ترغب بنشر مسار تعليمي؟ اضغط هنا

The first integral cohomology of pure mapping class groups

178   0   0.0 ( 0 )
 نشر من قبل Nicholas Vlamis
 تاريخ النشر 2017
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surfaces simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.



قيم البحث

اقرأ أيضاً

142 - Nicholas G. Vlamis 2020
We prove that the mapping class group of a surface obtained from removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk has no proper countable-index subgroups. The proof is an application of the automatic con tinuity of these groups, which was established by Mann. As corollaries, we see that these groups do not contain any proper finite-index subgroups and that each of these groups have trivial abelianization.
We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt using different methods). As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG(S); a characterization of the image of the curve-complex projection map from MCG(S) to the product of the curve complexes of essential subsurfaces of S; and a construction of Sigma-hulls in MCG(S), an analogue of convex hulls.
We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of th e paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the circle. The third part focuses on some fruitful relations between the dynamics of the two actions. For example, we prove that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). In addition, we are able to construct non trivial quasimorphisms on many subgroups of big mapping class groups, even if they are not acylindrically hyperbolic.
We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا