Related papers: The fundamental group of random 2-complexes
For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf,…
We study the homological properties of random simplicial complexes. In particular, we obtain the asymptotic behavior of lifetime sums for a class of increasing random simplicial complexes; this result is a higher-dimensional counterpart of…
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links,…
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…
Random walks on regular bounded degree expander graphs have numerous applications. A key property of these walks is that they converge rapidly to the uniform distribution on the vertices. The recent study of expansion of high dimensional…
For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation…
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…
In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this group is isomorphic to the edge group of the clique complex of the digital image…
We study hyperbolic cohomology classes in the general context of simplicial complexes and prove homological invariance statements for them. We relate the existence of hyperbolic cohomology classes to the non-amenability of the fundamental…
Recently there has been growing interest in discrete homotopies and homotopies of graphs beyond treating graphs as 1-dimensional simplicial spaces. One such type of homotopy is $\times$-homotopy. Recent work by Chih-Scull has developed a…
A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension $2$ is closed under taking finitely presented (or more generally $FP_2$) subgroups. We prove the analogous result for relatively hyperbolic…
We prove that 2-dimensional simplicial complexes whose first homology group is trivial have topological embeddings in 3-space if and only if there are embeddings of their link graphs in the plane that are compatible at the edges and they…
In this work we give precise asymptotic expressions on the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.
Consider a random uniform sample of $n$ points in a compact region $A$ of Euclidean $d$-space, $d \geq 2$, with a smooth or (when $d=2$) polygonal boundary. Fix $k \in {\bf N}$. Let $T_{n,k}$ be the threshold $r$ at which the geometric…
Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPKVB10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model:…
This paper discusses the enumeration of two-terminal series-parallel networks, i.e. the number of electrical networks built with n identical elements connected in series or parallel with two-terminal nodes. They frequently occur in applied…
In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a generalization of the constructions of right-angled…
We consider random right-angled Coxeter groups, $W_{\Gamma}$, whose presentation graph $\Gamma$ is taken to be an Erd\H{o}s--R\'enyi random graph, i.e., $\Gamma\sim \mathcal{G}_{n,p}$. We use techniques from probabilistic combinatorics to…
We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the…
General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the…