMS measure for SPR groups in higher rank Lie groups.
problem Finite measure for certain groups in higher rank Lie groups.
method Developed SPR property and proved finite BMS measure.
result Finite Bowen-Margulis-Sullivan measure for SPR groups in higher rank Lie groups.
We construct a covering of Culler-Vogtmann Outer space by the Teichmuller spaces of punctured surfaces. By considering the equivariant homology for the action of Out(F_n) on this covering, we construct a spectral sequence converging to the homology of Out(F_n) that has E^1 terms given by the homology of mapping class g…
Generic elements of hyperbolic groups act as loxodromics.
problem Understanding the behavior of loxodromic elements in hyperbolic group actions.
method Utilizing automatic structure, Patterson-Sullivan measure, and ergodic theory of random walks.
result The proportion of loxodromic elements in a ball of radius n about the identity in G approaches 1 as no∞. The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.
problem Exploring acylindrical actions on trees and their properties.
method Demonstrates criteria for preserving acylindrical hyperbolicity and analyzes the outer automorphism group of Baumsligar-Solitar groups.
result Proves acylindrical hyperbolicity of non-solvable Baumsligar-Solitar groups.
Given a general pseudo-Anosov flow in a three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to …
We construct a graph complex calculating the integral ho- mology of the bordered mapping class groups. We compute the ho- mology of the bordered mapping class groups of various surfaces. Using the circle action on this graph complex, we build a double complex and a spectral sequence converging to the homology of the un…
The study examines largest hyperbolic actions in groups and finds many do not exist.
problem Identifying largest hyperbolic actions in groups.
method Analysis of equivalence classes of cobounded actions on hyperbolic metric spaces.
result Many families of groups, including 3-manifold groups and mapping class groups, do not have largest hyperbolic actions.
Finite group actions on smooth 3-manifolds can be smoothed.
problem Finite group actions on smooth 3-manifolds.
method Uniform limit of smooth actions.
result Every continuous action of a finite group on a smooth 3-manifold is a uniform limit of smooth actions.
New stability theorem for non-hyperbolic group actions.
problem Structural stability of non-hyperbolic group actions.
method Introducing 'meandering hyperbolicity' for group actions on geodesic metric spaces.
result Meandering-hyperbolic actions are structurally stable.
The paper establishes a new link between isometries and scaling nonvanishing property in manifolds with Ricci curvature.
problem Lack of links between large-scale and small-scale geometry in manifolds with Ricci curvature.
method Investigates isometric actions with scaling nonvanishing property and their implications on the structure of manifolds.
result Establishes a dimension monotonicity on the limit group associated with rescaling sequences of universal covers.
Study finite group actions on vector bundle moduli spaces.
problem Understanding fixed-point sets of finite group actions on moduli spaces.
method Relate fixed-point sets to representation varieties of orbifold fundamental groups.
result Established a connection between fixed-point sets and representation varieties.
Extends group actions on metric spaces, preserving properties.
problem Natural extension of group actions on metric spaces.
method Formalizing the problem, constructing induced actions, using hyperbolically embedded subgroups.
result Induced actions can preserve properties of original actions.
New criteria for non-isometric group actions in metric spaces.
problem Understanding group actions on non-isometric spaces.
method Generalizing results from isometric to continuous group actions.
result Criterion for cocompact cyclic groups to be inessential.
New method for natural policy gradients converges linearly.
problem Improving natural policy gradient methods for better convergence.
method Fisher-Rao gradient flow applied to state-action distributions.
result Linear convergence rate with geometry-dependent factor.
Non-proper surface group action on product of trees found.
problem Proper surface group action on product of trees proposed, but not proper.
method Demonstrated non-properness of the surface group action.
result Surface group action on product of trees is not proper.
Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.
problem Characterizing and understanding group actions on surfaces, especially maximal handlebody and Hurwitz groups.
method Analyzing various group actions, comparing Hurwitz and handlebody groups, and examining bounding actions.
result Relationship between Hurwitz groups and maximal handlebody groups, and insights into geometric bounding actions.
Conditions for reducing quasi-actions to tree actions and group properties.
problem Conditions for reducing quasi-actions to tree actions.
method Reduction to cobounded isometric actions on trees.
result Groups with quasi-orbits quasi-isometric to trees are virtually free.
Let G be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space Z so that there exists a continuous G-equivariant map i:∂G→Z, which we call a \emph{Cannon-Thurston map}. We obtain two characterzations (a dynamical one and a geometric one) of conical limit p…