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Projective aspects of the Higman-Thompson group. (English) Zbl 0860.57038

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. 633-644 (1991).
The author studies some subgroups of the Higman-Thompson group \(G\), a group of homeomorphisms of the circle \(S^1\). Using the fact that \(S^1= \mathbb{R}\cup \{\infty\}\) and \(\text{PSL}_2 \mathbb{Z}\) acts on \(S^1\) by linear fractional transformations, the author considers \(G\) as the group of homeomorphisms \(g: S^1\to S^1\) such that the circle can be split into some arcs upon which \(g\) acts via elements of \(\text{PSL}_2 \mathbb{Z}\). The Higman-Thompson group \(G\) contains the subgroup \(F\) consisting of elements fixing \(\infty\). For any subgroup \(\Gamma\) of \(\text{PSL}_2 \mathbb{Z}\), the author defines subgroups \(G_\Gamma\) and \(F_\Gamma\) of \(G\) assuming that on the arcs in question \(g\) acts via elements of \(\Gamma\). Then the author defines a tree on which \(G_\Gamma\) acts “at infinity”. The main application of the developed ideas concerns the free action of the lattice \(\Gamma\) on the hyperbolic plane \(\mathbb{H}\). The result asserts that if the genus of the quotient \(\mathbb{H}/\Gamma\) is positive, \(G_\Gamma\) and \(F_\Gamma\) are not finitely generated. If the genus of the quotient \(\mathbb{H}/\Gamma\) vanishes, the question whether \(G_\Gamma\) and \(F_\Gamma\) are finitely generated remains open.
For the entire collection see [Zbl 0809.00017].

MSC:

57S20 Noncompact Lie groups of transformations
57S10 Compact groups of homeomorphisms
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