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Register a new userFinite rigid sets and the separating curve complex
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For each closed surface of genus $gge3$, we find a finite subcomplex of the separating curve complex that is rigid with respect to incidence-preserving maps.
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Aramayona and Leininger have provided a finite rigid subset $mathfrak{X}(Sigma)$ of the curve complex $mathscr{C}(Sigma)$ of a surface $Sigma = Sigma^n_g$, characterized by the fact that any simplicial injection $mathfrak{X}(Sigma) to mathscr{C}(Sigma)$ is induced by a unique element of the mapping class group $mathrm{Mod}(Sigma)$. In this paper we prove that, in the case of the sphere with $ngeq 5$ marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a $mathrm{Mod}(Sigma)$-module generator for the reduced homology of the curve complex $mathscr{C}(Sigma)$, answering in the affirmative a question posed by Aramayona and Leininger. For the surface $Sigma = Sigma_g^n$ with $ggeq 3$ and $nin {0,1}$ we find that the finite rigid set $mathfrak{X}(Sigma)$ of Aramayona and Leininger contains a proper subcomplex $X(Sigma)$ whose reduced homology class is a $mathrm{Mod}(Sigma)$-module generator for the reduced homology of $mathscr{C}(Sigma)$ but which is not itself rigid.
By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve complex.
In this note we make progress toward a conjecture of Durham--Fanoni--Vlamis, showing that every infinite-type surface with finite-invariance index 1 and no nondisplaceable compact subsurfaces fails to have a good curve graph, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and where the natural mapping class group action has infinite diameter orbits. Our arguments use tools developed by Mann--Rafi in their study of the coarse geometry of big mapping class groups.
We introduce the separating semigroup of a real algebraic curve of dividing type. The elements of this semigroup record the possible degrees of the covering maps obtained by restricting separating morphisms to the real part of the curve. We also introduce the hyperbolic semigroup which consists of elements of the separating semigroup arising from morphisms which are compositions of a linear projection with an embedding of the curve to some projective space. We completely determine both semigroups in the case of maximal curves. We also prove that any embedding of a real curve to projective space of sufficiently high degree is hyperbolic. Using these semigroups we show that the hyperbolicity locus of an embedded curve is in general not connected.
Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics, respectively. In the first part of this paper, we present a result on separation of simplices and balls (up to affine equivalence) among all compact convex sets in two- and three-dimensional Euclidean spaces, which focuses on the set of extreme points and the action of affine transformations on it. Regarding the above-mentioned axiomatization of quantum physics, our result corresponds to the case of simplest (2-level) quantum system. We also discuss a possible extension to higher dimensions. In the second part, towards generalizations of the framework of general probabilistic theories and several existing results including ones in the first part from the case of compact and finite-dimensional physical systems as in most of the literatures to more general cases, we study some fundamental properties of convex sets and affine maps that are relevant to the above subject.
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