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.

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.

Reviewer: Vahagn H. Mikaelian (Yerevan)

##### 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 |