Articles/Preprints

Let S be an orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S), and the Schmutz graph G(S) of S. When all topological ends of S carry genus, we show that all elements in the automorphism groups Aut(C(S)), Aut(N(S)), and Aut(G(S)) are geometric, i.e. these groups are naturally isomorphic to the extended mapping class group MCG*(S) of the infinite surface S. Finally, we study rigidity phenomena within Aut(C(S)) and Aut(N(S)).

For an orientable surface S of finite topological type with genus g ≥ 3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph C(S). The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger’s finite rigid set in [Aramayona and Leininger, J. Topology Anal. 5(2) (2013), 183–203] and [Aramayona and Leininger, Pac. J. Math. 282(2) (2016), 257–283], and in fact a consequence of our proof is that Aramayona and Leininger’s set also exhausts the curve graph via rigid expansions.

Suppose S1 and S2 are orientable surfaces of finite topological type such that S1 has genus at least 3 and the complexity of S1 is an upper bound of the complexity of S2 . Let φ : C(S1) → C(S2) be an edge-preserving map; then S1 is homeomorphic to S2 , and in fact φ is induced by a homeomorphism. To prove this, we use several simplicial properties of rigid expansions, which we prove here.

Let S1 and S2 be connected orientable surfaces of genus g1, g2 ≥ 3, n1 , n2 ≥ 0 punctures, and empty boundary. Let also φ : HT(S1) → HT(S2) be an edge-preserving alternating map between their Hatcher–Thurston graphs. We prove that g1 ≤ g2 and that there is also a multicurve of cardinality g2 − g1 contained in every element of the image. We also prove that if n1 = 0 and g1 = g2 , then the map φ obtained by filling the punctures of S2 , is induced by a homeomorphism of S1.

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 an 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.

Let φ : C(S) → C(S′) be a simplicial isomorphism between curve graphs of infinite-type surfaces. In this paper, we show that, in this situation, S and S′ are homeomorphic and φ is induced by a homeomorphism h : S → S′.

Let S1 and S2 be orientable surfaces of finite topological type with empty boundary, both of genus at least 1 and n1, n2 ≥ 0 punctures. We define the nonseparating and outer curve graph NO(Si): a particular induced subgraph of the curve graph, which has the same large-scale geometry. We prove (under certain conditions on the complexity of S1 and S2) that if φ : NO(S1) → NO(S2) is a superinjective map, then S1 is homeomorphic to S2 and φ is induced by a homeomorphism.

Let Sg,n be an orientable surface of genus g with n punctures. We identify a finite rigid subgraph Xg,n of the pants graph P(Sg,n), that is, a subgraph with the property that any simplicial embedding of Xg,n into any pants graph P(Sg',n') is induced by an embedding Sg,n → Sg',n'. This extends results of the third author for the case of genus zero surfaces.

We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface Σ on itself. Our main results are:

(1) All conjugacy classes of MCG(Σ) are meager for every Σ,

(2) MCG(Σ) has a somewhere dense conjugacy class if and only if Σ has at most two maximal ends and no non-displaceable finite-type subsurfaces,

(3) MCG(Σ) has a dense conjugacy class if and only if Σ has a unique maximal end and no non-displaceable finite-type subsurfaces.

Our techniques are based on model-theoretic methods developed by Kechris, Rosendal and Truss. 

We explore the use of Costa and Farber model for random simplicial complexes to give probabilistic evidence for versions of Ivanov's Rigidity metaconjecture of the Mapping Class Group. 

In this work we compute the first integral cohomology of the pure mapping class group of a non-orientable surface of infinite topological type and genus at least 3. To this purpose, we also prove several other results already known for orientable surfaces such as the existence of an Alexander method, the fact that the mapping class group is isomorphic to the automorphism group of the curve graph along with the topological rigidity of the curve graph, and the structure of the pure mapping class group as both a Polish group and a semi-direct product.

This work is the extension of the results by the author in [7] and [6] for low-genus surfaces. Let S be an orientable, connected surface of finite topological type, with genus g≤2, empty boundary, and complexity at least 2; as a complement of the results of [6], we prove that any graph endomorphism of the curve graph of S is actually an automorphism. Also, as a complement of the results in [6] we prove that under mild conditions on the complexity of the underlying surfaces any graph morphism between curve graphs is induced by a homeomorphism of the surfaces.

To prove these results, we construct a finite subgraph whose union of iterated rigid expansions is the curve graph (S). The sets constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid sets in [2]. Similarly to [7], a consequence of our proof is that Aramayona and Leininger's rigid set also exhausts the curve graph via rigid expansions, and the combinatorial rigidity results follow as an immediate consequence, based on the results in [6]. 

We show that the open genus 2 handlebody admits uncountably-many fibrations over the circle with fiber homeomorphic to the Cantor tree surface with non-conjugate monodromies in the mapping class group. The construction generalizes to produce uncountably many fibrations of other tame 3-manifolds with infinite type fibers, including the blooming Cantor tree and many other types of surfaces.