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We classify static manifolds which admit more than one static decomposition whenever a condition on the curvature is fullfilled. For this, we take a standard static vector field and analyze its associated one parameter family of projections…
The mapping class group of a compact oriented surface of genus greater than one with boundary acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group, for every choice of…
We give an infinite presentation for the mapping class group of a non-orientable surface with boundary components. The presentation is a generalization of the presentation given by the second author [15].
It is proved that the mapping class group of any closed surface with finitely many marked points is quasiisometric to a CAT(0) cube complex. We provide two distinct proofs, one tailored to mapping class groups, and one applying to a larger…
In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between…
We show that the mapping class group of any closed connected orientable surface of genus at least five is generated by only two commutators, and if the genus is three or four, by three commutators.
We compute the mapping class group-valued monodromy of any sufficiently ample linear system on any smooth simply connected projective surface, identifying this with the r-spin mapping class group associated to a maximal root of the adjoint…
We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of…
Eight different refinements of trapped surfaces are proposed, of three basic types, each intended as potential stability conditions. Minimal trapped surfaces are strictly minimal with respect to the dual expansion vector. Outer trapped…
For $\Sigma$ an orientable surface of finite topological type having genus at least 3 (possibly closed or possibly with any number of punctures or boundary components), we show that the mapping class group $Mod(\Sigma)$ has no faithful…
We show that the extended based mapping class group of an infinite-type surface is naturally isomorphic to the automorphism group of the loop graph of that surface. Additionally, we show that the extended mapping class group stabilizing a…
We study the action of the mapping class group M(F) on the complex of curves of a non-orientable surface F. We obtain, by using a result of K. S. Brown, a presentation for M(F) defined in terms of the mapping class groups of the…
A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $\Pi$ is considered. A problem of determination of such surfaces when both principal curvatures are given…
Let $S$ be a hyperbolic oriented Riemann surface of finite type. The main purpose of this paper is to show that non-trivial geometric intersection between closed curves on $S$ is detected by some symplectic submodules they naturally…
Omori and the author have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group.
We characterize helix surfaces (constant angle surfaces) in the special linear group $\mathrm{SL}(2,\r)$. In particular, we give an explicit local description of these surfaces in terms of a suitable curve and a 1-parameter family of…
We show that several families of asymptotically rigid mapping class groups arise as explicit quotients of the fundamental group of a graph of groups, with mapping class groups as vertex and edge stabilizers. Using this description, and…
We show that the mapping class group of a handlebody of genus at least 2 (with any number of marked points or spots) is exponentially distorted in the mapping class group of its boundary surface. The same holds true for solid tori with at…
We produce a sequence of finite dimensional representations of the fundamental group $\pi_1(S)$ of a closed surface where all simple closed curves act with finite order, but where each non--simple closed curve eventually acts with infinite…
We define the notion of a hierarchically cocompact classifying space for a family of subgroups of a group. Our main application is to show that the mapping class group $\mbox{Mod}(S)$ of any connected oriented compact surface $S$, possibly…