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 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.
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.
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.
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.
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}.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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…
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.
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…
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. 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…
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…
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 method connects CAT(0) spaces to hyperbolic spaces.
problem Injecting sublinear Morse boundaries into Gromov boundaries.
method Developed curtain machinery to characterize sublinear Morse properties.
result Continuous injection of sublinear Morse boundaries into Gromov boundaries.
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 …
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.
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.
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.
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…
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.
CAT(0) properties extended to a class of Shephard groups.
problem Extending CAT(0) properties to Shephard groups.
method Introduced new criteria for CAT(1) complexes and extended results for Coxeter groups to Shephard groups.
result Shephard groups with enough finite parabolic subgroups are CAT(0).
We study commensurating actions of groups and the associated properties FW and PW, in connection with wallings, median graphs, CAT(0) cubings and multi-ended Schreier graphs.
Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set, together with a pairwise-disjoint collection of subcomplexes corresponding, in the…
Uniform criterion for vanishing products in bounded cohomology.
problem Vanishing of cup products and Massey products in bounded cohomology.
method Uniform vanishing criterion for products in bounded cohomology.
result Reproved and extended previous vanishing results.
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…
Given a CAT(0) cube complex X, we show that if Aut(X) = Isom(X) then there exists a full subcomplex of X which decomposes as a product with Rn. As applications, we prove that if X is δ-hyperbolic, cocompact and 1-ended, then Aut(X) = Isom(X) unless X is quasi-isometric to H2, and extend…
We describe a correspondence between spaces with walls and CAT(0) cube complexes.
In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups,…
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.
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.
Locally finite complexes with polyhedral CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) metrics are arborescent. 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…
We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of "n-tracks of length n", which is a structure like Z+Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, …
Stable cubulations and bicombings in mapping class groups and Teichmüller spaces.
problem Understanding geometric structures in mapping class groups and Teichmüller spaces.
method Proving stably approximated by CAT(0) cube complexes, applying to broader colorable hierarchically hyperbolic spaces and groups.
result Stable cubulations and bicombings in mapping class groups and Teichmüller spaces, with stable coarse barycenters.
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.
Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) …
We show that group actions on irreducible CAT(0) cube complexes with no free faces are uniquely determined by their ℓ1 length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first …
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.
Clarifies boundary criterion for non-one-ended subgroups in cubulation theory.
problem Boundary criterion for relative cubulation in non-one-ended subgroups.
method Showed that if boundary criterion is satisfied for a relatively hyperbolic group, the group admits a relatively geometric action on a CAT(0) cube complex.
result The refinement of the boundary criterion is useful for constructing new relative cubulations.