Let be Legendrian submanifolds in the cosphere bundle . Given a Lagrangian cobordism of Legendrians from to , we construct a functor
We show that, for , any action of a finite cyclic group by -bilipschitz homeomorphisms on a closed 3-manifold is conjugated to a smooth action.
The Torelli group of a genus oriented surface is the subgroup of the mapping class group consisting of all mapping classes that act trivially on . The quotient group is isomorphic to the symplectic group . The cohomological dimension of the group equ
We give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger–Guéritaud–Kassel: we show that convex cocompactness in is equivalent to an expansion property of the group about its limit set, occurring in different Grassmannians. As an application, we give a sufficient and necessary condition for convex cocompactness for groups that are hyperbolic relative to a collection of convex cocompact subgroups. We show that convex cocompactness in this situation is equivalent to the existence of an equivariant homeomorphism from the Bowditch boundary to the quotient of the limit set of the group by the limit sets of its peripheral subgroups.
In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting.
Let be a compact orientable surface of genus with marked points in the interior. Franks–Handel (Proc. Amer. Math. Soc. 141 (2013) 2951–2962) proved that if then the image of a homomorphism from the mapping class group of to is trivial if and is finite cyclic if . The first result is our own proof of this fact. Our second main result shows that for up to conjugation there are only two homomorphisms from to
Let be a Kähler form on the real 4-torus . Suppose that satisfies an irrationality condition that can be achieved by an arbitrarily small perturbation of . This note shows that the smoothly trivial symplectic mapping class group of the one-point symplectic blowup of is infinitely generated.
Let be an orientable connected -dimensional manifold with and let be a two-sided closed connected incompressible hypersurface that does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of and are either both spin or both nonspin. Using Gromov's -bubbles, we show that does not admit a complete metric of psc. We provide an example showing that the spin/nonspin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension 7, a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension 2. We deduce as special cases that, if does not admit a metric of psc and , then
We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the “dimension problem” by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary the torsion of the homotopy group into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, as for every prime , there is some -torsion in by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group
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