In the first two lectures, I showed that property (T) was related to exponential convergence of averages in Hilbert space, a fact that can alternatively be formulated as a Poincaré inequality. And I explained how this extends to certain non linear metric spaces, like spaces.
Today, I will describe random groups, a machine that produces groups which have fixed point properties.
1. From groups actions to random walks
1.1. Invariant random walks
Let a group act isometrically on metric spaces and . Assume that the action on is free and properly discontinuous.
Let be a -invariant Markov chain on , i.e. a -equivariant map from to the set of probability measures on . Assume is reversible with respect to , i.e. for every -invariant function ,
Idea: Study . Find a constant function in .
Assume that is a finite polyhedron (this guarantees that is non empty).
Definition 1 If is -equivariant, set
The connection with our previous discussion of isometric group actions on is provided by the following
Example 1 is the Cayley graph, is the nearest neighbour random walk.
Then mapping identifies with and energy
1.2. Assumptions on
Definition 2 Say a metric space is if it is geodesic and the following inequality holds for all , , .
1.3. Center of mass
In a space, given a probability measure , there is a unique point , called the center of mass of , where the function
attains its minimum.
Proof: Let . Let . Let , . Integrating the inequality yields
Take , find
Which implies that . By completeness, the intersection of ‘s is non empty, whence the minimum is attained.
The proof shows that
Back to our setting: acts on and , -equivariant. When is , one can define
We need a lazy variant of this.
Definition 4 Given , define by
Example 2 . Let be the barycentric subdivision of . On , alternate two operations,
- Set midpoint of ends of edge.
- Do nearest neighbour averaging on .
Operation (1) is an embedding . Thanks to reversibility, it reduces energy on
Remark 1 Again, by an ultrafilter argument, fixed point property in is equivalent to decreasing energy property for this averaging procedure.
In our non linear setting, this is not quite the same as .
Note that, due to non linearity, . Nevertheless, it turns out that the adequate substitute for powers of is .
For every , ,
by reversibility. We are facing a difficulty due to parity. We solve it here by using .
From now on, assume that is a tree, i.e. the Cayley graph of the free group (this is a covering space of ). A direct calculation shows that, on a tree,
where , are numerical functions,
- is exponentially small.
On a tree,
By triangle inequality,
By definition of average,
Assume that we also have a Poincaré inequality
with independant of . Then, for large enough,
with . Therefore converges to a constant map as tends to infinity, providing a fixed point.
Conclusion: To show that where is a family of spaces, it is enough to establish the follwing Poincaré inequality. There exist , such that for all , for all , there exists such that
2. Random groups in the graph model
Due to Gromov, 2003.
Let be a graph, let be a symmetric labelling (i.e. switching an oriented edge gives inverse label). extends to a multiplicative map on paths, with values in the free group . Words read along cycles of form a normal subgroup of . We are interested in the group and its Cayley graph .
There is a projection from to . Take base points , . Set
where is a path from in . Using this projection, one can try to simulate random walk on by random walk on .
2.2. Basic results
Theorem 5 (Gromov 2003) Assume that the girth is large (say ). Choose uniformly at random. With high probability, one can effectively simulate steps of random walk on by random walk on .
See also my appendix to Gromov’s GAFA paper.
Theorem 6 (Gromov 2003, Ollivier, Arzhantseva, Delzant) Assume that the girth is large. Choose uniformly at random. With high probability, is infinite and Gromov-hyperbolic.
These results reduce the problem of finding groups with to finding graphs with large girth having Poincaré inequality for -valued maps. Specifically,
Problem: We want a constant and a family of graphs with increasing girth such that for all graphs in the family, for all , for all ,
where is random walk on and a probability measure on .
Remark 2 Restriction to maps with values in a geodesic shows that Poincaré inequality must hold for -valued functions, i.e. the family must be a family of expanders.
The resulting groups will automatically have property (T).
3. Non linear Poincaré inequalities
Remember that the constant in the scalar Poincaré inequality is the inverse of the spectral gap . For non linear spaces, try to get a constant of the form .
3.1. The tangent cone
We saw that
(equality in Hilbert spaces), and by triangle inequality, the reverse inequality holds up to a factor of ,
Definition 7 Let be a space. Let . Define a new metric on by
where is the angle at between the geodesics and . Then is again . Its completion is the tangent cone of at .
The inequality implies that radial distances are preserved and , i.e. the identity map is distance non increasing. Therefore, to get a Poincaré inequality for -valued maps, it is sufficient to prove one for -valued maps (up to a factor of ). This observation is due to Mu-Tao Wang.
Corollary 8 (Wang 1998) If all tangent cones embed isometrically in Hilbert spaces, then Poincaré inequality holds with constant .
Example 3 Lattices in higher rank semisimple Lie groups have property (T), so they have a spectral gap. Nevertheless, they act isometrically on a space, their symmetric space, without fixed point.
It is the factor that makes the Poincaré inequality too weak to imply fixed point property. This loss has been studied in detail by Kondo, Nayatani, Ozeki. They could
3.2. Building-valued Poincaré inequalities
Theorem 9 (Lang, Schlichenmaier 2004) Buildings of linear groups over local fields have finite Assouad-Nagata dimension.
It follows (Assouad, Lee-Naor) that buildings have snowflakes which bilipschitz embed in Hilbert space. Now Hilbert space has a linear isometric embedding in . So to prove a -valued Poincaré inequality, it suffices to prove a scalar -Poincaré inequality, i.e.
Proposition 10 (Matousek) If -Poincaré inequality holds for a graph , then -Poincaré inequalities holds for up to a constant which depends only on and .
This allows to upgrade a spectral gap into a building-valued Poincaré inequality.
Corollary 11 (Naor, Silberman 2010) For every , there exist a graph such that, with high probability, random groups modelled on have fixed point preperty on all buildings of rank .
3.3. Banach space-valued Poincaré inequalities
Lafforgue, using representation theory of , and later ??, using zig-zag products, have found graphs which satisfy Poincaré inequalities with values in uniformly convex Banach spaces, but they do not have large girth.
3.4. An application
Let be a smooth compact manifold, equipped with a smooth volume form . Consider the space of all smooth Riemannian metrics on with volume form pointwise equal to . The -metric on is defined as follows. The tangent space to at is the space of fields of trace free quadratic forms. Set
View this as a Riemannian metric on the infinite dimensional manifold , and consider the corresponding distance. Every smooth diffeomorphism of acts isometrically on .
Here is an alternative definition. Throw away a set of measure zero to make contractible, so that the tangent bundle becomes trivial. Note that is the canonical metric on the symmetric space of . So embeds isometrically into the space of Borel maps equipped with the -metric. Let be the completion of with respect to the -metric. Then is a space (an infinite product of space).
Corollary 12 Let be the class of spaces with Hilbertian tangent cones. Let be a group having property . If acts smoothly on a compact manifold , preserving a smooth volume form. Then fixes an Riemannian metric on .
Theorem 13 (Zimmer) If is a smooth compact manifold, and an Riemannian metric on . Then embeds in a compact group.
It is easy to make groups which do not embed into any compact group: add to the relations of an infinite sequence of long random words. So the resulting group (which is a limit of hyperbolic groups but is not hyperbolic itself) cannot act non trivially on any smooth manifold.
Theorem 14 (Fisher, Silberman 2010) There exist torsion free finitely generated groups such that every homorphism is trivial.