Related papers: Graphical C(3)-T(6) implies CAT(0)

We explain and generalise a construction due to Gromov to realise geometric small cancellation groups over graphs of groups as fundamental groups of non-positively curved 2-dimensional complexes of groups. We then give conditions so that…

We provide a systematic description of the automorphism groups of specially cocompact CAT(0) cube complexes. We show that these groups are topologically finitely generated, present a method to explicitly obtain generating sets, and prove a…

We construct an uncountable family of groups of type $FP$. In contrast to every previous construction of non-finitely presented groups of type $FP$ we do not use Morse theory on cubical complexes; instead we use Gromov's graphical small…

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…

We study the collapsibility of finite simplicial complexes of dimension 3 endowed with a CAT(0) metric. Our main result states that, under an additional hypothesis, finite simplicial 3-complexes endowed with a CAT(0) metric collapse to a…

Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning…

We prove that infinitely presented graphical $Gr(7)$ small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical $C(7)$-groups and, hence, classical $C'(\frac{1}{6})$-groups are acylindrically…

Gromov (2003) constructed finitely generated groups whose Cayley graphs contain all graphs from a given infinite sequence of expander graphs of unbounded girth and bounded diameter-to-girth ratio. These so-called Gromov monster groups…

We study actions of (infinitely presented) graphical small cancellation groups on the Gromov boundaries of their coned-off Cayley graphs. We show that a class of graphical small cancellation groups, including (infinitely presented)…

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 the Haagerup property (= Gromov's a-T-menability) for finitely generated groups defined by infinite presentations satisfying the graphical C'(lambda)-small cancellation condition with respect to graphs endowed with a compatible…

We prove the arborescence of any locally finite complex that is $CAT(0)$ with a polyhedral metric for which all vertex stars are convex. In particular locally finite $CAT(0)$ cube complexes or equilateral simplicial complexes are…

We investigate the structure of the minimal displacement set in CAT(0) cubical complexes. We show that such set is convex, it is locally endowed with a CAT(0) metric and it is simply connected.

We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the…

S. Gersten and H. Short have proved that if a group has a presentation which satisfies the algebraic C(4) and T(4) small-cancellation condition then the group is automatic. Their proof contains a gap which we aim to close. To do that we…

Shephard groups are common generalizations of Coxeter groups, Artin groups, and graph products of cyclic groups. Their definition is similar to that of a Coxeter group, but generators may have arbitrary order rather than strictly order 2.…

We prove a Tits alternative theorem for groups acting on CAT(0) cubical complexes. Namely, suppose that $G$ is a group for which there is a bound on the orders of its finite subgroups. We prove that if $G$ acts properly on a…

We obtain a sufficient condition for lattices in the automorphism group of a finite dimensional CAT(0) cube complex to have infinite girth. As a corollary, we get a version of Girth Alternative for groups acting geometrically: any such…

We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations for which the…

If G is a locally essential subgroup of a compact abelian group K, then: (i) t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K must be metrizable; (iii) G contains a super-sequence S converging to 0 such that…