9 jun 2022 -- 17:00
Aula Tricerri, DIMaI, Firenze
Seminario di Geometria del Dini
Abstract.
It is a classical problem to determine the relations between the geometry and topology of manifolds or, more specifically in our case, whether a certain geometric property imposes constraints on the fundamental group of a manifold. A standard instance is represented by the restrictions prescribed by positive curvature properties. A Sasaki structure consists of a contact form together with a compatible Riemannian metric and a transverse complex structure, which define a transverse K\"ahler structure for the Reeb foliation. Therefore, these structures are regarded as the odd-dimensional analogs to K\"ahler structures. In light of this, it is natural to ask whether fundamental groups of Sasakian manifolds form a special class of groups, as it happens in the K\"{a}hler setting. During this talk we will give an affirmative answer to this question and focus on certain properties of fundamental groups of Sasakian manifolds (Sasaki groups). Namely, we will see that these groups share several restrictions with fundamental groups of K\"{a}hler manifolds (K\"{a}hler groups). On the other hand, arguably more interestingly, we will show that Sasaki groups do not satisfy some of the most immediate properties of K\"{a}hler groups.