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**Voronoi tessellations associated to the Cremona group.
(Pavage de Voronoï associé au groupe de Cremona.)**
*(French.
English summary)*
Zbl 1447.20013

The Cremona group \(\operatorname{Bir}(\mathbb{P}^2)\) is the group of birational transformations of the projective plane over a field \(k\). A faithful action of the Cremona group on an infinite dimensional hyperbolic space \(\mathbb{H}^{\infty}\) is introduced by Manin and Picard. This action has been very useful for studying various properties of the Cremona group. In particular it allows S. Cantat et al. [Acta Math. 210, No. 1, 31–94 (2013; Zbl 1278.14017)] to prove that the Cremona group is not simple. The article under review studies the geometry of this action on \(\mathbb{H}^{\infty}\), more precisely the Voronoï tesselation of \(\mathbb{H}^{\infty}\) associated with this action.

Let \(l\in \mathbb{H}^{\infty}\) be the point corresponding to a line in \(\mathbb{P}^2\). A Voronoï cell associated to a point \(p\) in the orbit of \(l\) is the subset of \(\mathbb{H}^{\infty}\) whose points are closer to \(p\) than any other points in the orbit of \(l\). The stabilizer of \(l\) is \(\operatorname{PGL}_3(k)\). The Voronoï cells are thus indexed by the cosets of \(\operatorname{PGL}_3(k)\) in \(\operatorname{Bir}(\mathbb{P}^2)\). Actually, to see the geometry of the action it suffices to study the Voronoï tesselation in the convex hull generated by the orbit of \(l\). For technical reasons, the author consider in this paper the Voronoï tesselation of a convex subset slightly larger than that convex hull.

For a point \(x\) to be in the cell associated to \(l\), a priori \(x\) needs to satisfy \(d(x,f(p))\geq d(x,p)\) for all \(f\in\operatorname{Bir}(\mathbb{P}^2)\) and it is proved that in fact we just need \(d(x,f(p))\geq d(x,p)\) for all \(f\) in the Jonquières subgroup. The cells that are adjacent to the cell associated to \(l\) are determined, as well as the cells that are quasi-adjacent to the cell associated to \(l\) (quasi-adjacent means adjacent in the boundary of \(\mathbb{H}^{\infty}\)).

One of the motivations for studying the Voronoï tesselation is to look at the action of the Cremona group on the graph of adjacency of the tesselation. This is studied in a second paper of the author where the graph of quasi-adjacency is proved to be Gromov-hyperbolic.

Let \(l\in \mathbb{H}^{\infty}\) be the point corresponding to a line in \(\mathbb{P}^2\). A Voronoï cell associated to a point \(p\) in the orbit of \(l\) is the subset of \(\mathbb{H}^{\infty}\) whose points are closer to \(p\) than any other points in the orbit of \(l\). The stabilizer of \(l\) is \(\operatorname{PGL}_3(k)\). The Voronoï cells are thus indexed by the cosets of \(\operatorname{PGL}_3(k)\) in \(\operatorname{Bir}(\mathbb{P}^2)\). Actually, to see the geometry of the action it suffices to study the Voronoï tesselation in the convex hull generated by the orbit of \(l\). For technical reasons, the author consider in this paper the Voronoï tesselation of a convex subset slightly larger than that convex hull.

For a point \(x\) to be in the cell associated to \(l\), a priori \(x\) needs to satisfy \(d(x,f(p))\geq d(x,p)\) for all \(f\in\operatorname{Bir}(\mathbb{P}^2)\) and it is proved that in fact we just need \(d(x,f(p))\geq d(x,p)\) for all \(f\) in the Jonquières subgroup. The cells that are adjacent to the cell associated to \(l\) are determined, as well as the cells that are quasi-adjacent to the cell associated to \(l\) (quasi-adjacent means adjacent in the boundary of \(\mathbb{H}^{\infty}\)).

One of the motivations for studying the Voronoï tesselation is to look at the action of the Cremona group on the graph of adjacency of the tesselation. This is studied in a second paper of the author where the graph of quasi-adjacency is proved to be Gromov-hyperbolic.

Reviewer: Shengyuan Zhao (Stony Brook)