Related papers: Effective flipping, skewering and rank rigidity fo…

We introduce two new tools for studying CAT(0) spaces: \emph{curtains}, versions of cubical hyperplanes; and the \emph{curtain model}, a counterpart of the curve graph. These tools shed new light on CAT(0) spaces, allowing us to prove a…

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow…

We prove that a K\"ahler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free…

In this paper, we prove two main theorems: conformal measure rigidity and ergodicity of horospherical foliations, especially in higher rank. Both theorems are new even for relatively Anosov groups. First, we establish a higher rank…

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,…

The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a…

We show that group actions on irreducible ${\rm CAT(0)}$ cube complexes with no free faces are uniquely determined by their $\ell^1$ length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and…

We generalize the concept of rigid inner forms, defined by Kaletha in [Kal16], to the setting of a local function field $F$ in order state the local Langlands conjectures for arbitrary connected reductive groups over $F$. To do this, we…

In this paper we study hyperbolic groups acting on CAT(0) cube complexes. The first main result (Theorem A) is a structural result about the Sageev construction, in which we relate quasi-convexity of hyperplane stabilizers with…

Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that X has a factor system in the sense of [BHS14]. We also prove that one of these…

We introduce a class of spaces, called real cubings, and study the stucture of groups acting nicely on these spaces. Just as cubings are a natural generalisation of simplicial trees, real cubings can be regarded as a natural generalisation…

We show that colorable hierarchically hyperbolic groups (HHGs) admit asymptotically CAT(0) metrics, that is, roughly, metrics where the CAT(0) inequality holds up to sublinear error in the size of the triangle. We use the asymptotically…

We use an algorithm by Ozsvath and Szabo to find closed formulae for the ranks of the hat version of the Heegaard Floer homology groups for non-zero Dehn surgeries on knots in the 3-sphere. As applications we provide new bounds on the…

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 study and classify free actions of compact quantum groups on unital C*-algebras in terms of generalized factor systems. Moreover, we use these factor systems to show that all finite coverings of irrational rotation C*-algebras are cleft.

Let $F$ be any finite-rank free group, and $R$ be any finite subset of $\{g, [g]: g \in F-\{1\}\}$, where $[g]:= \{fgf^{-1}:f\in F\}$. By an $R$-allocating $F$-factorization we mean a set $\mathcal{H}$ of nontrivial subgroups of $F$ such…

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…

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues…

Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…

We prove a strengthened sector lemma for irreducible, finite-dimensional, locally finite, essential, cocompact CAT(0) cube complexes under the additional hypothesis that the complex is \emph{hyperplane-essential}; we prove that every…