Commensurating actions of groups of birational transformations
Joint work with Serge Cantat.
1. Birational geometry
This is more than a century old. Let be irreducible varieties over . A rational map 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 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 , is birational. Therefore and 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 forms a birational map to iff it is algebraically free over and generates the field .
Notation. is the group of birational equivalences of , i.e. the automorphism group of the field of rational functions on .
Example. It turns out that . On the other hand, contains but also . However, contains an abelian group of infinite rank, the set of maps , where .
The difficulty of 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 , and it is still possible (but unlikely) that every finite group occurs in .
2. Commensurating actions
2.1. Commensurated sets
Say a group acting on a set . Say a subset is commensurated if , (Stallings, Dunwoody,…). Say that is transfixed if there exists a -invariant subset such that . It turns out that it is equivalent to being bounded (proved by several authors in the 1960’s, with good bounds finally obtained by W. Neumann).
Example. Let act on a Schreier graph . 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 act on set , and . Let is an affine Hilbert space. It is invariant iff is commensurated. This provides an affine isometric action of .
Example. Cubulation. Define a graph whose vertices are subset such that . Put an edge between and . This a median graph, therefore an action on a cube complex arises from it. Conversely, every action of on a cube complex induces a commensurating action on the set of half-spaces, with 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 has property FW if for every commensurating action of on , is transfixed.
Say has property FW relative to subgroup if for every commensurating action of on , is transfixed by .
Example. For every cyclic distorted subgroup , has property FW relative to (Haglund).
This can be used to prove that , , or , , has property FW (using bounded generation by exponentially distorted unipotents).
Property (T) implies property FW (use action).
Proposition 2 or any group , the following are equivalent:
- has property FW.
- Every isometric action of on a cube complex has a fixed point.
- Every isometric action of on a median graph has a finite orbit.
- If is finitely generated, every Schreier graph of has at most 1 end.
- For all actions of on sets , (functions with finite support).
3. Birational groups
Fix a projective variety . Let be the set of irreducible hypersurfaces of . Every subgroup acts on a set which commensurates as follows. The crucial point is the following. If is smooth (in fact, normal is sufficient) and is a birational morphism, inverse images of hypersurfaces are well defined: among the several subvarieties whose union is , there is exactly one which maps onto . Therefore one can define as the inverse limit of over all birational morphisms . Birational self-maps of act on and the subset is commensurated.
If acts by automorphism of a Zariski open set of , then transfixes . This is also true for pseudo-automorphisms, i.e. isomorphisms between proper Zariski open subsets of .
Theorem 3 Let . Then transfixes iff the action on is conjugate to an action by pseudo-automorphisms.
Example. The monomial action of on raises each coordinate to a power given by a matrix coefficient. These are pseudo-automorphisms.
Theorem 4 Let have dimension 2. Let have property FW. Then the action of on is conjugate to an action by automorphisms on some variety , 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.