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Superrigidity for the commensurability group of tree lattices. (English) Zbl 0839.22011
If $$T = T_k$$ is a regular $$k$$-tree $$(k \geq 3)$$, then its automorphism group $$A$$ is a locally compact group which has a number of properties which are analogous to properties enjoyed by simple Lie groups, particularly with respect to lattice subgroups $$\Gamma$$. In H. Bass and R. Kulkarni [J. Am. Math. Soc. 3, 843-902 (1990; Zbl 0734.05052)], it was shown that any two uniform lattices are commensurable up to conjugation, so that the commensurability subgroup $$C$$ of such a lattice is uniquely determined by $$A$$ up to conjugation. Furthermore $$C$$ is dense in $$A$$.
In this paper, the main result is to show that if $$\rho : C \to \operatorname{Aut} (T')$$ defines a minimal action on some tree $$T'$$, then either $$\rho$$ can be extended to an action of $$A$$ or the image of each element of $$\Gamma$$ fixes an edge or a vertex so that $$\overline {\rho (\Gamma)}$$ is compact. In addition, by exhibiting, for each automorphism $$\rho$$ of $$C$$, a uniform lattice $$\Gamma$$ such that $$\overline {\rho (\Gamma)}$$ cannot be compact, it is shown that each automorphism of $$C$$ is geometric, i.e. induced by an element of $$A$$.
The authors offer two proofs of the main result. The first is a graph-theoretic proof, while the second uses methods of ergodic theory. The second proof gives an extension to non-uniform lattices. A number of other consequences of these results is given, together with very clear indications of what remains to be proved and how recent results in this area fit together.

##### MSC:
 2.2e+41 Discrete subgroups of Lie groups 2e+09 Groups acting on trees
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