Dendrology and its applications.

*(English)*Zbl 0843.20018
Ghys, É. (ed.) et al., Group theory from a geometrical viewpoint. Proceedings of a workshop, held at the International Centre for Theoretical Physics in Trieste, Italy, 26 March to 6 April 1990. Singapore: World Scientific. 543-616 (1991).

The author gives a series of lectures about group actions on real trees in the spirit of his earlier expository article [in Publ., Math. Sci. Res. Inst. 8, 265-319 (1987; Zbl 0649.20033)].

Section 1 includes Gupta and Sidki’s construction of finitely generated infinite \(p\)-groups using simplicial trees, and Brown’s interpretation of the Bieri-Neumann invariant in terms of real trees. In Section 2 he describes Bestvina and Handel’s work on outer automorphisms of free groups and its connection with Culler-Vogtmann and Gersten’s outer space and with exotic free actions of free groups on trees; one example is worked out in considerable detail. Section 2 also includes a sketch of Skora’s proof that every small action of a surface groups on a real tree is dual to a measured foliation.

In Section 3 is given an account of the Bruhat-Tits tree for \(\text{SL}_2\) of a valued field, and its application by Lubotzky, Phillips and Sarnak in their work on Ramanujan graphs. He also discusses in some depth his work with Culler on trees associated to ideal points of curves in the character variety of a group and their applications in 3-manifold theory, including the proof of the Cyclic Surgery Theorem by Culler, Gordon, Luecke and himself. In Section 4 he explains the connection between trees and hyperbolic geometry in much greater depth than in [op. cit.], and includes accounts of both the original approach used in his work with Morgan and based on the Bruhat-Tits tree, and the approach of Bestvina and Paulin based on Gromov’s notions of convergence of metric spaces. There is also a brief discussion of Paulin’s work on finiteness of outer automorphism groups. The section concludes with some idle speculations.

In Section 5 the emphasis is on the aspects of his work with Gillet on rank-2 trees that were left out of [op. cit.], especially the notion of strong convergence. This section includes a brand-new, and therefore somewhat tentative, conjecture about how to extend the theory to arbitrary rank. He also discusses some surprising connections between the notion of strong convergence on the one hand, and both Bestvina-Handel theory and the contractibility of outer space on the other.

For the entire collection see [Zbl 0809.00017].

Section 1 includes Gupta and Sidki’s construction of finitely generated infinite \(p\)-groups using simplicial trees, and Brown’s interpretation of the Bieri-Neumann invariant in terms of real trees. In Section 2 he describes Bestvina and Handel’s work on outer automorphisms of free groups and its connection with Culler-Vogtmann and Gersten’s outer space and with exotic free actions of free groups on trees; one example is worked out in considerable detail. Section 2 also includes a sketch of Skora’s proof that every small action of a surface groups on a real tree is dual to a measured foliation.

In Section 3 is given an account of the Bruhat-Tits tree for \(\text{SL}_2\) of a valued field, and its application by Lubotzky, Phillips and Sarnak in their work on Ramanujan graphs. He also discusses in some depth his work with Culler on trees associated to ideal points of curves in the character variety of a group and their applications in 3-manifold theory, including the proof of the Cyclic Surgery Theorem by Culler, Gordon, Luecke and himself. In Section 4 he explains the connection between trees and hyperbolic geometry in much greater depth than in [op. cit.], and includes accounts of both the original approach used in his work with Morgan and based on the Bruhat-Tits tree, and the approach of Bestvina and Paulin based on Gromov’s notions of convergence of metric spaces. There is also a brief discussion of Paulin’s work on finiteness of outer automorphism groups. The section concludes with some idle speculations.

In Section 5 the emphasis is on the aspects of his work with Gillet on rank-2 trees that were left out of [op. cit.], especially the notion of strong convergence. This section includes a brand-new, and therefore somewhat tentative, conjecture about how to extend the theory to arbitrary rank. He also discusses some surprising connections between the notion of strong convergence on the one hand, and both Bestvina-Handel theory and the contractibility of outer space on the other.

For the entire collection see [Zbl 0809.00017].

##### MSC:

20E08 | Groups acting on trees |

57M50 | General geometric structures on low-dimensional manifolds |

57M07 | Topological methods in group theory |