## Notes of Yves Cornulier’s Cambridge lecture 16-05-2017

Commensurating actions of groups of birational transformations

Joint work with Serge Cantat.

1. Birational geometry

This is more than a century old. Let ${X,Y}$ be irreducible varieties over ${{\mathbb C}}$. A rational map ${X\rightarrow Y}$ is a regular (given by the ratio of two polynomials) function between Zariski dense open subsets, modulo coincidence on a dense open subset. A birational map ${X\rightarrow Y}$ is a regular function between Zariski dense open subsets which is an isomorphism of varieties, again modulo coincidence on a dense open subset.

Example. The map ${{\mathbb C}^2\rightarrow P^1\times P^1}$, ${(x,y)\mapsto ((x:1);(y:1))}$ is birational. Therefore ${P^1\times P^1}$ and ${P^2}$ are birationally equivalent. Therefore a rational map comes in several different models. One often needs to consider all of them simultaneously.

A set of rational functions on ${{\mathbb C}^d}$ forms a birational map to ${{\mathbb C}^d}$ iff it is algebraically free over ${{\mathbb C}}$ and generates the field ${{\mathbb C}(t_1,\ldots,t_d)}$.

Notation. ${Bir(X)}$ is the group of birational equivalences of ${X}$, i.e. the automorphism group of the field ${{\mathbb C}(X)}$ of rational functions on ${X}$.

Example. It turns out that ${Bir({\mathbb C})=Bir(P^1)=Aut(P^1)=PGl(2,{\mathbb C})}$. On the other hand, ${Bir(P^2)}$ contains ${Aut(P^2)=PGl(3,{\mathbb C})}$ but also ${Aut(P^1\times P^1)=PGl(2,{\mathbb C})\times PGl(2,{\mathbb C})}$. However, ${Bir(P^2)}$ contains an abelian group of infinite rank, the set of maps ${(x,y)\mapsto (x+P(y),y)}$, where ${P\in{\mathbb C}[t]}$.

The difficulty of ${Bir}$ is that it comes with quasi-actions which are not quite actions. Some results have been obtained in dimension 2, nearly nothing is known in higher dimensions. For instance, one does not know which finite groups occur in ${Bir(P^3)}$, and it is still possible (but unlikely) that every finite group occurs in ${Bir(P^4)}$.

2. Commensurating actions

2.1. Commensurated sets

Say a group ${G}$ acting on a set ${E}$. Say a subset ${A\subset E}$ is commensurated if ${\forall g\in G}$, ${\ell_A(g):=|A\Delta gA|<\infty}$ (Stallings, Dunwoody,…). Say that ${A}$ is transfixed if there exists a ${G}$-invariant subset ${A'}$ such that ${|A\Delta A'|<\infty}$. It turns out that it is equivalent to ${\ell_A}$ being bounded (proved by several authors in the 1960’s, with good bounds finally obtained by W. Neumann).

Example. Let ${G}$ act on a Schreier graph ${G/H}$. Then a subset is commensurated iff its boundary is finite. A subset is transfixed iff it is finite or its complement is finite. In particular, there exists a commensurated, non-transfixed subset iff the Schreier graph has at least 2 ends.

Example. Let ${G}$ act on set ${E}$, and ${A\subset E}$. Let ${\ell_A^2(E)=1_A+\ell^2(E)\subset{\mathbb R}^E}$ is an affine Hilbert space. It is ${G}$ invariant iff ${A}$ is commensurated. This provides an affine isometric action of ${G}$.

Example. Cubulation. Define a graph whose vertices are subset ${B\subset E}$ such that ${|B\Delta A|<\infty}$. Put an edge between ${B}$ and ${B\cup\{b\}}$. This a median graph, therefore an action on a ${CAT(0)}$ cube complex arises from it. Conversely, every action of ${G}$ on a ${CAT(0)}$ cube complex induces a commensurating action on the set ${E}$ of half-spaces, with ${A}$ being the subset of half-spaces contaning a fixed vertex. This gives huge cube complexes or actions, far from being optimal.

2.2. Property FW

Definition 1 Say ${G}$ has property FW if for every commensurating action of ${G}$ on ${(E,A)}$, ${A}$ is transfixed.

Say ${G}$ has property FW relative to subgroup ${H}$ if for every commensurating action of ${G}$ on ${(E,A)}$, ${A}$ is transfixed by ${H}$.

Example. For every cyclic distorted subgroup ${H, ${G}$ has property FW relative to ${H}$ (Haglund).

This can be used to prove that ${Sl(d,{\mathbb Z})}$, ${d\geq 3}$, or ${Sl(d,{\mathbb Z}[\sqrt{2}])}$, ${d\geq 2}$, has property FW (using bounded generation by exponentially distorted unipotents).

Property (T) implies property FW (use ${\ell^2}$ action).

Proposition 2 or any group ${G}$, the following are equivalent:

1. ${G}$ has property FW.
2. Every isometric action of ${G}$ on a ${CAT(0)}$ cube complex has a fixed point.
3. Every isometric action of ${G}$ on a median graph has a finite orbit.
4. If ${G}$ is finitely generated, every Schreier graph of ${G}$ has at most 1 end.
5. For all actions of ${G}$ on sets ${E}$, ${H^1(G,{\mathbb Z}^{(E)})=0}$ (functions with finite support).

3. Birational groups

Fix a projective variety ${X}$. Let ${Hyp(X)}$ be the set of irreducible hypersurfaces of ${X}$. Every subgroup ${G acts on a set ${E}$ which commensurates ${A=Hyp(X)}$ as follows. The crucial point is the following. If ${X}$ is smooth (in fact, normal is sufficient) and ${f:Y\rightarrow X}$ is a birational morphism, inverse images of hypersurfaces are well defined: among the several subvarieties whose union is ${f^{-1}(H)}$, there is exactly one which maps onto ${H}$. Therefore one can define ${E=\widetilde{Hyp}(X)}$ as the inverse limit of ${Hyp(Y)}$ over all birational morphisms ${Y\rightarrow X}$. Birational self-maps of ${X}$ act on ${E}$ and the subset ${A=Hyp(X)}$ is commensurated.

If ${G}$ acts by automorphism of a Zariski open set of ${X}$, then ${G}$ transfixes ${A}$. This is also true for pseudo-automorphisms, i.e. isomorphisms between proper Zariski open subsets of ${X}$.

Theorem 3 Let ${G\subset Bir(X)}$. Then ${G}$ transfixes ${Hyp(X)}$ iff the ${G}$ action on ${X}$ is conjugate to an action by pseudo-automorphisms.

Example. The monomial action of ${Sl(n,{\mathbb Z})}$ on ${{\mathbb C}^{n^2}}$ raises each coordinate to a power given by a matrix coefficient. These are pseudo-automorphisms.

Theorem 4 Let ${X}$ have dimension 2. Let ${G\subset Bir(X)}$ have property FW. Then the action of ${G}$ on ${X}$ is conjugate to an action by automorphisms on some variety ${Y}$, with a short list of exceptions.

This fails in higher dimensions (see the monomial action). There is a corresponding statement for groups with relative property FW.