Notes of Lior Silberman’s lecture nr 3

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 {CAT(0)} 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 {\Gamma} act isometrically on metric spaces {X} and {Y}. Assume that the action on {X} is free and properly discontinuous.

Let {\mu:x\mapsto \mu_x} be a {\Gamma}-invariant Markov chain on {X}, i.e. a {\Gamma}-equivariant map from {X} to the set of probability measures on {X}. Assume {\mu} is reversible with respect to {\nu}, i.e. for every {\Gamma}-invariant function {f:X\times X\rightarrow {\mathbb R}},

\displaystyle  \begin{array}{rcl}  \int_{\Gamma\setminus X}d\nu(x)\int d\mu(x\rightarrow y)f(x,y)=\int_{\Gamma\setminus X}d\nu(y)\int d\mu(y\rightarrow x)f(x,y). \end{array}

Idea: Study {B=B^{\Gamma}(X,Y)=\{f:X\rightarrow Y\,;\,\Gamma-\textrm{equivariant, finite energy}\}}. Find a constant function in {B}.

Assume that {\Gamma\setminus X} is a finite polyhedron (this guarantees that {B} is non empty).

Definition 1 If {f\in B(X,Y)} is {\Gamma}-equivariant, set

\displaystyle  \begin{array}{rcl}  E_{\mu}(f)=\frac{1}{2}\int_{\Gamma\setminus X}d\nu(x)\int_{X}d\mu(x\rightarrow y)d_Y (f(x),f(y))^2 , \end{array}

The connection with our previous discussion of isometric group actions on {Y} is provided by the following

Example 1 {X=Cay(\Gamma,S)} is the Cayley graph, {\mu} is the nearest neighbour random walk.

Then mapping {f\mapsto f(e)} identifies {B(X,Y)} with {Y} and energy

\displaystyle  \begin{array}{rcl}  E_{\mu}(f)=\frac{1}{2|S|}\int_{s\in S}d_Y (y,sy)^2 =E_S (y),\quad y=f(e). \end{array}

1.2. Assumptions on {Y}

Definition 2 Say a metric space {Y} is {CAT(0)} if it is geodesic and the following inequality holds for all {x}, {y}, {z\in Y}.

\displaystyle  \begin{array}{rcl}  d(x,m(y,z))^2 \leq \frac{1}{2}d(x,y)^2 +\frac{1}{2}d(x,z)^2 -\frac{1}{4}d(y,z)^2 . \end{array}

1.3. Center of mass

In a {CAT(0)} space, given a probability measure {\sigma}, there is a unique point {c(\sigma)}, called the center of mass of {\sigma}, where the function

\displaystyle  \begin{array}{rcl}  y\mapsto d^{2}(y,\sigma):=\int_{Y}d(y,y')^{2}\,d\sigma(y')=\mathop{\mathbb E}_{\sigma}(d_y^2). \end{array}

attains its minimum.

Proof: Let {m=\inf d^2 (\cdot,\sigma)}. Let {C_{\epsilon}=\{y\,;\,d^{2}(y,\sigma)\leq m+\epsilon\}}. Let {y_0}, {y_1 \in C_{\epsilon}}. Integrating the {CAT(0)} inequality yields

\displaystyle  \begin{array}{rcl}  m\leq d([y_0 ,y_1]_t ,\sigma)\leq (1-t)d^2 (y_0 ,\sigma) +t d^2 (y_1 ,\sigma)-t(1-t)d(y_0 ,y_1)^2 . \end{array}

Take {t=\frac{1}{\epsilon}}, find

\displaystyle  \begin{array}{rcl}  m\leq m+\epsilon -\frac{1}{4}d(y_0 ,y_1)^2 \end{array}

Which implies that {diameter(C_{\epsilon})\leq 2\sqrt{\epsilon}}. By completeness, the intersection of {C_{\epsilon}}‘s is non empty, whence the minimum is attained. \Box

The proof shows that

\displaystyle  \begin{array}{rcl}  d^2 (y,\sigma)\geq d(y,c(\sigma))^2 +d^2 (c(\sigma),\sigma). \end{array}

1.4. Averaging

Back to our setting: {\Gamma} acts on {X} and {Y}, {f:X\rightarrow Y} {\Gamma}-equivariant. When {Y} is {CAT(0)}, one can define

Definition 3

\displaystyle  \begin{array}{rcl}  (A_{\mu}f)=c(f_{*}\mu_x), \end{array}

We need a lazy variant of this.

Definition 4 Given {x\in X}, define {g_x : X\rightarrow Y} by

\displaystyle  \begin{array}{rcl}  g_x (x')=[f(x),f(x')]_{1/2}. \end{array}

Then set

\displaystyle  \begin{array}{rcl}  (\tilde{A}_{\mu}f)=c((g_{x})_{*}\mu_x), \end{array}

Example 2 {X=Cay(\Gamma,S)}. Let {\tilde{X}} be the barycentric subdivision of {X}. On {\tilde{X}}, alternate two operations,

  1. Set {f(\textrm{midpoint})=} midpoint of ends of edge.
  2. Do nearest neighbour averaging on {Y}.

Operation (1) is an embedding {B(X,Y)\rightarrow B(\tilde{X},Y)}. Thanks to reversibility, it reduces energy on

Remark 1 Again, by an ultrafilter argument, fixed point property in {Y} is equivalent to decreasing energy property for this averaging procedure.

In our non linear setting, this is not quite the same as {A_{\mu\star\mu}}.

1.5. Powers

Note that, due to non linearity, {A_{\mu\star\mu}\not=A_{\mu}\circ A_{\mu}}. Nevertheless, it turns out that the adequate substitute for powers of {A_{\mu}} is {A_{\mu^{\ell}}}.

For every {x\in X}, {y\in Y},

\displaystyle  \begin{array}{rcl}  d((A_{\mu^{\ell}}f)(x),y)^2 \leq d^2 (y,f_{*}\mu_x)-d^2 ((A_{\mu^{\ell}}f)(x),f_{*}\mu_x)^2 , \end{array}


\displaystyle  \begin{array}{rcl}  E_{\mu^{\ell}}(A_{\mu^{\ell}}f) &\leq& \frac{1}{2}\int_{\Gamma\setminus X}d\nu(x)\int_{X}d\mu(x\rightarrow x')\left(d^2 (y,f_{*}\mu_x)-d^2 ((A_{\mu^{\ell}}f)(x),f_{*}\mu_x)^2 \right)\\ &=& \frac{1}{2}\int_{\Gamma\setminus X}d\nu(x)\int_{X}d\mu(x\rightarrow x') \int_{X}d\mu^{\ell}(x\rightarrow x'')\,d((A_{\mu^{\ell}}f)(x'),f(x''))^2 \\ &&-\frac{1}{2}\int_{\Gamma\setminus X}d\nu(x)\int_{X}d\mu(x\rightarrow x') \int_{X}d\mu^{\ell}(x\rightarrow x'')\,d((A_{\mu^{\ell}}f)(x),f(x''))^2 \\ &=& \frac{1}{2}\int_{\Gamma\setminus X}d\nu(x)\int_{X}d\mu^{\ell+1}(x\rightarrow x')\,d((A_{\mu^{\ell}}f)(x),f(x'))^2\\ &&-\frac{1}{2}\int_{\Gamma\setminus X}d\nu(x)\int_{X}d\mu^{\ell}(x\rightarrow x')\,d((A_{\mu^{\ell}}f)(x),f(x'))^2 . \end{array}

by reversibility. We are facing a difficulty due to parity. We solve it here by using {E_{\mu^2}}.

\displaystyle  \begin{array}{rcl}  E_{\mu^2}(A_{\mu^{2\ell}}f)\leq\frac{1}{2}\int_{\Gamma\setminus X}d\nu(x)\int_{X}\left(d\mu^{2\ell+2}(x\rightarrow x')-d\mu^{2\ell}(x\rightarrow x')\right)\,d((A_{\mu^{\ell}}f)(x),f(x'))^2 \end{array}

From now on, assume that {X} is a tree, i.e. the Cayley graph of the free group {F_S} (this is a covering space of {Cay(\Gamma,S)}). A direct calculation shows that, on a tree,

\displaystyle  \begin{array}{rcl}  d\mu^{2\ell+2}(x\rightarrow x')-d\mu^{2\ell}(x\rightarrow x') \leq g(\ell)d\mu^{2\ell}(x\rightarrow x') +h(\ell), \end{array}

where {g}, {h} are numerical functions,

  1. {g(t)\sim \sqrt{\frac{\log\log t}{\log t}}},
  2. {h} is exponentially small.

On a tree,

\displaystyle  \begin{array}{rcl}  E_{\mu^2}(f)\sim\max_{\gamma\in S^2} d(f(x),f(x\gamma))^2 . \end{array}

By triangle inequality,

\displaystyle  \begin{array}{rcl}  d((A_{\mu^{\ell}}f)(x),f(x'))^2 \leq \ell^{C} E_{\mu^2}(f). \end{array}

This yields

\displaystyle  \begin{array}{rcl}  E_{\mu^2}(A_{\mu^{2\ell}}f)\leq g(\ell)\int_{\Gamma\setminus X\times X}d\nu(x)d\mu^{2\ell}(x\rightarrow x')d((A_{\mu^{\ell}}f)(x),f(x'))^2 +h(\ell) E_{\mu^2}(f). \end{array}

By definition of average,

\displaystyle  \begin{array}{rcl}  \frac{1}{2}\int_{\Gamma\setminus X\times X}d\nu(x)d\mu^{\ell}(x\rightarrow x')d((A_{\mu^{\ell}}f)(x),f(x'))^2 &\leq& \frac{1}{2}\int_{\Gamma\setminus X\times X}d\nu(x)d\mu^{\ell}(x\rightarrow x')d(f(x),f(x'))^2 \\ &=& E_{\mu^{\ell}}(f). \end{array}


\displaystyle  \begin{array}{rcl}  E_{\mu^2}(A_{\mu^{2\ell}}f)\leq g(\ell)E_{\mu^{2\ell}}(f)+h(\ell)E_{\mu^{\ell}}(f). \end{array}

Assume that we also have a Poincaré inequality

\displaystyle  \begin{array}{rcl}  E_{\mu^{2\ell}}(f)\leq C\,E_{\mu^2}(f), \end{array}

with {C} independant of {\ell}. Then, for {\ell} large enough,

\displaystyle  \begin{array}{rcl}  E_{\mu^2}(A_{\mu^{2\ell}}f)\leq r E_{\mu^{2}}(f), \end{array}

with {r<1}. Therefore {A_{\mu^{2\ell}}f} converges to a constant map as {\ell} tends to infinity, providing a fixed point.

Conclusion: To show that {\Gamma\in F\mathcal{C}} where {\mathcal{C}} is a family of {CAT(0)} spaces, it is enough to establish the follwing Poincaré inequality. There exist {r<1}, {C>0} such that for all {Y\in\mathcal{C}}, for all {f\in B^{\Gamma}(X,Y)}, there exists {\ell} such that

\displaystyle  \begin{array}{rcl}  E_{\mu^{2\ell}}(f)\leq C\,E_{\mu^2}(f) \quad \textrm{and}\quad Cg(\ell)+h(\ell)\leq r. \end{array}

2. Random groups in the graph model

Due to Gromov, 2003.

2.1. Idea

Let {G=(V,E)} be a graph, let {\alpha:\vec{E}\rightarrow S} be a symmetric labelling (i.e. switching an oriented edge gives inverse label). {\alpha} extends to a multiplicative map on paths, with values in the free group {F_S}. Words read along cycles of {G} form a normal subgroup {R_{\alpha}} of {F_S}. We are interested in the group {\Gamma_{\alpha}=\langle S\,|\,R_{\alpha} \rangle} and its Cayley graph {X_{\alpha}}.

There is a projection from {X_{\alpha}} to {G}. Take base points {v_0 \in V}, {x_0 \in X_{\alpha}}. Set

\displaystyle  \begin{array}{rcl}  \pi_{v_0 \rightarrow x_0}(v)=x_0 \alpha(\vec{p}), \end{array}

where {\vec{p}} is a path from {v_0} in {v}. Using this projection, one can try to simulate random walk on {X_{\alpha}} by random walk on {G}.

2.2. Basic results

Theorem 5 (Gromov 2003) Assume that the girth {g(G)} is large (say {\geq 4\ell}). Choose {\alpha} uniformly at random. With high probability, one can effectively simulate {\ell} steps of random walk on {X_{\alpha}} by random walk on {G}.

See also my appendix to Gromov’s GAFA paper.

Theorem 6 (Gromov 2003, Ollivier, Arzhantseva, Delzant) Assume that the girth {g(G)} is large. Choose {\alpha} uniformly at random. With high probability, {\Gamma_{\alpha}} is infinite and Gromov-hyperbolic.

These results reduce the problem of finding groups with {F\mathcal{C}} to finding graphs with large girth having Poincaré inequality for {Y}-valued maps. Specifically,

Problem: We want a constant {C} and a family of graphs with increasing girth such that for all graphs in the family, for all {Y\in\mathcal{C}}, for all {f:V\rightarrow Y},

\displaystyle  \begin{array}{rcl}  E_{\nu_{G}\star \nu_{G}}(f)\leq C\,E_{\mu_G}(f), \end{array}

where {\mu_G} is random walk on {G} and {\nu_G} a probability measure on {V}.

Remark 2 Restriction to maps with values in a geodesic shows that Poincaré inequality must hold for {{\mathbb R}}-valued functions, i.e. the family must be a family of expanders.

The resulting groups {\Gamma_{\alpha}} 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 {\frac{1}{1-\lambda(G)}}. For non linear spaces, try to get a constant of the form {C_Y \frac{1}{1-\lambda(G)}}.

3.1. The tangent cone

We saw that

\displaystyle  \begin{array}{rcl}  \int_{V\times V} \nu_G (x) \nu_G (x')\,d(f(x),f(x') \geq \int_{V}\nu_G (x) d(f(x),c(f_* \nu_G))^2 , \end{array}

(equality in Hilbert spaces), and by triangle inequality, the reverse inequality holds up to a factor of {2},

\displaystyle  \begin{array}{rcl}  \int_{V\times V} \nu_G (x) \nu_G (x')\,d(f(x),f(x') \leq 2\int_{V}\nu_G (x) d(f(x),c(f_* \nu_G))^2 . \end{array}

Definition 7 Let {Y} be a {CAT(0)} space. Let {p\in Y}. Define a new metric {\bar{d}} on {Y} by

\displaystyle  \begin{array}{rcl}  \bar{d}(y,y')^2 =d(p,y)^2 +d(p,y')^2 -2d(p,y)d(p,y')\cos(\theta_p (y,y')), \end{array}

where {\theta_p (y,y')} is the angle at {p} between the geodesics {[p,y]} and {[p,y']}. Then {(Y,\bar{d})} is again {CAT(0)}. Its completion is the tangent cone {T_p Y} of {Y} at {p}.

The {CAT(0)} inequality implies that radial distances are preserved and {\bar{d}\leq d}, i.e. the identity map {(Y,d)\rightarrow (Y,\bar{d})} is distance non increasing. Therefore, to get a Poincaré inequality for {Y}-valued maps, it is sufficient to prove one for {T_p Y}-valued maps (up to a factor of {2}). This observation is due to Mu-Tao Wang.

Corollary 8 (Wang 1998) If all tangent cones {T_p Y} embed isometrically in Hilbert spaces, then Poincaré inequality holds with constant {\frac{2}{1-\lambda(G)}}.

Example 3 Lattices in higher rank semisimple Lie groups have property (T), so they have a spectral gap. Nevertheless, they act isometrically on a {CAT(0)} space, their symmetric space, without fixed point.

It is the factor {2} 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 {(Y,d)} have snowflakes {(Y,\sqrt{d})} which bilipschitz embed in Hilbert space. Now Hilbert space has a linear isometric embedding in {L^4}. So to prove a {Y}-valued Poincaré inequality, it suffices to prove a scalar {\ell_4}-Poincaré inequality, i.e.

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}_{\nu_G \times \nu_G}|f(x)-f(x')|^4 \leq C\,\mathop{\mathbb E}_{\mu_G}|f(x)-f(x')|^4 . \end{array}

Proposition 10 (Matousek) If {\ell_p}-Poincaré inequality holds for a graph {G}, then {\ell_q}-Poincaré inequalities holds for {G} up to a constant which depends only on {p} and {q}.

This allows to upgrade a spectral gap into a building-valued Poincaré inequality.

Corollary 11 (Naor, Silberman 2010) For every {n}, there exist a graph {G} such that, with high probability, random groups modelled on {G} have fixed point preperty on all buildings of rank {\leq n}.

3.3. Banach space-valued Poincaré inequalities

Lafforgue, using representation theory of {PSl(3,\mathbb{Q}_p)}, 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 {M} be a smooth compact manifold, equipped with a smooth volume form {\mu}. Consider the space {Y'} of all smooth Riemannian metrics on {M} with volume form pointwise equal to {\mu}. The {L_2}-metric on {Y'} is defined as follows. The tangent space to {Y'} at {g} is the space of fields of trace free quadratic forms. Set

\displaystyle  \begin{array}{rcl}  G_g (h)=\int_{M}\mathrm{Trace}((g(x)^{-1}h(x))^2)\,d\mu(x). \end{array}

View this as a Riemannian metric on the infinite dimensional manifold {Y'}, and consider the corresponding distance. Every smooth diffeomorphism of {M} acts isometrically on {Y'}.

Here is an alternative definition. Throw away a set of measure zero to make {M} contractible, so that the tangent bundle becomes trivial. Note that {\mathrm{Trace}((g(x)^{-1}h(x))^2)} is the canonical metric on the symmetric space {S} of {Sl(n,{\mathbb R})}. So {Y'} embeds isometrically into the space of Borel maps {M\rightarrow S} equipped with the {L_2}-metric. Let {Y} be the completion of {Y'} with respect to the {L_2}-metric. Then {Y=L_2 (M,S)} is a {CAT(0)} space (an infinite product of {CAT(0)} space).

Corollary 12 Let {\mathcal{C}} be the class of {CAT(0)} spaces with Hilbertian tangent cones. Let {\Gamma} be a group having property {F\mathcal{C}}. If {\Gamma} acts smoothly on a compact manifold {M}, preserving a smooth volume form. Then {\Gamma} fixes an {L_2} Riemannian metric on {M}.

Theorem 13 (Zimmer) If {M} is a smooth compact manifold, and {g} an {L_2} Riemannian metric on {M}. Then {Isom(M,g)} embeds in a compact group.

It is easy to make groups which do not embed into any compact group: add to the relations of {\Gamma} 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 {\Gamma} such that every homorphism {\Gamma:Diff_{\mu}(M)} is trivial.


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