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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/223485
Large and iterated finite group actions on aspherical manifolds
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[eng] Finite transformation group theory investigates the finite symmetries of topological objects, such as manifolds or CW-complexes. In this thesis, we focus on actions on closed topological manifolds and adopt the following approach: instead of directly studying the action properties of a finite group G to a manifold M, we focus on the properties of action restricted to certain subgroups H of bounded index. Several problems align with this philosophy, such as determining whether the group of homeomorphisms of a manifold is Jordan, calculating the discrete degree of symmetry of a manifold, determining whether a manifold is quasi-asymmetric, and studying the number and size of isotropy subgroups for finite group actions on manifolds. In the first part of the thesis, we provide solutions to these problems for two general classes of manifolds, namely: (1) Closed, connected and aspherical manifolds, whose fundamental group has a group of external Minkowski automorphisms (a group G is Minkowski if there exists a constant C such that every finite subgroup H of G has order at most C). (2) Closed, connected and orientable manifolds that admit a non-zero degree application to a nilmanifold. We show that the group of external automorphisms of a lattice of a connected Lie group is Minkowski, which allows us to apply our results to locally homogeneous aspherical closed manifolds. In addition, we provide the earliest known examples of manifolds M and M' with isomorphic cohomology rings such that Homeo(M) is Jordan but Homeo(M') is not. We establish two stiffness results for the discrete degree of symmetry: if M is a closed, connected, aspherical manifold and the external automorphism group of the fundamental group of M is Minkowski, or if M admits a non-zero degree application to a nilmanifold and its fundamental group is virtually solvable, then M is homeomorphic to a torus if its discrete degree of symmetry is equal to the dimension of M. In the second part, we refine the concept of group actions to explore in greater depth the topological and cohomology rigidity of closed and connected manifolds. This framework allows us to analyze in more detail the structure of closed aspherical manifolds and those that admit a non-zero degree application to a nilmanifold. We define new invariants, such as the iterated length of a manifold, which is closely related to its self-coatings, and introduce a refined version of the discrete degree of symmetry, called the discrete degree of iterate symmetry. We show that if M is a closed, oriented manifold that admits a non-zero degree application to a nilmanifold of nilpotency class 2, and both manifolds have the same discrete degree of iterated symmetry, then the rational cohomology of M is isomorphic to that of the nilmanifold. Furthermore, if the fundamental group of M is virtually solvable, then M is homeomorphic to the nilmanifold. We also prove that if M is a locally homogeneous closed aspherical manifold with a discrete degree of iterated symmetry equal to its dimension, then M is homeomorphic to a nilmanifold of nilpotency class 2.
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Matèries (anglès)
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DAURA SERRANO, Jordi. Large and iterated finite group actions on aspherical manifolds. [consulta: 30 de novembre de 2025]. [Disponible a: https://hdl.handle.net/2445/223485]