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…
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48 results for “dual cube complexes”
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…
Study provides bounds for longest curves with intersections on Teichmüller space.
problem Finding the longest curve with a given number of intersections on Teichmüller space.
method Uses lower bounds for the infimum of length functions associated with curve collections and dual cube complexes.
result Obtains estimates for the longest curve with k self-intersections.
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.
Characterizes alternating link exteriors using cubed complexes.
problem Understanding the structure of alternating link exteriors.
method Introducing signed BW cubed-complexes and characterizing their homeomorphism to link exteriors.
result Characterization of alternating link exteriors in terms of cubed complexes.
Generalizes Leighton's theorem to cube complexes.
problem Extending graph covering theorem to cube complexes.
method Generalizes Leighton's theorem to a family of cube complexes.
result Cube complexes have common finite covers.
New invariant characterizes cube complexes with abelian fundamental group.
problem Characterizing cube complexes with abelian fundamental groups.
method Defining genus and using it to prove properties of cube complexes.
result Fundamental group of special cube complexes is either free abelian or surjects onto a non-cyclic free group.
Spatial Cube Complexesmath.GR
New geometric spine for Artin groups defined by cube complexes.
problem Geometric characterization of Artin group spines.
method Category of spatial cube complexes for geometric simplicial complex.
result Realized spine as natural simplicial complex.
Cube complexes' boundaries determine their structure.
problem Understanding the structure of cube complexes.
method Introducing a cross-ratio on Roller boundaries and showing its uniqueness in preserving cubical isomorphisms.
result Every cross-ratio preserving bijection of Roller boundaries extends to a cubical isomorphism.
Study mapping class groups on CAT(0) cube complexes.
problem Asymptotically rigid mapping class groups of infinitely-punctured surfaces.
method Introduced CAT(0) cube complexes and determined their properties.
result Determined CAT(0) properties of cube complexes.
Groups act on cube complexes with varying dimensions.
problem Finding groups with different minimal cube complex dimensions.
method Proving groups acting on CAT(0) cube complexes split as products.
result Groups can have minimal cube complex dimensions strictly larger than their CAT(0) space dimensions.
Finite stature proven for cube complexes with cyclonormal edges.
problem Understanding the structure of cube complexes with specific edge properties.
method Analyzing the fundamental groups of edge and vertex spaces, showing cyclonormality and virtual specialness.
result The fundamental group of a cube complex has finite stature with respect to vertex groups.
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.
A curiously cubulated groupmath.GR
New group acts on complex but not in lower dimensions.
problem Understanding group actions on cube complexes.
method Constructed a specific group acting on a locally finite CAT(0) cube complex.
result Group does not act properly on a finite dimensional CAT(0) cube complex.
We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex.
Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
problem Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
method Proving non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
result Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groupsmath.GR
Characterizes contracting isometries in CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups.
problem Characterizing contracting isometries in CAT(0) cube complexes and their relation to acylindrical hyperbolicity.
method Characterization of contracting isometries without local finiteness assumption, combinatorial boundary introduction, and application to diagram groups.
result Determine precise conditions for acylindrical hyperbolicity of diagram groups.
Extends folding techniques to study subgroups of CAT(0) cube complexes.
problem Studying subgroups of non-positively curved cube complexes.
method Folding-like techniques applied to CAT(0) cube complexes.
result Effective algorithmic representation of subgroups.
Lecture notes on CAT(0) cube complexes for beginners.
problem Introduction to CAT(0) cube complexes for students.
method Elementary and self-contained lectures.
result Accessible introduction to advanced topics for students.
The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.
problem Conditions for groups acting on CAT(0) cube complexes to have infinite girth.
method Analyzes lattices in automorphism groups of finite dimensional CAT(0) cube complexes.
result Groups either have infinite girth or are {locally finite}-by-{virtually abelian}.
The paper proves effective bounds on curve complex distances and their relation to covers of surfaces.
problem Effective bounds on curve complex distances and their relation to covers of surfaces.
method Proves effective version of a theorem relating curve complex distance to electric distance in hyperbolic 3-manifolds.
result Effective proof of a result regarding maps between curve complexes of surfaces induced by finite covers.
Toric Cubesmath.CO
A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells…
Develops theory of relatively geometric actions on CAT(0) cube complexes.
problem Understand actions of relatively hyperbolic groups on CAT(0) cube complexes.
method Introduces and studies relatively geometric actions, proving key results.
result Proves full relatively quasi-convex subgroups are convex compact.
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.
Borel asymptotic dimension of the Roller boundary of finite dimensional CAT(0) cube complexesmath.GT
Study shows Roller compactification's median graph has limited asymptotic dimension.
problem Understanding the asymptotic dimension of Roller compactifications.
method Proved using finite dimensional CAT(0) cube complexes and Borel median graph.
result Borel asymptotic dimension is bounded by the complex's dimension.
Study cup products on CAT(0) cube complexes, proving quasimorphisms' vanishing results.
problem Understanding cup products in higher bounded cohomology groups.
method Extending vanishing results from Brooks quasimorphisms to median quasimorphisms on CAT(0) cube complexes.
result Vanishing results for cup products of median quasimorphisms on CAT(0) cube complexes.
Not finitely presented group from CAT(0) cube complexes.
problem Identifying finitely presented groups.
method Developing techniques for proving non-simple connectedness of subcomplexes of CAT(0) cube complexes.
result Basilica Thompson group is not finitely presented.
Classifies 3-manifolds from cube identifications.
problem Classifying non-orientable 3-manifolds.
method Identifying cube faces to form manifolds, classifying based on surface-complexity.
result Identifies four flat non-orientable 3-manifolds with surface-complexity one.
Homotopy equivalent boundaries of cube complexes are studied.
problem The equivalence of different boundaries of cube complexes.
method Using a partial order on a quotient of the Roller boundary, we obtain the simplicial Roller boundary and show homotopy equivalence among the Tits, simplicial, and simplicial Roller boundaries.
result The Tits, simplicial, and simplicial Roller boundaries are homotopy equivalent.
The study examines geometric conditions on CAT(0) cube complexes and Coxeter groups.
problem Classifying and understanding divergence in CAT(0) cube complexes and Coxeter groups.
method Geometric conditions on hyperplanes of CAT(0) cube complexes.
result Quadratic divergence for all right-angled Coxeter groups and no divergence between quadratic and cubic.
New quasimorphisms show right-angled Artin groups have elements with high commutator length.
problem Estimating commutator lengths in right-angled Artin groups.
method Constructing quasimorphisms on CAT(0) cube complexes, using essential characteristic set and equivariant Euclidean embeddings.
result All non-trivial elements of right-angled Artin groups have stable commutator length at least 1/24.
The paper examines cone-offs of CAT(0) cube complexes and their geometric properties.
problem Characterizing the hyperbolicity of certain geometric structures.
method Study of cone-offs over combinatorially convex subcomplexes, focusing on Gromov-hyperbolicity.
result Direct proof of relative hyperbolicity for right-angled Coxeter groups and acylindrical hyperbolicity for specific quotients.
Researchers solve 3D cube complex boundary rigidity problem.
problem Determining the combinatorial type of a 3D CAT(0) cube complex from boundary distances.
method Discrete version of boundary rigidity problem, focusing on CAT(0) cube complexes.
result The combinatorial type of a finite CAT(0) cube complex can be reconstructed from its boundary distances.
New conditions ensure hierarchical hyperbolicity of cube complexes.
problem Ensuring hierarchical hyperbolicity of cube complexes.
method Three conditions on a group action ensure hierarchical hyperbolicity.
result Hierarchical hyperbolicity of groups is confirmed under these conditions.
A cubical Rips constructionmath.GR
New group constructed from cube complex properties.
problem Creating a new group from cube complex properties.
method Cubical Rips construction for finitely presented groups.
result New group surjects onto a given group with specific properties.
Groups acting on CAT(0) cube complexes have hyperfinite boundary actions.
problem Hyperfiniteness of boundary actions for special groups.
method Proving hyperfiniteness for groups acting on CAT(0) cube complexes.
result Boundary actions of virtually special groups are hyperfinite.
Automorphisms of contact graphs match those of cube complexes under weak conditions.
problem Understanding automorphisms of cube complexes and their contact graphs.
method Analyzing the relationship between automorphisms of cube complexes and their contact graphs.
result The automorphism groups of cube complexes and their contact graphs coincide under weak assumptions.
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.
Study shows how knot Floer cube relates to HOMFLY-PT homology.
problem Understanding the relationship between knot Floer cube and HOMFLY-PT homology.
method Using a filtration induced by additional basepoints on the Heegaard diagram, the filtered complex decomposes into HOMFLY-PT homologies of subdiagrams.
result The graded Euler characteristic of the direct sum of HOMFLY-PT homologies equals the HOMFLY-PT polynomial of the knot.
Cube complexes allow hyperbolic groups to have Anosov representations.
problem Understanding representations of hyperbolic groups in higher dimensions.
method Analyzing representations of hyperbolic groups acting on CAT(0) cube complexes.
result Generic representations of certain groups are Anosov.
Proof shows local convexity implies global convexity in special geometric spaces.
problem Proving convexity in CAT(0) cubed complexes from local convexity.
method Analyzes vertex link structures to determine convexity.
result Local combinatorial properties determine global convexity.
The study connects the Furstenberg-Poisson boundary to CAT(0) cube complexes under specific conditions.
problem Understanding the Furstenberg-Poisson boundary in CAT(0) cube complexes.
method Analyzing a random walk on a group acting on a CAT(0) cube complex.
result The Roller boundary of a CAT(0) cube complex is the Furstenberg-Poisson boundary for a specific random walk.
New examples of normal currents with purely 2-unrectifiable supports are constructed.
problem Constructing normal currents with specific geometric properties.
method Using inverse systems of cube complexes to create examples of normal currents.
result Examples of -dimensional normal currents with purely 2-unrectifiable supports and simple vector fields.
New method identifies unique group actions on CAT(0) cube complexes.
problem Identifying group actions on CAT(0) cube complexes.
method Developed cross-ratio on Roller boundaries and extended cross-ratio preserving maps.
result Group actions on irreducible CAT(0) cube complexes are uniquely determined by their length function.
Grupos y geometría cubularmath.GR
Survey on non-positively curved cube complexes and geometric group theory.
problem None explicitly stated, focuses on introduction.
method Lecture notes and mini-course teaching.
result Introduction to non-positively curved cube complexes and geometric group theory.
New groups act on cube complexes without compact cubulation.
problem Triangle-free Artin groups without compact cubulation.
method Proved proper actions on CAT(0) cube complexes.
result First examples of non-cocompactly cubulated groups.
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 bottleneck conjecturemath.MG
The Mahler volume of a centrally symmetric convex body K is defined as M(K)= (Vol K)(Vol K^dual). Mahler conjectured that this volume is minimized when K is a cube. We introduce the bottleneck conjecture, which stipulates that a certain convex body K^diamond subset K X K^dual has least volume when K is an ellipsoid. If…