About this book
Group actions on trees furnish a unified geometric way of recasting the chapter of combinatorial group theory dealing with free groups, amalgams, and HNN extensions. Some of the principal examples arise from rank one simple Lie groups over a non-archimedean local field acting on their Bruhat—Tits trees. In particular this leads to a powerful method for studying lattices in such Lie groups.
This monograph extends this approach to the more general investigation of X-lattices G, where X-is a locally finite tree and G is a discrete group of automorphisms of X of finite covolume. These "tree lattices" are the main object of study. Special attention is given to both parallels and contrasts with the case of Lie groups. Beyond the Lie group connection, the theory has application to combinatorics and number theory.
The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems. Non-uniform tree lattices are much more complicated than uniform ones; thus a good deal of attention is given to the construction and study of diverse examples. The fundamental technique is the encoding of tree action in terms of the corresponding quotient "graphs of groups."
Tree Lattices should be a helpful resource to researcher sin the field, and may also be used for a graduate course on geometric methods in group theory.
- Book Title Tree Lattices
- Series Title Progress in Mathematics
- Series Abbreviated Title Progress in Mathematics(Birkhäuser)
- DOI https://doi.org/10.1007/978-1-4612-2098-5
- Copyright Information Birkhäuser Boston 2001
- Publisher Name Birkhäuser Boston
- eBook Packages Springer Book Archive
- Hardcover ISBN 978-0-8176-4120-7
- Softcover ISBN 978-1-4612-7413-1
- eBook ISBN 978-1-4612-2098-5
- Series ISSN 0743-1643
- Series E-ISSN 2296-505X
- Edition Number 1
- Number of Pages XIII, 233
- Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
Group Theory and Generalizations
Order, Lattices, Ordered Algebraic Structures
Topological Groups, Lie Groups
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"The book is a helpful resource to researchers in the field and students of geometric methods in group theory."
--Educational Book Review