Study expands on Dunwoody's and Sageev's work on group actions.
problem Group actions and related structures.
method Extended study of Dunwoody's and Sageev's constructions.
result Expands understanding of group actions and related structures.
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48 results for “Sageev's theorem”
We prove that a hyperplane in a CAT(0) cubical complex X has no self-intersections and separates X into two convex complementary components. These facts were originally proved by Sageev. Our argument shows that his theorem is a corollary of Gromov's link condition. We also give new arguments establishing some combinato…
Almost Invariant Setsmath.GR
A short proof of a conjecture of Kropholler is given. This gives a relative version of Stallings' Theorem on the structure of groups with more than one end. A generalisation of the Almost Stability Theorem is also obtained, that gives information about the structure of the Sageev cubing.
Theory of structure trees applied to network max-flow min-cut theorem and group theory.
problem Max-Flow Min-Cut Theorem and its generalizations for infinite networks and groups.
method Theory of structure trees and edge cuts in networks.
result Generalization of Max-Flow Min-Cut Theorem to infinite networks and new insights into group structure.
Minimal cubingsmath.GR
We combine ideas of Scott and Swarup on good position for almost invariant subsets of a group with ideas of Sageev on constructing cubings from such sets. We construct cubings which are more canonical than in Sageev's original construction. We also show that almost invariant sets can be chosen to be in very good positi…
Effective rank rigidity proved for cubulated groups with factor systems.
problem Rank rigidity in cubulated groups with factor systems.
method Exhibiting special pairs of hyperplanes and curtains for skewering.
result Effective form of rank rigidity proved for cubulated groups.
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 .
Let denote the closed orientable surface of genus . We construct exponentially many mapping class group orbits of collections of simple closed curves on which pairwise intersect exactly once, extending a result of the first author and further answering a question of Malestein-Rivin-Theran. To dist…
The contact graph of a CAT(0) cubical complex has unbounded structure and a Gaussian CLT for random walks.
problem Understanding the structure and behavior of random walks on CAT(0) cubical complexes.
method Proved the contact graph is unbounded and homeomorphic to the boundary. Reformulated Caprace-Sageev's theorem. Proved a Central Limit Theorem for random walks.
result A Central Limit Theorem for random walks on CAT(0) cubical complexes, with a non-degenerate Gaussian distribution.
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…
New examples show some convex-cocompact subgroups are separable.
problem Whether all convex-cocompact subgroups are separable.
method Using Manning-Mj-Sageev construction, examples of separable subgroups of arbitrary finite rank are given.
result Examples of separable convex-cocompact subgroups of arbitrary finite rank exist.
In the presence of certain topological conditions, we provide lower bounds for the infimum of the length function associated to a collection of curves on Teichmüller space that depend on the dual cube complex associated to the collection, a concept due to Sageev. As an application of our bounds, we obtain estimates for…
We explain how to adapt a construction of M. Sageev's to construct a proper action on a CAT(0) cube complex starting from a proper action on a wall space, and use this to deduce that if G is a group containing an amenable subgroup H of super-polynomial growth and G acts properly on a space with walls then there are arb…
Proves Tits alternative for finite rank median spaces, restricting group actions.
problem Understanding group actions on median spaces.
method Extending Caprace-Sageev machinery and Hagen's theory to median spaces.
result Groups acting on finite rank median spaces are more restricted.
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.
CAT(0) cube complexes have special isometry groups.
problem Characterizing isometry groups of CAT(0) cube complexes.
method Analyzing subcomplexes and using hyperbolicity properties.
result If Aut(X) ≠ Isom(X), X has a product structure.
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space , we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathc…
In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersectio…
Stable cylinders found in hyperbolic groups and curve graphs.
problem Torsionfree hyperbolic groups and curve graphs of surfaces have globally stable cylinders.
method Generalised Sageev's construction to improve fine properties of hyperbolic spaces.
result Proved curve graphs of surfaces admit equivariant quasi-isometric embeddings in finite products of quasitrees.
We give an example of two JSJ decompositions of a group that are not related by conjugation, conjugation of edge-inclusions, and slide moves. This answers the question of Rips and Sela stated in "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition," Ann. of Math. 146 (1997), 53-109. On th…
In this paper we initiate a study of the topological group of pattern-preserving quasi-isometries for a hyperbolic Poincare duality group and an infinite quasiconvex subgroup of infinite index in . Suppose admits a visual metric with , where is the Hausd…
Characterizes ends and coends of graph pairs using quasi-median graphs.
problem Understanding the number of ends and coends in graph pairs.
method Characterizes ends and coends using quasi-median graphs.
result Characterizes ends and coends of graph pairs in terms of quasi-median graphs.
Groups on CAT(0) cube complexes grow exponentially uniformly.
problem Uniform exponential growth of groups acting on CAT(0) cube complexes.
method Study groups acting without global fixed points on CAT(0) square complexes.
result Groups with uniform exponential growth or stabilize Euclidean subcomplexes.
Study on connectivity and geometry of random Coxeter groups.
problem Connectivity threshold for square percolation on random graphs.
method Probabilistic combinatorics and techniques from geometric group theory.
result Determines connectivity threshold and cubical coarse median structure for random Coxeter groups.
Study boundary actions on CAT(0) spaces, proving topological freeness.
problem Understanding boundary actions on CAT(0) spaces.
method Verification of freeness of Myrberg points on boundaries.
result Large class of boundary actions are topologically free.
Characterizes when the Roller boundary equals the Poisson boundary of CAT(0) cube complexes.
problem Understanding the Roller boundary of CAT(0) cube complexes.
method Characterizes the Roller boundary and Poisson boundary equivalence.
result Characterizes when the Roller boundary equals the Poisson boundary of CAT(0) cube complexes.
For , let be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension , . Let be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of is a codimension one topological sphere. 2) limit set of is an e…
Study boundary actions of CAT(0) spaces and their -algebras.
problem Investigate boundary actions of CAT(0) spaces and their associated -algebras.
method Topological dynamics and -algebras, focusing on actions of specific groups and their properties.
result Established (strongly) pure infiniteness results for reduced crossed product -algebras of boundary actions.
The study describes quasiflats in 2D Artin groups and their properties.
problem Understanding the structure and properties of quasiflats in 2D Artin groups.
method Metric systolicity and combinatorial analysis of tilings.
result Precise description of building blocks (atomic sectors) for quasiflats in 2D Artin groups.
Theorem analogues proven using Artin's approximation theorem.
problem Proving analogues of Moser's Theorem.
method Using Artin's approximation theorem.
result Few analogues of Moser's Theorem proven.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Fixed-point theorems for set-valued maps using homological methods.
problem Finding fixed points for specific types of set-valued maps.
method Homological selection theorems applied to finite-dimensional spaces.
result Established fixed-point theorems for usco homologically UV^n set-valued maps.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
New measure proves Poncelet-type theorems.
problem Proving Poncelet-type theorems.
method Introducing a new invariant measure on the circle.
result Simple proof of Emch closing theorem.
New theorem for doodles on sphere, similar to Markov's.
problem Understanding doodles on a sphere.
method Description of twins with equivalent closures.
result Analogous to Markov's theorem for doodles.
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Analyzes Saito vanishing theorem using methods.
problem Proving the Saito vanishing theorem.
method Uses -methods to prove the theorem.
result Analytic proof of the Saito vanishing theorem.
Extends calculus theorem to higher dimensions.
problem Calculus theorem limitations in higher dimensions.
method Type Stokes' theorem for type -chains.
result Extends fundamental theorem of calculus.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.