Abstract. We study arc graphs and curve graphs for surfaces of infinite topological type. First, we define an arc graph relative to a finite number of (isolated) punctures and prove that it is a connected, uniformly hyperbolic graph of infinite diameter; this extends a recent result of J. Bavard to a large class of punctured surfaces.Second, we study the subgraph of the curve graph spanned by those elements which intersect a fixed separating curve on the surface. We show that this graph has infinite diameter and geometric rank 3, and thus is not hyperbolic.

Abstract. Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions.First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a consequence, we show that for any λ ∈ (0, 1) there exists a constant C λ such that every closed Riemannian surface of genus g whose area is normalized at 4π(g − 1) has at least [λg] homologically independent loops of length at most C λ log(g). This result extends Gromov's asymptotic log(g) bound on the homological systole of genus g surfaces. We construct hyperbolic surfaces showing that our general result is sharp. We also extend the upper bound obtained by P. Buser and P. Sarnak on the minimal norm of nonzero period lattice vectors of Riemann surfaces in their geometric approach of the Schottky problem to almost g homologically independent vectors.Then, we consider the lengths of pants decompositions on complete Riemannian surfaces in connexion with Bers' constant and its generalizations. In particular, we show that a complete noncompact Riemannian surface of genus g with n ends and area normalized to 4π(g + n 2 −1) admits a pants decomposition whose total length (sum of the lengths) does not exceed Cg n log(n + 1) for some constant Cg depending only on the genus.Finally, we obtain a lower bound on the systolic area of finitely presentable nontrivial groups with no free factor isomorphic to Z in terms of its first Betti number. The asymptotic behavior of this lower bound is optimal.

We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature (g, n). This maximum is shown to be strictly increasing in terms of the number of cusps for small values of n.We also show that this function is greater than a function that grows logarithmically in function of the ratio g/n.Date: November 2, 2018.2010 Mathematics Subject Classification. Primary: 30F10. Secondary: 32G15, 53C22.

The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichmüller space and is quasi-isometric to the underlying mapping class group. We study this space in two main directions. We first show that strata corresponding to triangulations containing a same multiarc are strongly convex within the whole space and use this result to deduce properties about the mapping class group. We then focus on the quotient of this space by the mapping class group to obtain a type of combinatorial moduli space. In particular, we are able to identity how the diameters of the resulting spaces grow in terms of the complexity of the underlying surfaces.