# Tree lattices by Hyman Bass

By Hyman Bass

Team activities on bushes provide a unified geometric approach of recasting the bankruptcy of combinatorial team thought facing loose teams, amalgams, and HNN extensions. a few of the imperative examples come up from rank one basic Lie teams over a non-archimedean neighborhood box performing on their Bruhat--Tits timber. particularly this ends up in a robust procedure for learning lattices in such Lie teams. This monograph extends this method of the extra normal research of $X$-lattices $\Gamma$, the place $X$ is a in the neighborhood finite tree and $\Gamma$ is a discrete crew of automorphisms of $X$ of finite covolume. those "tree lattices" are the most item of analysis. certain consciousness is given to either parallels and contrasts with the case of Lie teams. past the Lie workforce connection, the speculation has purposes to combinatorics and quantity thought. The authors current a coherent survey of the consequences on uniform tree lattices, and a (previously unpublished) improvement of the speculation of non-uniform tree lattices, together with a few primary and lately proved lifestyles theorems. Non-uniform tree lattices are even more complex than unifrom ones; hence a great deal of awareness is given to the development and examine of various examples. a few fascinating new phenomena are saw right here which can't ensue when it comes to Lie teams. the elemental approach is the encoding of tree activities by way of the corresponding quotient "graph of groups." {\it Tree Lattices} will be a worthwhile source to researchers within the box, and will even be used for a graduate direction in geometric crew concept.

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