Researchers compute BNSR-invariants for surface Houghton groups.
problem Computing BNSR-invariants for surface Houghton groups.
method Proved Stein--Farley cube complex is CAT(0), adapted Zaremsky's method.
result Computed BNSR-invariants of surface Houghton groups.
Bieri, Geoghegan and Kochloukova computed the BNSR-invariants Σm(F) of Thompson's group F for all m. We recompute these using entirely geometric techniques, making use of the Stein--Farley CAT(0) cube complex on which F acts.
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 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.
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.
We give a characterization of alternating link exteriors in terms of cubed complexes. To this end, we introduce the concept of a "signed BW cubed-complex", and give a characterization for a signed BW cubed-complex to have the underlying space which is homeomorphic to an alternating link exterior.
We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non…
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.
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.
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.
We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady's classifying space for braid groups and the Stein--Farley complexes used for various relatives of Thompson's groups. It recovers the fact th…
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}.
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.
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…
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.
We introduce a Z-valued cross ratio on Roller boundaries of CAT(0) cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional…
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.
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.
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 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 mCAT(0) cube complexes under weak conditions.
problem Understanding automorphisms of mCAT(0) cube complexes and their contact graphs. method Analyzing the relationship between automorphisms of mCAT(0) cube complexes and their contact graphs. result The automorphism groups of mCAT(0) cube complexes and their contact graphs coincide under weak assumptions. The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and we show that contracting isometries are strongly related to isolated points at …
We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they "see" all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions…
Let Sg denote the closed orientable surface of genus g. We construct exponentially many mapping class group orbits of collections of 2g+1 simple closed curves on Sg which pairwise intersect exactly once, extending a result of the first author and further answering a question of Malestein-Rivin-Theran. To dist…
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.
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 study proves properties of specific groups acting on cube complexes.
problem Proving properties of specific groups acting on cube complexes.
method Elementary construction of a contractible cube complex.
result Proves T♯,T∗ are of type F∞ and brHn is of type Fn−1 but not of type Fn. Extends graph factor system to quasi-median graphs.
problem Constraint relaxation for combinatorial HHS machinery.
method Relaxing domain constraints on combinatorial HHS machinery and extending factor system to quasi-median graphs.
result Factor system applied to quasi-median graphs.
We define a homology theory for a certain class of posets equipped with a representation. We show that when restricted to Boolean lattices this homology is isomorphic to the homology of the "cube" complex defined by Khovanov.
Explicit polynomial bound found for subgroup Dehn function.
problem Finding explicit bounds on Dehn functions of subgroups of hyperbolic groups.
method Constructing a specific example of a non-hyperbolic subgroup and analyzing its Dehn function.
result Explicit polynomial upper bound n96 on the Dehn function of a non-hyperbolic subgroup. The Roller boundary is a well-known compactification of a CAT(0) cube complex X. When X is locally finite, essential, irreducible, non-Euclidean and admits a cocompact action by a group G, Nevo-Sageev show that a subset, B(X), of the Roller boundary is the realization of the Poisson boundary and that the action of G on…
We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cu…
These notes grew out of two lectures I have given on CAT(0) cube complexes. I've tried to keep the material elementary and self-contained in order to keep the material easily accessible and to provide an elementary introduction on the topic for advanced bachelor students or early master students with little to no previ…
Develops geometric foundations for sublinear Morse boundaries in mapping class groups and Teichmüller spaces.
problem Capturing generic directions in mapping class groups and Teichmüller spaces.
method Develops tools for modeling hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes.
result Sublinear Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph.
Decomposes axis bundles into cubist structures for fully irreducible outer automorphisms.
problem Understanding the geometry and structure of axis bundles for fully irreducible outer automorphisms.
method Develops a 'cubist' decomposition into branched cubes with special combinatorics.
result Locates a canonical finite collection of periodic fold lines in each axis bundle.
We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any N≥2 we provide examples of N-dimensional normal currents whose associated vector fields are simple, and whose supports are purely 2-unrectifiable and have Nagata dimension N. We show that in l∞ norm…
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…
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.