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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:
22E40 Discrete subgroups of Lie groups
20E08 Groups acting on trees
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