We prove that for any infinite-type orientable surface S there exists a collection of essential curves Γ in S such that any homeomorphism that preserves the isotopy classes of the elements of Γ is isotopic to the identity. The collection Γ is countable and has infinite complement in C(S), the curve complex of S. As a consequence we obtain that the natural action of the extended mapping class group of S on C(S) is faithful.

Abstract. Consider a connected orientable surface S of infinite topological type, i.e. with infinitely-generated fundamental group.Our main purpose is to give a description of the geometric structure of an arbitrary subgraph of the arc graph of S, subject to some rather general conditions. As special cases, we recover the main results of J. Bavard [2] and Aramayona-Fossas-Parlier [1].In the second part of the paper, we obtain a number of results on the geometry of connected, Mod(S)-invariant subgraphs of the curve graph of S, in the case when the space of ends of S is homeomorphic to a Cantor set.

Abstract. We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a topological space L(x) which captures all the ways that x can be "approached linearly". The homeomorphism type of L(x) is an affine invariant. When x is not a cone point or an infinite-angle singularity, we say it is wild; in this case it is necessary to add further metric data to L(x) to get a quantitative description of the surface near x.The study of flat surfaces, appearing under different guises (quadratic differentials, abelian differentials, translation surfaces, measured foliations, F-structures, and so on), reaches back at least to the 1970-80s, when seminal work of Thurston, Masur, Veech, and others uncovered fundamental connections among surface automorphisms, flat surface geometry, and billiard dynamics. However, their origins go back much further to the 1930-40s, with Nielsen's classification of torus automorphisms and Fox-Kerschner's association of a Riemann surface to billiards in a polygon [Fox36], sometimes called the Katok-Zemlyakov unfolding construction. Throughout much of the history of flat surfaces, the focus has been on compact flat surfaces, having so-called "cone-type" singularities, with non-compact surfaces appearing only sporadically. In this way, researchers could bring to bear the considerable power of finite-dimensionality in Teichmüller theory and in algebraic constructions such as homology groups.In recent years, increasing attention has been paid to the study of non-compact flat surfaces, or more precisely surfaces of infinite type. Several treatments deal with classes of examples such as covers of compact surfaces [HWS] or surfaces arising from certain dynamical systems (wind-tree models [HLT], irrational billiards [Val], exchanges of infinitely many intervals [Hoo10], etc.). In a similar vein, de CarvalhoHall have initiated a study of dynamical systems on genus-zero surfaces with infinitely many singularities [dCH11]. These studies have necessitated the adaptation of tools from the theory of compact flat surfaces, but have so far remained fuzzy on the local, intrinsic behavior of a surface near its singular points. The simple description via cone points becomes inadequate when the metric structure imposed on a surface can allow for essentially arbitrary topological complication within a bounded region. In our opinion, this constitutes an important lack and an obstacle to properly understanding basic notions such as straight-line flow and deformations of flat surfaces. Here we present a method for studying the local behavior of singularities of topologically infinite flat surfaces. For the most part, we restrict our attention to isolated singularities in the metric completion of a flat surface. These singularities do not, in general, have an analytic description parallel to the description of cone points as zeroes of holomorphic diffe...

We prove that the natural invariant surface associated with the billiard game on an irrational polygonal table is homeomorphic to the Loch Ness monster, that is, the only orientable infinite genus topological real surface with exactly one end.

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of GL + (2, R) avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surface is either countable, or one of three specific types. * Partially supported by MNiSW grant N201 012 32/0718 and the Foundation for Polish Science.† Partially supported by Landesstiftung Baden-Württemberg.‡ Partially supported by Sonderforschungsbereich/Transregio 45 and ANR Symplexe.