Hyman Bass Livres

The book explores group actions on trees, offering a geometric perspective on combinatorial group theory, particularly concerning free groups, amalgams, and HNN extensions. It focuses on "tree lattices," which are discrete groups acting on locally finite trees, and examines their parallels and differences with Lie groups. The authors provide a comprehensive overview of uniform and non-uniform tree lattices, highlighting complex examples and new phenomena unique to this study. This resource is aimed at researchers and can serve as a graduate course text in geometric group theory.

The theme of the monograph is an interplay between dynamical systems and group theory. The authors formalize and study "cyclic renormalization", a phenomenon which appears naturally for some interval dynamical systems. A possibly infinite hierarchy of such renormalizations is naturally represented by a rooted tree, together with a "spherically transitive" automorphism; the infinite case corresponds to maps with an invariant Cantor set, a class of particular interest for its relevance to the description of the transition to chaos and of the Mandelbrot set. The normal subgroup structure of the automorphism group of such "spherically homogeneous" rooted trees is investigated in some detail. This work will be of interest to researchers in both dynamical systems and group theory.