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Über dieses Buch

This volume presents the lecture notes from the authors’ three summer courses offered during the program “Automorphisms of Free Groups: Geometry, Topology, and Dynamics,” held at the Centre de Recerca Matemàtica (CRM) in Bellaterra, Spain.

The first two chapters present the basic tools needed, from formal language theory (regular and context-free languages, automata, rewriting systems, transducers, etc) and emphasize their connections to group theory, mostly relating to free and virtually-free groups. The material covered is sufficient to present full proofs of many of the existing interesting characterizations of virtually-free groups. In turn, the last chapter comprehensively describes Bonahon’s construction of Thurston’s compactification of Teichmüller space in terms of geodesic currents on surfaces. It also includes several intriguing extensions of the notion of geodesic current to various other, more general settings.



Chapter 1. An Automata-Theoretic Approach to the Study of Fixed Points of Endomorphisms

This chapter contains an extended version of the contents of the three hour course Fixed points of virtually free group endomorphisms of the Summer School on Automorphisms of Free Groups, held at the Centre de Recerca Matemàtica (CRM), Bellaterra, Barcelona, from the 25th to the 29th of September 2012.
Pedro Silva

Chapter 2. Context-Free Groups and Bass–Serre Theory

The word problem of a finitely generated group is the set of words over the generators that are equal to the identity in the group. The word problem is therefore a formal language. If this language happens to be context-free, then the group is called context-free. Finitely generated virtually free groups are context-free. In the seminal paper Muller–Schupp [38] the converse was shown: every context-free group is virtually free. Over the past decades a wide range of other characterizations of context-free groups have been found. It underlines that context-free groups play a major role in combinatorial group theory.
Volker Diekert, Armin Weiß

Chapter 3. Hyperbolic Structures on Surfaces and Geodesic Currents

This chapter contains the lecture notes from the course “Hyperbolic structures on surfaces and geodesic currents”, given by the authors during the summer school on Automorphisms of Free Groups: Geometry, Topology, and Dynamics, held at the CRM (Barcelona) in September 2012. The main objective of the notes is to give an account of Bonahon’s description [4] of Thurston’s compactification of Teichmüller space in terms of geodesic currents on surfaces. The plan of the chapter is as follows. Section 3.2 deals with hyperbolic structures on surfaces, explaining why a surface equipped with a complete hyperbolic structure is isometric to the quotient of H2 by a Fuchsian group. In Section 3.3 we will review some basic features of Teichmüller spaces and measured geodesic laminations, ending with some words about the “classical” construction of Thurston’s compactification. In Section 3.4, we will introduce geodesic currents, and explain Bonahon’s interpretation of the compactification of Teichmüller space. Finally, in Section 3.5 we will present some generalizations of the notion of geodesic currents to other settings, such as negatively curved metrics on surfaces, flat metrics on surfaces, and free groups.
Javier Aramayona, Christopher J. Leininger
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