Definable Equivariant Retractions in Non-Archimedean Geometry

Hils, Martin; Hrushovski, Ehud; Simon, Pierre

Abstract

For G an algebraic group definable over a model of ACVF, or more generally a definable subgroup of an algebraic group, we study the stable completion Gˆ of G, as introduced by Loeser and the second author. For G connected and stably dominated, assuming G commutative or that the valued field is of equicharacteristic 0, we construct a pro-definable G-equivariant strong deformation retraction of Gˆ onto the generic type of G. For G=S a semiabelian variety, we construct a pro-definable S-equivariant strong deformation retraction of Sˆ onto a definable group which is internal to the value group. We show that, in case S is defined over a complete valued field K with value group a subgroup of R, this map descends to an S(K)-equivariant strong deformation retraction of the Berkovich analytification San of S onto a piecewise linear group, namely onto the skeleton of San. This yields a construction of such a retraction without resorting to an analytic (non-algebraic) uniformization of S. Furthermore, we prove a general result on abelian groups definable in an NIP theory: any such group G is a directed union of ∞-definable subgroups which all stabilize a generically stable Keisler measure on G.