Subset Currents on Surfaces

• Record Nr.: UNINA9910957081003321
• Autore: Sasaki Dounnu
• Pubbl/distr/stampa: Providence : , : American Mathematical Society, , 2022; ©2022
• ISBN: 9781470471682; 9781470453435
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (178 pages)
• Collana: Memoirs of the American Mathematical Society ; ; v.278
• Classificazione: 20F6730F35
• Disciplina: 515/.9; 515.9
• Soggetti: Fuchsian groups; Riemann surfaces; Hyperbolic groups; Ergodic theory; Group theory and generalizations -- Special aspects of infinite or finite groups -- Hyperbolic groups and nonpositively curved groups; Functions of a complex variable -- Riemann surfaces -- Fuchsian groups and automorphic functions
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Cover -- Title page -- Chapter 1. Introduction -- 1.1. Background -- 1.2. Main results -- 1.3. Future study -- 1.4. Organization of article -- 1.5. Acknowledgements -- Chapter 2. Subset Currents on Hyperbolic Groups -- 2.1. Space of subset currents on hyperbolic group -- 2.2. Measure theory background -- Chapter 3. Volume Functionals on Kleinian Groups -- Chapter 4. Subgroups, Inclusion Maps and Finite Index Extension -- 4.1. Natural continuous linear maps between subgroups -- 4.2. Finite index extension of functionals -- Chapter 5. Intersection Number -- 5.1. Intersection number of closed curves -- 5.2. Intersection number of surfaces -- 5.3. Continuous extension of intersection number -- Chapter 6. Intersection Functional on Subset Currents -- Chapter 7. Projection from Subset Currents onto Geodesic Currents -- 7.1. Construction of projection -- 7.2. Application of projection -- Chapter 8. Denseness Property of Rational Subset ; Currents -- 8.1. Denseness property of free groups -- 8.2. Approximation by a sequence of subgroups -- 8.3. Denseness property of surface groups -- Bibliography -- Index -- Back Cover.
• Sommario/riassunto: "Subset currents on hyperbolic groups were introduced by Kapovich and Nagnibeda as a generalization of geodesic currents on hyperbolic groups, which were introduced by Bonahon and have been successfully studied in the case of the fundamental group 1() of a compact hyperbolic surface . Kapovich and Nagnibeda particularly studied subset currents on free groups. In this article, we develop the theory of subset currents on 1(), which we call subset currents on . We prove that the space SC() of subset currents on is a measure-theoretic completion of the set of conjugacy classes of non-trivial finitely generated subgroups of 1(), each of which geometrically corresponds to a convex core of a covering space of . This result was proved by Kapovich-Nagnibeda in the case of free groups, and is also a generalization of Bonahon's result on geodesic currents on hyperbolic groups. We will also generalize several other results of them. Especially, we extend the (geometric) intersection number of two closed geodesics on to the intersection number of two convex cores on and, in addition, to a continuous R0-bilinear functional on SC()"--