Related papers: Curve complexes and Garside groups

Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study the finite extensions $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ such that the number of $\mathbb{F}_{q^n}$-rational points on $E$ attains the Hasse upper bound. We obtain an…

Recently, Marin and Gonz\'alez-Meneses introduced a class of ``parabolic'' subgroups for generalized braid groups associated to arbitrary complex reflection groups. Using notably Garside group structures on these generalized braid groups,…

We consider three kinds of quotients of the curve complex which are obtained by coning off uniformly quasi-convex subspaces: symmetric curve sets, non-maximal train track sets, and compression body disc sets. We show that the actions of the…

Let $\Sigma$ be a compact, orientable surface of genus $g$, and let $\Gamma$ be a relation on $\pi_0(\partial \Sigma)$ such that the prescribed arc graph $\mathcal{A}(\Sigma,\Gamma)$ is Gromov-hyperbolic and non-trivial. We show that…

We motivate the study of metric spaces with a unique convex geodesic bicombing, which we call CUB spaces. These encompass many classical notions of nonpositive curvature, such as CAT(0) spaces and Busemann-convex spaces. Groups having a…

Let $(X, d)$ be an unbounded metric space and let $\tilde r=(r_n)_{n\in\mathbb N}$ be a sequence of positive real numbers tending to infinity. A pretangent space $\Omega_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity is a limit of the…

We study fundamental groups of clique complexes associated to random graphs. We establish thresholds for their cohomological and geometric dimension and torsion. We also show that in certain regime any aspherical subcomplex of a random…

This article resolves several long-standing conjectures about Artin groups of euclidean type. In particular, we prove that every irreducible euclidean Artin group is a torsion-free centerless group with a decidable word problem and a…

We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its…

Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $\le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as…

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each…

Recently George Bergman proved that the symmetric group of an infinite set possesses the following property which we call by the {\it universality of finite width}: given any generating set $X$ of the symmetric group of an infinite set…

Skeletal polyhedra and polygonal complexes are finite or infinite periodic structures in 3-space with interesting geometric, combinatorial, and algebraic properties. These structures can be viewed as finite or infinite periodic graphs…

Consider a connected orientable surface $S$ of infinite topological type, i.e. with infinitely-generated fundamental group. We describe the large-scale geometry of arbitrary connected subgraphs of the arc complex $A(S)$ and curve complex…

In \cite{Oh22}, the second author defined a complex of groups decomposition of the fundamental group of a finitely generated 2-dimensional special group, called an \emph{intersection complex}, which is a quasi-isometry invariant. In this…

We show that, for all odd natural numbers $N$, the $N$-torsion points on an elliptic curve may be placed in an $N \times N$ grid such that the sum of each column and each row is the point at infinity.

Every nontrivial abelian variety over a Hilbertian field in which the weak Mordell-Weil theorem holds admits infinitely many torsors with period any $n > 1$ which is not divisible by the characteristic. The corresponding statement with…

Given an elliptic curve $E/\mathbb{Q}$ with torsion subgroup $G = E(\mathbb{Q})_{\rm tors}$ we study what groups (up to isomorphism) can occur as the torsion subgroup of $E$ base-extended to $K$, a degree 6 extension of $\mathbb{Q}$. We…

A number of properties of spherical Artin groups extend to Garside groups, defined as the groups of fractions of monoids where least common multiples exist, there is no nontrivial unit, and some additional finiteness conditions are…

We introduce the universal Euler characteristic of orbit space definable groupoids, a class of groupoids containing cocompact proper Lie groupoids as well as translation groupoids associated to proper definable group actions. We show that…