## Notes of Lior Silberman’s lecture nr 1

Fixed points for groups actions

Fix notation. ${\Gamma}$ will always be a discrete countable group, although many considerations extend to general locally compact groups. An action of ${\Gamma}$ on ${Y}$ by isometries is a homomorphism ${\rho:\Gamma\rightarrow Isom(Y)}$. Say action fixes a point if there exists ${y_0 \in Y}$ such that for all ${\gamma \in \Gamma}$, ${\rho(\gamma)y_0 =y_0}$.

Definition 1 Given a class of metric spaces ${\mathcal{C}}$, say that ${\Gamma}$ has Property ${F\mathcal{C}}$ if every isometric action of ${\Gamma}$ on a space belonging to ${\mathcal{C}}$ has a fixed point.

Questions:

1. Does ${\Gamma}$ have property ${F\mathcal{C}}$ ?
2. Make groups with property ${F\mathcal{C}}$.
3. Use them.

We shall see that property ${F\mathcal{C}}$ is rather a source of examples and counterexamples than a proving tool.

After some generalities, we shall study the case of linear (affine) actions, where it is natural to average, and then extend the idea to certain non linear actions. A leitmotive will be Poincaré inequalities and embeddings of metric spaces.

1. Plan

1. Introduction
• Fixed point properties
• Interaction between algebra and geometry
2. Linear actions: Kazhdan’s property (T)
• Spectral criteria
• Metric criteria
• Averaging and random groups
3. Non-linear averaging
• Uniform convexity
• ${CAT(0)}$ spaces
4. Random groups with property ${F\mathcal{C}}$

See the blog: I posted a bibliography last night. I will update it along the way. A good place to start is Bekka, de la Harpe and Valette’s book on property (T) and Serre’s book on Trees, amalgams, ${SL(2,{\mathbb R})}$.

3. Fixed point properties

3.1. Example: The ${2}$ element group

The ${2}$ element group ${C_2}$ has fixed points whenever it acts on a uniquely geodesic space, i.e. a metric space in which every two points are connected by a unique unit speed minimizing geodesic.

Proof: Let ${\sigma}$ be the non trivial element of ${C_2}$. Choose arbitrary ${y\in Y}$. If ${\sigma(y)=y}$, we are done. Otherwise, let ${z}$ be the midpoint of the unique geodesic from ${y}$ to ${\sigma(y)}$. Then ${\sigma(z)=z}$. $\Box$

The point of the proof is that the “average” ${z}$ was constructed from ${y}$ and ${\sigma(y)}$ using the metric alone.

3.2. Compact groups

Definition 2 Let ${\mathcal{A}}$ be the class of real trees. Let ${\mathcal{H}}$ be the class of Hilbert spaces (seen as metric spaces).

Note that every isometry of Hilbert space is affine.

Proposition 3 Compact groups belong to ${F\mathcal{H}}$.

Proof: For infinite groups, one must assume the action is continuous (in fact, merely that the orbit maps ${\gamma\mapsto \rho(\gamma)y}$ are continuous). Use left Haar probability measure ${d\gamma}$. Let ${Y}$ be a Hilbert space. Take arbitrary ${y \in Y}$. Then

$\displaystyle \begin{array}{rcl} y_0 =\int_{\Gamma}\rho(\gamma)y_0 \,d\gamma \end{array}$

is a fixed point. Indeed, given ${\eta\in\Gamma}$, since ${\rho(\eta)}$ is affine,

$\displaystyle \begin{array}{rcl} \rho(\eta)(y_0)&=&\int_{\Gamma}\rho(\eta)\rho(\gamma)y_0 \,d\gamma\\ &=&\int_{\Gamma}\rho(\eta\gamma)y_0 \,d\gamma\\ &=&\int_{\Gamma}\rho(\gamma')y_0 \,d\gamma'=y_0. \end{array}$

$\Box$

Note that when ${\Gamma=C_2}$, the averaging construction gives the midpoint. Furthermore, the midpoint can be viewed as the average of the measure ${\frac{1}{2}(\delta_{y}+\delta_{\sigma(y)}}$. Formulated that way, we see a possibility to generalize the argument to those Banach spaces which admit non affine isometries. It suffice to have a purely metric notion of average of a probability measure. We shall see one later, which need not coincide with the affine average.

3.3. Actions on trees

Definition 4 Let ${\mathcal{A}}$ be the class of real trees.

We first relate property ${F\mathcal{A}}$ to finite generation. An intermediate step is provided by cofinality.

Definition 5 Say ${\Gamma}$ has cofinality ${\omega}$ if ${\Gamma}$ is the countable union of proper subgroups.

This notion comes from set theory: certain cardinals (e.g. ${\aleph_0}$) have cofinality ${\omega}$, others (e.g. finite ones, ${\aleph_{\omega}}$) do not.

Given a cofinal group ${\Gamma=\bigcup_{n\in\omega}\Gamma_n}$, consider the following graph: vertex set is ${V=\coprod_n \Gamma/\Gamma_n}$, edge set is ${\vec{E}=\{(\gamma\Gamma_n ,\gamma\Gamma_{n+1})\,;\,\gamma\in\Gamma, \,n\in\omega\}}$. Cofinality implies that the obtained graph ${G}$ is connected. It is in fact a tree. ${\Gamma}$ acts on it by simplicial maps. All stabilizers are ${\Gamma_n}$, so all ${\Gamma}$-orbits are sets ${\Gamma/\Gamma_n}$, they are infinite, so no fixed points. No bounded orbits either.

Note: ${\exists}$ bounded orbits iff all orbits are bounded.

Corollary 6 (Serre) If a countable group ${\Gamma}$ has property ${F\mathcal{A}}$, then ${\Gamma}$ is finitely generated.

Proof: Enumerate the elements of ${\Gamma}$ and let ${\Gamma_n}$ be the subgroup generated by the first ${n}$ of them. If ${\Gamma}$ is not finitely generated, then all ${\Gamma_n}$ are proper, and ${\Gamma}$ is cofinal. Thus it does not have property ${F\mathcal{A}}$. $\Box$

We have proved an algebraic property of ${\Gamma}$ using a geometric action. This is typical of how fixed point properties work. They prevent something algebraic to happen.

3.4. Relating ${F\mathcal{A}}$ to ${F\mathcal{H}}$

Proposition 7 If ${\Gamma}$ is countable and has property ${F\mathcal{H}}$, then ${\Gamma}$ is finitely generated.

Proof: Fact 1: The metric of a tree is of negative type.

Fact 2: For any subset ${A}$ of a Hilbert space, any isometric ${\Gamma}$-action on ${A}$ extends to an isometric action on the whole Hilbert space.

Combining these to facts, we see that property ${F\mathcal{H}}$ implies boundedness of orbits in any action on a tree. $\Box$

We shall see later that ${F\mathcal{H}\Rightarrow F\mathcal{A}}$. What misses now is the fact that bounded orbits in a tree implies existence of fixed point. This will follow from an averaging procedure in a tree.

3.5. Linear versus non linear actions

The above argument originates in David Kazhdan’s 4 page paper introducing property (T) in 1967. There, he considers a countable group ${\Gamma}$, assumes that it is not finitely generated, enumerates,…. Then he considers

$\displaystyle \begin{array}{rcl} H=\bigoplus \ell^2 (\Gamma/\Gamma_n). \end{array}$

This is a unitary representation of ${\Gamma}$, without fixed vectors. But the function ${\delta_1 \in \ell^2 (\Gamma/\Gamma_n)}$ is ${\Gamma_n}$-invariant. So every finite subset of ${\Gamma}$ fixes some non zero vector in ${H}$.

Theorem 8 (Kazhdan) Let ${G}$ be an almost simple Lie group of rank ${\geq 2}$ (e.g. ${Sl(3,{\mathbb R})}$). Let ${\Gamma\subset G}$ be a lattice, i.e. a discrete subgroup of finite covolume. Let ${H}$ be a unitary representation of ${\Gamma}$, without fixed vectors. Then there are no almost fixed vectors either.

Almost fixed vectors is weaker than the property that every finite subset has a non zero fixed vector. Kazhdan calls this property (T), we shall give a precise definition in the next section.

Instead of using unitary representations, Kazhdan could have used the isometric action of ${G}$ (and ${\Gamma}$) on the symmetric space ${G/K}$, ${K}$ a maximal compact subgroup of ${G}$. But the geometry of fundamental domains of ${\Gamma}$ was not fully understood at that time (it was in special cases, like ${SL(2,{\mathbb R})}$, since Siegel, and possibly, Poincaré), an a proof of finite generation along these lines was completed only a bit later (by Borel and Serre), and is more complicated (see also recent preprint by Tsachik Gelander). Unitary representations can be viewed as a linearization of the situation. They are very flexible and allow for soft arguments.

Theorem 9 (Delorme, Guichardet) For locally compact and ${\sigma}$-compact groups, property (T) and property ${F\mathcal{H}}$ coincide.

Later, we shall give a proof based on ultrafilters.

4. Kazhdan’s property (T)

4.1. Definition

Definition 10 (Kazhdan) Let ${Y}$ be a Hilbert space. Let ${\pi:\Gamma\rightarrow U(Y)}$ be a unitary representation. Say that ${\pi}$ almost has invariant vectors if for all finite sets ${S\in\Gamma}$, for all ${\epsilon>0}$, there is a ${y\in Y}$ such that ${|y|=1}$ and (10)

$\displaystyle \begin{array}{rcl} \forall s\in S,~ |\pi(s)y-y|<\epsilon. \end{array}$

Be careful with terminology: we indeed mean “almost has invariant vectors” and not “has almost invariant vectors”.

Note: ${|\pi(s)y-y|}$ is small iff the matrix coefficient ${\xi:\gamma\mapsto\pi(\gamma)y\cdot y}$ is close to the constant function ${1}$ uniformly on ${S}$. Indeed,

$\displaystyle \begin{array}{rcl} |\pi(s)y-y|^2 =2-2\pi(s)y\cdot y. \end{array}$

So ${\pi}$ almost has invariant vectors iff in the topology of uniform convergence on compact sets, the constant function is in the closure of the matrix coefficients of ${\pi}$.

Definition 11 (Kazhdan) Say ${\Gamma}$ has property (T) if for every unitary representation ${\pi}$ of ${\Gamma}$, ${\pi}$ almost has invariant vectors implies that ${\pi}$ has invariant vectors.

We already saw that for discrete groups, property (T) implies finite generation.

4.2. Kazhdan constants

Lemma 12 Let ${\Gamma}$ have a finite generating set ${S}$. It is enough to check inequality (10) with that ${S}$.

Proof: By triangle inequality, if (10) holds with constant ${\epsilon}$, (10) holds too for set ${S^k}$ with constant ${k\epsilon}$. $\Box$

Fixing ${S}$, one can say that ${y\in Y}$ is ${(S,\epsilon)}$-invariant if

$\displaystyle \begin{array}{rcl} \forall s\in S,\quad |\pi(s)y-y|<\epsilon. \end{array}$

One says that ${\Gamma}$ has almost invariant vectors if ${\forall \epsilon>0}$, ${\exists y\in Y}$, ${|y|=1}$, which is ${(S,\epsilon)}$-invariant.

Definition 13 Let ${\Gamma}$ have a finite generating set ${S}$. Let ${Y}$ be a Hilbert space. Let ${\pi:\Gamma\rightarrow U(Y)}$ be a unitary representation. Define, for unit vectors ${y\in Y}$,

$\displaystyle \begin{array}{rcl} E_S (y)=\frac{1}{2|S|}\sum_{s\in S}|\pi(s)y-y|^2 . \end{array}$

And property (T) can be reformulated as follows.

Proposition 14 For a countable discrete group ${\Gamma}$ and a finite generating set ${S}$, the following are equivalent.

1. ${\Gamma}$ has property (T).
2. For every unitary representation ${\pi}$, there exists ${\epsilon>0}$ such that existence of a unit vector with ${E_S (y)<\epsilon}$ implies existence of non zero invariant vectors.
3. There exists ${\epsilon>0}$ such that for all unitary representations ${\pi}$, existence of a unit vector with ${E_S (y)<\epsilon}$ implies existence of non zero invariant vectors.

A number ${\epsilon}$ as in the proposition is called a Kazhdan constant for ${\Gamma}$ and ${S}$.

Proof: To interchange ${\exists}$ and ${\forall}$, construct the giant unitary representation which contains at least one copy of every irreducible representation. Since ${\Gamma}$ is countable, each lives in a separable Hilbert space, fix one once and for all. Then unitary representations of ${\Gamma}$ in that space form a set, and we can sum then up. $\Box$

Theorem 15 (Gelander, Zuk) ${\Gamma=SO(5,{\mathbb Z}[\frac{1}{5}])}$ has property (T). Nevertheless, it has finite generating sets with arbitrarily small Kazhdan constants.

It is not known wether ${SL(3,{\mathbb Z})}$ has the same property. This issue is related to recent work by Helfgott, Bourgain, Gamburd on expansion of Cayley graphs of finite quotients of such groups with respect to (nearly) arbitrary generating systems.

4.3. Spectral formulation of property (T)

For unit ${y}$, let us calculate

$\displaystyle \begin{array}{rcl} E_S (y)&=&\frac{1}{2|S|}\sum_{s\in S}|\pi(s)y-y|^2 \\ &=&\frac{1}{2|S|}\sum_{s\in S}2-2\pi(s)y\cdot y \\ &=&(I-\pi(A_S))y\cdot y, \end{array}$

where ${A}$ denotes the averaging operator

$\displaystyle \begin{array}{rcl} A_S =\frac{1}{|S|}\sum_{s\in S}s, \end{array}$

viewed as an element of the group ring ${{\mathbb Z}\Gamma}$.

For any unitary representation ${\pi}$, ${\pi(A_S)}$ is a self adjoint operator or norm ${\leq 1}$. Its spectrum is contained in ${[-1,1]}$, so ${spec(I-\pi(A_S))\subset[0,2]}$. Saying that for all unit ${y}$, ${E_S (y)\geq\epsilon}$ is equivalent to ${spec(I-\pi(A_S))\in[\epsilon,2]}$. Invariant vectors (if any) form the ${0}$-eigenspace. This proves that

Proposition 16 For a discrete group ${\Gamma}$ and a finite generating set ${S}$, the following are equivalent.

1. ${\Gamma}$ has property (T).
2. There exists ${\epsilon>0}$ such that for all unitary representations ${\pi}$,$\displaystyle \begin{array}{rcl} spec(\pi(A_S))\in[-1,1-\epsilon]\cup\{1\}. \end{array}$

4.4. Bipartiteness

Note that the spectral criterion is one sided. We would prefer two sided gaps, since these would be equivalent to powers of ${A_S}$ tending to ${0}$. To get around this annoying point, two ways.

1. Replace ${A_S}$ with ${\frac{1}{2}(I+A_S)}$.
2. Replace ${A_S}$ with ${A_S^2 =\frac{1}{|S|^2}(I+A_{S^{(2)}})}$, where ${S^{(2)}}$ is the sphere of radius ${2}$ (with multiplicities).

In other words, the second choice differs from the first only in that it amounts to working with the subgroup ${\Gamma^2}$ generated by ${S^2}$, which has index at most ${2}$ in ${\Gamma}$. We shall freely use either trick.

Lemma 17 The following are equivalent.

1. ${spec(A_S)\subset[-1,1-\epsilon]\cup\{1\}}$.
2. For all ${y\in Y}$, (17)$\displaystyle \begin{array}{rcl} E_{S^{(2)}}(y)\leq (2-\epsilon)E_S (y), \end{array}$where

$\displaystyle \begin{array}{rcl} E_{S^{(2)}}(y)=\frac{1}{2|S|^2}\sum_{s_1 ,\,s_2}|\pi(s_1 s_2) y-y|^2 =\frac{1}{2}\mathop{\mathbb E}_{s_1 ,\, s_2}|\pi(s_1 s_2) y-y|^2 . \end{array}$

We call inequality (17) a Poincaré inequality. It is nice since it is purely metric. It will make sense in a non linear setting. It is annoying that the precise value of the constant (being less than ${2}$) be essential. We shall see later situations where a mere uniform constant suffices to draw conclusions.

4.5. Decidability issues

Since ${\Gamma}$ acts by isometries, ${|\pi(s_1 s_2) y-y|^2 =|\pi(s_1) y-\pi( s_2)y|^2}$, and one can rewrite the Poincaré inequality (17) in the form

$\displaystyle \begin{array}{rcl} \frac{1}{2}\mathop{\mathbb E}_{s_1 ,\, s_2}|\pi(s_1) y-\pi(s_2)y|^2 \leq C\,\mathop{\mathbb E}_{s}|\pi(s) y-y|^2 . \end{array}$

This is reminiscent of hypercontractivity (see Bernard Maurey’s 4th course).

View it as a statement about functions ${f:S\rightarrow Y}$ (in fact, only those functions which can be extended to actions). Take an ${r}$-ball ${B\in Cay(\Gamma,S)}$. Consider those functions ${f:B\rightarrow{\mathbb R}}$ such that

$\displaystyle \begin{array}{rcl} |f(\gamma\gamma_1)-f(\gamma\gamma_2)|=|f(\gamma_1)-f(\gamma_2)|. \end{array}$

If (17) holds for such functions, a fortiori it holds for functions coming from orbits. In fact, to control the constant up to a factor of ${\frac{1}{R}}$, one needs only check it for functions on ${B}$ with values in the finite set ${\{-1, -\frac{R-1}{R},\cdots,1\}}$. A computer can do that.

If it fails for all ${r}$, one can pass to a limiting function ${f}$ defined on all of ${Cay(\Gamma,S)}$, with values in some Hilbert space ${H}$, arising from an orbit of an isometric ${\Gamma}$-action on ${H}$. Since ${E_{S^{(2)}}(f)+2E_S (f)}$, ${\Gamma}$ does not have property (T).

This is an algorithm which can enumerate all group presentations giving groups having property (T). Thus we have shown

Corollary 18 (Silberman) The set of presentations of Kazhdan groups is recursively enumerable.

Corollary 19 (Shalom) If ${\Gamma}$ has property (T), then ${\Gamma}$ is the quotient of a finitely presented group which has property (T).

Proposition 20 Property (T) is undecidable.

Proof: Since free products act on trees without fixed points, ${\Gamma\star\Gamma}$ has property (T) iff ${\Gamma}$ is trivial. So an algorithm deciding if a presentation produces a group with property (T), applied to ${\Gamma\star\Gamma}$, would decide if ${\Gamma}$ is non trivial, this does not exist. $\Box$