Related papers: Random surfaces with large systoles

Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant length, as the order of the graphs tends to infinity.

In this article, we construct a new simplicial complex for infinite-type surfaces, which we call the grand arc graph. We show that if the end space of a surface has at least three different self-similar equivalence classes of maximal ends,…

Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called {\it…

We construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by…

We consider the asymptotic behavior of properly embedded minimal surfaces in the product of the hyperbolic plane with the line, taking into account the fact that there is more than one natural compactification of this space. This provides a…

Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on…

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface $S$ of genus $g$, we get a geometric lower bound on ${\lambda_{2g-2}}(S)$: ${\lambda_{2g-2}}(S) > 1/4 +…

In this paper, we obtain an improved upper bound involving the systole and area for the volume entropy of a Riemannian surface. As a result, we show that every orientable and closed Riemannian surface of genus $g\geq 18$ satisfies Loewner's…

We prove that random triangulations of high genus contain very large expander subgraphs, answering a question of Benjamini. Our approach relies on new general criteria for arbitrary graphs to contain large expander subgraphs.

We show that any infinite-type surface without planar ends admits arbitrarily large families of length isospectral hyperbolic structures. If the surface has infinite genus and its space of ends is self-similar, we construct an uncountable…

Random geometric graphs result from taking $n$ uniformly distributed points in the unit cube, $[0,1]^d$, and connecting two points if their Euclidean distance is at most $r$, for some prescribed $r$. We show that monotone properties for…

For any fixed integer $R \geq 2$ we characterise the typical structure of undirected graphs with vertices $1, ..., n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled…

In this article we consider surfaces in the product space $\h^2\times \r$ of the hyperbolic plane $\h^2$ with the real line. The main results are: a description of some geometric properties of minimal graphs; new examples of complete…

We give the formula for the maximal systole of the surface admits the largest $S^3$-extendable abelian group symmetry. The result we get is $2\mathrm{arccosh} K$. Here \begin{eqnarray*} K &=& \sqrt[3]{\frac{1}{216}L^3 +\frac{1}{8} L^2 +…

Since the work of Mirzakhani and Petri on random hyperbolic surfaces of large genus, length statistics of closed geodesics have been studied extensively. We focus on the case of random hyperbolic surfaces with cusps, the number of which…

We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random…

Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to…

We develop a theory for distributed branch points and investigate their role in determining the shape and influencing the mechanics of thin hyperbolic objects. We show that branch points are the natural topological defects in hyperbolic…

We investigate the geometry of $\pi_1$-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any $e>0$, if the manifold $M$ has sufficiently large systole $\sys_1(M)$, the genus of any such surface in $M$ is bounded…

A growth model for porous sedimentary rocks is proposed, using a simple computer simulation algorithm. We generate the structure by ballistic deposition of particles with a bimodal size distribution. Porosity and specific surface area are…