Topological simplicity of the Cremona groups

Abstract

The Cremona group is topologically simple when endowed with the Zariski or Euclidean topology, in any dimension ≥ 2 and over any infinite field. Two elements are moreover always connected by an affine line, so the group is path-connected.

Three plots about the Cremona groups

“…Linear connectedness of the group Cr n (k) implies its connectedness. Proof (different from the proof in [BZ18]).…”

“…For example (special case of Corollary 3.14), we prove that if k is an algebraically closed field of characteristic zero, and with each integer d > 0 any abstract group H d from the following list is associated: [BZ18], where the linear connectedness (and therefore the connectedness) of the group Cr n (k) is proved in the case of an infinite field k (for an algebraically closed field k, this was proved earlier in [Bl10]). We give a short new proof for the case of an infinite field k. It is based on an argument, ideologically close to that of Alexander, which he used in [Al23] in proving the connectedness of the homeomorphism group of the ball, and which was then adapted in [Sh82, Lem.…”

Properties of the Cremona group endowed with the Euclidean topology

“…Proof of Corollary 1.3. A morphism ρ : A 1 Bir(P n ) is continuous in the Euclidean topology [BZ16,Lemma 2.11]. Hence the sequence ρ(t m ) converges to id in the Euclidean topology and by Theorem 1.1 we have ρ(t m ) = id for m ≫ 0.…”

“…In fact, Bir(P 3 ) is not even generated by its algebraic subgroups [BY19,Theorem C]. The Euclidean topology on the Cremona groups is largely unstudied and some results found in [BF13,BZ16,UZ21,Z16].…”

Quotients of higher dimensional Cremona groups

“…In fact in the trivial case of dimension n = 1, we have Bir(P 1 ) = Aut(P 1 ) = PGL 2 (k), which is indeed a simple group when the ground field k is algebraically closed. Another evidence in favour of the simplicity of the Cremona groups is that one can endow Bir C (P n ) with two topologies: the Zariski or the Euclidean one (see [Bla10,BF13]), and that in both cases all closed normal subgroups are either trivial or the whole group, as proven in [Bla10] for n = 2 and generalised in [BZ18] to any dimension.…”