The purpose of this project is to study the mapping class group and its subgroups by looking at their actions on the Teichmueller space and on certain simplicial complexes encoding the combinatorics of ”nice” topological objects on the surfaces (curves, arcs and triangulations). The main protagonists will be the curve complex, the arc complex and the flip graph of topological and geometric surfaces. The curve complex is an important tool in the proof of Thurston’s Ending Lamination Conjecture and in the study of the coarse geometry of the mapping class group. We will be particularly interested in investigating the large-scale geometry and the model-theoretical properties of these graphs and their variations for surfaces endowed with ”good” geometric structures. In this project we want to establish the first bridge between geometric topology, model theory, geometric group theory and dynamics.
This project consists of three parts:
I. combinatorial actions of the mapping class group and rigidity;
II. large scale geometry of complexes of (multi-)arcs and interactions between geometric group theory and dynamics;
III.Thurston’s distance on the Teichmueller space and its generalizations in higher Teichmueller theory.
Part I deals with mapping class groups and the simplicial rigidity problem. In the 1990s Ivanov proved that the mapping class group can be represented as the automorphism group of the curve complex. Subsequently, many other simplicial complexes associated to a surface have exhibited this same feature. Understanding what all these objects have in common is still an open problem (Ivanov’s metaconjecture). Recently Brendle-Margalit characterize one entire family of complexes with this property and formulated a precise conjecture in this regard. I am attacking this problem for multi-arcs and studying the model theory of the curve complex and other graphs associated to mapping class groups together with Thomas Koberda and Javier de la Nuez-Gonzalez.
Part II deals with the geometric group theory of arc complexes, flip graphs, related algorithmic problems and a new application of these tools in dynamics. After foundational works by Masur-Minsky there has been a lot of interest in the large scale properties of the curve complex and other analogue combinatorial complexes. A recent trend in geometric group theory is to look at complexes of arcs and triangulations. Further motivations come from the many applications these objects have in other fields of mathematics (theoretical computer science, cluster algebras, quantum topology, Margulis space times...). I am interested in understanding the large scale geometry of the mapping class group via graphs of triangulations. I am developing a program with Anja Randecker and Robert Tang to employ techniques from geometric group theory in dynamics via the study of simplicial complexes built from saddle connections on a translation surface.
Part III deals with Thurston’s distance on Teichmueller space. Thurston’s distance is an analogue of Teichmueller distance defined by Thurston. Together with Daniele Alessandrini we are generalizing Thurston’s results to surfaces with boundary.
Publications
We prove that the saddle connection graph associated to any half-translation surface is 4–hyperbolic and uniformly quasi-isometric to the regular countably infinite-valent tree. Consequently, the saddle connection graph is not quasi-isometrically rigid. We also characterise its Gromov boundary as the set of straight foliations with no saddle connections. In our arguments, we give a generalisation of the unicorn paths in the arc graph which may be of independent interest.
Related project(s):
44Actions of mapping class groups and their subgroups
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston's stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodesics in the Teichmueller space of a closed/punctured surface that are not stretch lines in the sense of Thurston.
Related project(s):
44Actions of mapping class groups and their subgroups
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston's stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodesics in the Teichmueller space of a closed/punctured surface that are not stretch lines in the sense of Thurston.
Related project(s):
44Actions of mapping class groups and their subgroups
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston's stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodesics in the Teichmueller space of a closed/punctured surface that are not stretch lines in the sense of Thurston.
Related project(s):
44Actions of mapping class groups and their subgroups
For a half-translation surface , the associated saddle connection complex is the simplicial complex where vertices are the saddle connections on , with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that any simplicial isomorphism between saddle connection complexes is induced by an affine diffeomorphism . In particular, this shows that the saddle connection complex is a complete invariant of affine equivalence classes of half-translation surfaces. Throughout our proof, we develop several combinatorial criteria of independent interest for detecting various geometric objects on a half-translation surface.
Related project(s):
44Actions of mapping class groups and their subgroups
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston's stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodesics in the Teichmueller space of a closed/punctured surface that are not stretch lines in the sense of Thurston.
Related project(s):
44Actions of mapping class groups and their subgroups
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston's stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodesics in the Teichmueller space of a closed/punctured surface that are not stretch lines in the sense of Thurston.
Related project(s):
44Actions of mapping class groups and their subgroups
In this paper we develop a bridge between model theory, geometric topology, and geometric group theory. In particular, we investigate the Ivanov Metaconjecture from the point of view of model theory, and more broadly we seek to answer the general question: why does the curve graph of a surface play such a central role in the study of surfaces and mapping class groups?
More specifically, we consider a surface $\Sigma$ of finite type and its curve graph $C(\Sigma)$, and we investigate its first-order theory in the language of graph theory. Crucially, $C(\Sigma)$ is bi-interpretable with a certain object called the augmented Cayley graph of the mapping class group of the surface. We use this bi-interpretation to prove that the theory of the curve graph is \omega--stable, to compute its Morley rank, and to show that it has quantifier elimination with respect to the class of $\forall \exists$--formulae. We also show that many of the complexes which are naturally associated to a surface are interpretable in $C(\Sigma)$. This shows that these complexes are all \omega--stable and admit certain a priori bounds on their Morley ranks. We are able to use Morley ranks to prove that various complexes are not bi--interpretable with the curve graph. As a consequence of quantifier elimination, we show that algebraic intersection number is not definable in the first order theory of the curve graph. Finally, we prove that the curve graph of a surface enjoys a novel phenomenon that we call interpretation rigidity. That is, if surfaces $\Sigma_1$ and $\Sigma_2$ admits curve graphs that are mutually interpretable, then $\sigma_1$ and $\Sigma_2$ are homeomorphic to each other. Along the way, numerous technical results are obtained.
Related project(s):
44Actions of mapping class groups and their subgroups
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston's stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodesics in the Teichmueller space of a closed/punctured surface that are not stretch lines in the sense of Thurston.
Related project(s):
44Actions of mapping class groups and their subgroups
Team Members
Dr. Valentina Disarlo
Project leader
Institute of Science and Technology Austria (ISTA)