Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes

Abstract

We show that plane Cremona groups over finite fields embed into Neretin groups, i.e., groups of almost automorphisms of rooted trees. The image of this embedding is dense if our base field has odd characteristic or is equal to F2. This is no longer true, if the finite base field has even characteristic and contains at least 4 elements; for this case we show that the permutations induced by birational transformations on rational points of regular projective surfaces are always even. In a second part, we construct explicit locally compact CAT(0) cube complexes, on which Neretin groups act properly. This allows us to recover in a unified way various results on Neretin groups such as that they are of type F∞. We also prove a new fixed-point theorem for CAT(0) cube complexes without infinite cubes and use it to deduce a regularisation theorem for plane Cremona groups over finite fields.2010 Mathematics subject classification.