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Tree lattices. With appendices by H. Bass, L. Carbone, A. Lubotzky, G. Rosenberg, and J. Tits. (English) Zbl 1053.20026
Progress in Mathematics (Boston, Mass.) 176. Boston, MA: Birkhäuser (ISBN 0-8176-4120-3/hbk). xii, 233 p. (2001).
Let $$X$$ be a locally finite tree with the vertex set $$VX$$, and let for the vertex $$x\in VX$$ the group $$G_x$$ be its stabilizer in the locally compact group $$G=\operatorname{Aut}(X)$$. A subgroup $$\Gamma\leq G$$ is said to be discrete if $$\Gamma_x$$ is finite for some, and hence every vertex $$x\in VX$$. Then we can define the sum $$\text{Vol}(\Gamma\backslash\backslash X)=\sum_{x\in\Gamma\setminus VX}|\Gamma_x|^{-1}$$. $$\Gamma$$ is defined to be an $$X$$-lattice if $$\text{Vol}(\Gamma\backslash\backslash X)<\infty$$, and it is defined to be a uniform $$X$$-lattice if $$\Gamma\setminus X$$ is a finite graph. If $$G\setminus X$$ is finite, these are equivalent to $$\Gamma$$ being a lattice and uniform lattice, respectively. The current investigation studies these tree lattices, and it in some sense is a sequel to the work of H. Bass and R. Kulkarni [J. Am. Math. Soc. 3, No. 4, 843-902 (1990; Zbl 0734.05052)], where the study of the uniform tree lattices was initiated. Here the work is focused much more on the non-uniform case which is more complex. This approach is the generalization of the following. A non-compact simple Lie group $$H$$ in most cases can be realized as an essentially full isometry group of some contractible metric space $$X$$. The geometric action is a natural way to study $$H$$ and its discrete subgroups, in particular its lattices.
Chapter 1 provides a brief summary for basic facts about lattices in locally compact groups. Chapter 2 reviews some basic material on graphs of groups and edge-indexed graphs. Chapter 3 presents the notion of tree lattices and their basic properties. Chapter 4 shows that if $$X$$ is a locally finite tree and $$\Gamma$$ is a non-uniform $$X$$-lattice, then, even when $$X$$ is regular of degree $$\geq 3$$, $$\text{Vol}(\Gamma\backslash\backslash X)$$ can take any positive real value, that $$\Gamma\setminus X$$ can have any conceivable number of “cusps” (in some appropriate sense). Chapter 5 reviews the properties of hyperbolic length of tree automorphisms, and the information it carries about tree actions. Chapter 6 considers the centralizer $$Z_G(\Gamma)$$, the normalizer $$N_G(\Gamma)$$, and the commensurator $$C_G(\Gamma)$$ of the subgroup $$\Gamma$$ in the group $$G$$, where $$G$$ is a topological group. In particular, $$G=\operatorname{Aut}(X)$$, where $$X$$ is a tree. Chapter 7 considers the problem of existence of a uniform or non-uniform lattice for $$X$$. The answer is given in terms of properties of the edge-indexed quotient graph $$I(G\backslash\backslash X)$$. Chapter 8 considers non-uniform lattices on uniform trees and proves some special cases of Carbone’s Theorem. Chapter 9 studies specially defined parabolic trees, actions and lattices. Chapter 10 considers the lattices of Nagao type.
Three appendices conclude the volume. Appendix [BCR] determines conditions that ensure that (in notations used above) for the locally finite tree $$X$$ the group $$G=\operatorname{Aut}(X)$$ contains $$X$$-lattices. Appendix [BT] considers criteria for the cases when for the locally finite tree $$X$$ the locally compact group $$G=\operatorname{Aut}(X)$$ is discrete. Appendix [PN] studies a group constructed by Peter Neumann. It is used to produce certain self-normalizing non-uniform tree lattices.

##### MSC:
 20E08 Groups acting on trees 20-02 Research exposition (monographs, survey articles) pertaining to group theory 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F65 Geometric group theory 22E40 Discrete subgroups of Lie groups 22D05 General properties and structure of locally compact groups 57M15 Relations of low-dimensional topology with graph theory 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 57M07 Topological methods in group theory