Tree Lattices |
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| Author:
| Bass, Hyman
Lubotzky, Alexander |
| Appendix by:
| Carbone, L.
Rosenberg, G.
Tits, J. |
| Series title: | Progress in Mathematics Ser. |
| ISBN: | 978-1-4612-7413-1 |
| Publication Date: | Sep 2011 |
| Publisher: | Birkhäuser Boston
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| Imprint: | Birkhäuser |
| Book Format: | Paperback |
| List Price: | USD $54.99 |
Book Description:
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[UPDATED 6/6/2000]Group actions on trees furnish a unified geometric way of recastingthe chapter of combinatorial group theory dealing with free groups,amalgams, and HNN extensions. Some of the principal examples arisefrom rank one simple Lie groups over a non-archimedean local fieldacting on their Bruhat--Tits trees. In particular this leads to apowerful method for studying lattices in such Lie groups.This monograph extends this approach to the more general investigationof...
More Description[UPDATED 6/6/2000]Group actions on trees furnish a unified geometric way of recastingthe chapter of combinatorial group theory dealing with free groups,amalgams, and HNN extensions. Some of the principal examples arisefrom rank one simple Lie groups over a non-archimedean local fieldacting on their Bruhat--Tits trees. In particular this leads to apowerful method for studying lattices in such Lie groups.This monograph extends this approach to the more general investigationof $X$-lattices $\Gamma$, where $X$ is a locally finite tree and$\Gamma$ is a discrete group of automorphisms of $X$ of finitecovolume. These "tree lattices" are the main object of study.Special attention is given to both parallels and contrasts with thecase of Lie groups. Beyond the Lie group connection, the theory hasapplications to combinatorics and number theory.The authors present a coherent survey of the results on uniform treelattices, and a (previously unpublished) development of the theory ofnon-uniform tree lattices, including some fundamental and recentlyproved existence theorems. Non-uniform tree lattices are much morecomplicated than unifrom ones; thus a good deal of attention is givento the construction and study of diverse examples. Some interestingnew phenomena are observed here which cannot occur in the case of Liegroups. The fundamental technique is the encoding of tree actions interms of the corresponding quotient "graph of groups."{{\it Tree Lattices}} should be a helpful resource to researchers in thefield, and may also be used for a graduate course in geometric grouptheory.