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Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2001.
Subgroup growth studies the distribution of subgroups of finite
index in a group as a function of the index. In the last two
decades this topic has developed into one of the most active areas
of research in infinite group theory; this book is a systematic and
comprehensive account of the substantial theory which has emerged.
As well as determining the range of possible 'growth types', for
finitely generated groups in general and for groups in particular
classes such as linear groups, a main focus of the book is on the
tight connection between the subgroup growth of a group and its
algebraic structure. A wide range of mathematical disciplines play
a significant role in this work: as well as various aspects of
infinite group theory, these include finite simple groups and
permutation groups, profinite groups, arithmetic groups and Strong
Approximation, algebraic and analytic number theory, probability,
and p-adic model theory. Relevant aspects of such topics are
explained in self-contained 'windows'.
This monograph extends this approach to the more general
investigation of X-lattices, and these "tree lattices" are the main
object of study. The authors present a coherent survey of the
results on uniform tree lattices, and a (previously unpublished)
development of the theory of non-uniform tree lattices, including
some fundamental and recently proved existence theorems. Tree
Lattices should be a helpful resource to researchers in the field,
and may also be used for a graduate course on geometric methods in
group theory.
Robert J. Zimmer is best known in mathematics for the highly
influential conjectures and program that bear his name. Group
Actions in Ergodic Theory, Geometry, and Topology: Selected Papers
brings together some of the most significant writings by Zimmer,
which lay out his program and contextualize his work over the
course of his career. Zimmer's body of work is remarkable in that
it involves methods from a variety of mathematical disciplines,
such as Lie theory, differential geometry, ergodic theory and
dynamical systems, arithmetic groups, and topology, and at the same
time offers a unifying perspective. After arriving at the
University of Chicago in 1977, Zimmer extended his earlier research
on ergodic group actions to prove his cocycle superrigidity theorem
which proved to be a pivotal point in articulating and developing
his program. Zimmer's ideas opened the door to many others, and
they continue to be actively employed in many domains related to
group actions in ergodic theory, geometry, and topology. In
addition to the selected papers themselves, this volume opens with
a foreword by David Fisher, Alexander Lubotzky, and Gregory
Margulis, as well as a substantial introductory essay by Zimmer
recounting the course of his career in mathematics. The volume
closes with an afterword by Fisher on the most recent developments
around the Zimmer program.
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