## Notes of Emmanuel Breuillard’s informal talk in Cambridge 21-02-2017

Informal discussion on spectral gaps for isometric group actions

1. Finite Kazhdan data

1.1. Banach Kazhdan data ?

Cornelia Drutu wants a finite subset ${S}$ of a Lie group ${G}$ which is a Kazhdan set for every affine isometric action of ${G}$ on a Banach space from a certain class. This means there exists ${\epsilon>0}$ such that if there exists a point ${v}$ such that ${\max_S d(sv,v)<\epsilon}$, then there is a fixed point.

Shalom provides one for ${Sl(n,{\mathbb R})}$ which does the job for Hilbert spaces. Cornulier generalized this to all Kazhdan Lie groups.

Bader-Furman-Gelander-Monod establish ${F_{L^p}}$ for products of higher rank simple Lie groups, and this passes to lattices.

Here is how examples of lattices are obtained. Let ${\mathbb{G}}$ be an algebraic group defined over a number field ${K}$. Let ${K_i}$ be the Archimedean completions of ${K}$. ${\mathbb{G}(K_i)}$ can be compact or not. For instance, if ${\mathbb{G}=SO(q)}$, ${q}$ a quadratic form, then ${\mathbb{G}(K_i)}$ is compact iff ${q}$ is ${K_i}$-anisotropic, i.e. does not represent 0 over ${K_i}$.

Then (Borel-Harish Chandra) ${\Gamma=\mathbb{G}(\mathcal{O}_K)}$ is a lattice in the real Lie group

$\displaystyle \begin{array}{rcl} G=\prod_i \mathbb{G}(K_i) \end{array}$

and ${\Gamma}$ is cocompact iff one of the factors is compact. Remove the compact factors. This is a source of lattices. One can arrange that each remaining factor has higher rank. Then BFGM applies, a finite generating set of ${\Gamma}$ does the job for the family of ${L^p}$ spaces.

For a specific example, let ${K={\mathbb Q}[\sqrt{2}]}$ and

$\displaystyle \begin{array}{rcl} q(x_1,\ldots,x_4)=x_1^2+x_2^2+\sqrt{2}x_3^2+\sqrt{2}x_4^2. \end{array}$

Then ${\Gamma=SO({\mathbb Z}[\sqrt{2}])}$, up to finite index, is a lattice of rank 2 Lie group ${SO(2,2)}$.

PP: aren’t we making our life harder? If ${S}$ generates a dense subgroup, it should be easier ?

AV: Some of it remains for lattices in products of rank one groups ? Property ${\tau}$ is sometimes known.

1.2. Search for expanders

EB: In other words, what is looked for is a source of especially strong expanders.

CD: From Goulnara Arzhantseva’s point of view, a good expander should also have a large girth, proportional to diameter. This is required for counterexamples to Baum-Connes conjecture. Such graphs can be embedded into groups, by Gromov’s ansatz.

EB: Cayley graphs of finite simple groups are expected to behave in this way, for suitable generating sets. Babai’s conjecture states that for all generating sets, the diameter of the alternating group ${A_n}$ grows at most polynomially. On the other hand, for generic generating sets, girth is at least ${\log(n!)\sim n\log n}$. Babai’s conjecture is not even known for generic generating sets (best result by Helfgott and Seres is not far).

Babai’s conjecture for finite simple groups is

$\displaystyle \begin{array}{rcl} \sup_{S\, \mathrm{finite\,subset}}\mathrm{diameter}(Cay(G,S))=O(\log|G|^C). \end{array}$

For bounded rank, much better is expected,

$\displaystyle \begin{array}{rcl} \sup_{S\, \mathrm{finite\,subset}}\mathrm{diameter}(Cay(G,S))\leq \mathrm{Const}(\mathrm{rank}).\,\log|G|. \end{array}$

Known for ${Sl(2,p)}$. Furthermore, in this case, for random generators, both diameter and girth ${\sim\log p}$.

EB: Akhmedov shows that if ${\Gamma}$ is a linear group, not virtually solvable, then there exist finite generating sets with arbitrarily large girth.

Conjecture. For ${PSl(2,q)}$, for all generating sets, the Cayley graphs have a uniform lower bound on ${\lambda_1}$.

It is known for ${q}$ prime with a small family of exceptions.

Conjecture. ${Sl(3,{\mathbb Z})}$ has uniform property (T).

This would imply trivially a lot of recent results for ${Sl(3,p)}$, ${Sl(3,n)}$,… So it is pretty strong and should not generalize much. For instance, lattices in product cannot have this property, since they map onto dense subgroups of factors, which can be generated by small elements, and thus violate any uniformity.

It is hard since an arbitrary generating set need not contain unipotents, which are crucial to classical arguments (Shalom,…).

Fixed point properties. Is there a strong property ${\tau}$ ? Yes, see Lafforgue.

2. Finite simple groups

EB: This emerged from Lubotzky’s 1,2,3 question. 1 and 2 is classical, 3 is due to Bourgain-Gamburd. Historically, this goes back to Pimsker and Margulis. This question showed how few tools one had to obtain spectral gaps. Property (T), Selberg’s ${\frac{3}{16}}$ theorem (applies to congruence covers of the modular surface, but converts into an estimate for finite Cayley graphs, see my survey in Groups St Andrews 2013). Zig-zag products. Random graphs. Zuk’s criterion and applications by Dymara-Januskiewicz, Kassabov.

The Bourgain-Gamburd method was a revolution. It starts from the exponential mixing interpretation, i.e. behaviour of random walks. In short time, girth tells that things behave like on a tree. At larger time scales, additive combinatorics enters. The ${\ell^2}$ norm of the convolution product of measures must decay fast unless measures charge approximate subgroups. To conclude, use Sarnak-Xue’s multiplicity trick: because of symmetry, eigenspaces are representations, their dimensions are quite large.

The spectral gap for compact Lie groups was present in people’s minds, since Lubotzky-Philips-Sarnak, see Sarnak’s book. There is a motivation from quantum computation. One need to explore all of ${SU(2)}$ by combining a finite number of gates, i.e. multiplying elements from of fixed finite set. To get a ${\delta}$-dense subset of ${SU(2)}$, one needs words of length polylog in ${\delta}$, see the book of …

## Notes of Roland Bauerschmidt’s Cambridge lecture 7-02-2017

Local Kesten-McKay law for random regular graphs

Joint with Jiaohuang Huang ans Hong-Tzer Yau.

1. Motivation: quantum chaos

Consider billiard motion in a rectangle. It is a classically integrable system. The corresponding quantum problem consists in studying the Laplacian with Dirichlet boundary conditions. Eigenvalues can be explicitely computed. If square of ratio of sides is irrational, it is conjectured that

$\displaystyle \begin{array}{rcl} \frac{1}{N}\#\{j\leq N\,;\,\frac{\lambda_{j+1}-\lambda_j}{4\pi\mathrm{area}}\in[u,v]\}\rightarrow\int_u^v e^{-s}\,ds. \end{array}$

There are pretty close results (Sarnak). There are eigenfunctions which are localised near periodic orbits.

The Berry-Tabor conjecture is more general. In a hyperbolic rectangle, the classical dynamics is chaotic, the conjectured limit eigenvalue gap distribution is the eigenvalue gap distribution of a GOE matrix (standard Gaussian measure on space of symmetric ${N\times N}$ matrices with inner product ${Trace(XY)}$.

Bohigas-Giannoni-Schmit expect this to be true in general for chaotic systems.

The phenomenon of delocalised eigenfunctions is closely related.

It would be fun to study random surfaces, but hard. Therefore, we stick to graphs.

2. Random regular graphs

Let ${G_{N,d}}$ be the set of simple ${d}$-regular graphs on ${N}$ vertices. Equivalently, of 0-1 ${N\times N}$ symmetric matrices ${A}$ with rows summing to ${d}$. Use uniform probability measure on ${G_{N,d}}$, fix ${d}$ and let ${N}$ tend to infinity.

Locally, ${G_{N,d}}$ looks like a ${d}$-tree up to radius ${R=c\log_d N}$, ${c<1}$, i.e. ${R}$-balls of all vertices have a bounded number of cycles, no cycles at all for most vertices.

This implies that normalized eigenfunctions ${v}$ cannot be localized: for any ${d\geq 3}$,

$\displaystyle \begin{array}{rcl} \|v\|_\infty =O(\frac{1}{\log_d N}). \end{array}$

(Brooks-Lindenstrauss, Dumitriu-Pal, Geisinger).

Note that in the GOE, with high probability,

$\displaystyle \begin{array}{rcl} \|v\|_\infty =O(\sqrt{\frac{\log N}{N}}), \end{array}$

which is much smaller. Our result gets closer to this bound, except for the highest eigenvalues.

Theorem 1 For ${d\geq 10^{40}}$, then with high probability,

$\displaystyle \begin{array}{rcl} \|v\|_\infty \leq \frac{(\log N)^{100}}{\sqrt{N}}. \end{array}$

for all eigenvectors ${v}$ with eigenvalue in ${[-2\sqrt{d-1}+\epsilon,2\sqrt{d-1}-\epsilon]}$.

3. Green function

I.e. resolvent ${G(z)=(H-z)^{-1}}$, ${z\in{\mathbb C}}$, ${\Im(z)>0}$, where ${H=\frac{1}{\sqrt{d-1}}A}$. It allows to recover the spectral measure

$\displaystyle \begin{array}{rcl} \frac{1}{N}\Im Trace(G(E+i\eta) \end{array}$

and the eigenvectors: if ${Hv=Ev}$, then

$\displaystyle \begin{array}{rcl} \|v\|_\infty =\max_j \inf_{\eta>0}\sqrt{\eta\Im G_{ij}(E+i\eta)}. \end{array}$

The Green’s function of the infinite tree is obtained by summing over walks, removing vertices (${G}$ becomes ${G^{(i)}}$), leading to a functional equation

$\displaystyle \begin{array}{rcl} G_{jj}^{(i)}(z)=-\frac{1}{z+\frac{1}{d-1}\sum_{k\sim i}G_{kk}^{(j)}(z)}, \end{array}$

and to the value

$\displaystyle \begin{array}{rcl} G_{ii}(z)=m_d(z)=-\frac{1}{z+\frac{d}{d-1}m_{sc}(z)},\quad m_{sc}(z)=-\frac{1}{z+m_{sc}(z)}. \end{array}$

Le local tree-like structure gives that

$\displaystyle \begin{array}{rcl} \frac{1}{N}Trace G(z)\rightarrow m_d(z) \end{array}$

for any fixed ${z\in{\mathbb C}_+}$. It follows that the tends to an explicit law, th Kesten-McKay law.

4. Tree extension

Let us replace everything outside a ball of radius ${\ell}$ with trees. One gets a ${d}$-regular graph, eventually polluted by a few cycles.

Main estimate. Simultaneously for every ${z}$ with large imaginary part, all vertices ${i}$, ${j}$, with high probability,

$\displaystyle \begin{array}{rcl} G_{ij}(G,z)=G_{ij}(TE)+O(\log N)^{-c}, \end{array}$

where ${TE}$ is the tubular neighborhood of width ${r=\log\log N}$ of ${\{i,j\}}$ (empty if ${d(i,j)>r}$). The idea is that the right-hand side is deterministic and locally computable.

5. Proof ideas

The main estimate is not too hard if ${\Im(z)\geq 2d}$, by direct expansion.

To access global structure, we resample boundaries of large balls (radius ${\log\log N}$). Simultaneous switching of all pairs outside balls that do not collide is measure preserving. So starting from random graph ${G}$, this defines a new graph ${\tilde G}$ where the boundary gets random.

Use randomness of boundary to obtain two key improvements.

1. Better decay of ${G_{ij}(\tilde G)}$ for distinct vertices of the boundary.
2. Concentration estimate.

## Notes of Pierre Pansu’s informal Cambridge lecture 07-02-2017

Coarse spaces associated to dynamics and spectral gaps

As an incentive for a working seminar, I informally survey papers by Roe, Drutu-Nowak, Vigolo, Benoist-De Saxcé (+ Sawicki).

Goal: understand ways of encoding dynamical properties of group actions in a coarse metric space.

1. Examples of dynamics

1. Hyperbolic dynamics: ${\begin{pmatrix} 2 &1 \\ 1 & 1 \end{pmatrix}}$ acting on the 2-torus, negatively curved geodesic flow.
2. Isometric dynamics: action on ${G}$ compact Lie group of dense subgroup ${\Gamma< G}$, action of group on its profinite (resp. pro-${p}$) completion.

2. Topological entropy

If ${A\in\mathrm{Homeo}(Y)}$, ${Y}$ compact metrizable, pick arbitrary compatible background metric ${d_Y}$ on ${Y}$. Fot ${t\in{\mathbb N}}$, define ${Y_t}$ as the space of length ${t}$ orbits, i.e. ${Y}$ metrized by

$\displaystyle \begin{array}{rcl} d_t(y,y')=\sup_{0\leq s\leq t}d_Y(A^s(y),A^s(y')). \end{array}$

Fix ${\epsilon>0}$. Let ${N(Y_t,\epsilon)}$ be the number of ${\epsilon}$-balls needed to cover ${Y_t}$. Then topological entropy is

$\displaystyle \begin{array}{rcl} h(A):=\lim_{t\rightarrow\infty}\frac{1}{t}\log N(Y_t,\epsilon). \end{array}$

Result does not depend on choice of ${\epsilon}$, neither on choice of background metric ${d}$.

Definition 1 (Gromov ?) Let ${X}$ be the space of geodesic segments, i.e. ${X=Y\times{\mathbb N}}$ metrized by

$\displaystyle \begin{array}{rcl} \tilde d((y,t),(y',t'))=|t-t'|+\sup_{0\leq s\leq t}d_Y(A^s(y),A^s(y')). \end{array}$

Since, coarsely, this is a half-line, replace ${\tilde d}$ with the corresponding ${\epsilon}$-geodesic metric

$\displaystyle \begin{array}{rcl} d_\epsilon(x,x')=\inf\{\sum_{i=0}^{k-1}\tilde d(x_i,x_{i+1})\,;\,x_0=x,\,x_k=x',\,\sup_i \tilde d(x_i,x_{i+1})<\epsilon\}. \end{array}$

Question. Show that coarse space ${X_\epsilon}$ does not depend on ${\epsilon}$ nor on choice of background metric.

If so, every large scale invariant of ${X_\epsilon}$ become a dynamical invariant. For instance, topological entropy equals the exponential volume growth of ${X_\epsilon}$.

Question. If ${A}$ is the time 1 geodesic flow on unit tangent bundle ${Y=T_1 M}$ of a compact negatively curved manifold ${M}$, then ${X_\epsilon}$ is quasi-isometric to universal covering space ${\tilde M}$ ?

3. John Roe’s warped cone construction

${Y}$ Riemannian manifold, ${\mathrm{cone}(Y)=Y\times{\mathbb R}_+}$ with metric ${t^2g_Y+dt^2}$. Roe warps it with the ${\Gamma}$ action on ${Y\times{\mathbb R}_+}$ arising from ${\Gamma}$-action on ${Y}$.

Definition 2 ${X}$ metric space. ${\Gamma}$ acts by homeos of ${X}$ with finite generating set ${S}$. Define warped metric ${d_\gamma=}$ largest metric on ${X}$ with is ${\leq d_X}$ and such that for all ${s\in S}$, ${d_\Gamma(x,sx)\leq 1}$. Alternatively,

$\displaystyle \begin{array}{rcl} d_\Gamma(x,x')=\inf\{\sum_{i=0}^{k-1}d_X(x_i,x_{i+1})+|\gamma_i|\,;\,x_0=x,\,x_k=x',\,\gamma_i\in\Gamma\}. \end{array}$

Define warped cone ${O_\Gamma(Y)}$ as ${\mathrm{cone}(Y)}$ warped by ${\Gamma}$ action on ${Y\times{\mathbb R}_+}$ arising from ${\Gamma}$-action on ${Y}$.

As a coarse space, ${O_\Gamma (Y)}$ does not depend on choices of background metric on ${Y}$ and generating set of ${\Gamma}$. If ${\Gamma}$ acts by bi-Lipschitz homeos, ${O_\Gamma (Y)}$ has bounded geometry.

Example. Let ${Y={\mathbb R}/{\mathbb Z}}$ and ${\Gamma}$ cyclic group generated by translation ${y\mapsto y+\alpha}$. Then ${X=}$ Euclidean plane, ${\Gamma}$ is generated by the rotation of angle ${2\pi\alpha}$. The diameter of level ${t}$ seems to be governed by

$\displaystyle \begin{array}{rcl} \min_{0\leq s\leq t} 2\pi t\{s\alpha\}. \end{array}$

If ${|\alpha-\frac{p}{q}|\leq \frac{\epsilon}{q^2}}$, then ${q\{q\alpha\}\leq\epsilon}$. Thus if ${\alpha\notin{\mathbb Q}}$, there are infinitely many values of ${t}$ (Hurwitz) for which the diameter of ${t}$ level collapses to something pretty small (not smaller than ${\log t}$, though).

Question. Describe this example in more detail.

Remark. A smooth ${\Gamma}$-action on ${Y}$ can be suspended into a ${Y}$-bundle over ${B\Gamma}$ with a foliation ${\mathcal{F}}$ transverse to fibers. Pick a complement ${\nu}$ to ${T\mathcal{F}}$ in ambient tangent bundle. Warp metric on ${\mathrm{cone}(Y)}$ as follows:

$\displaystyle \begin{array}{rcl} g_{\mathcal{F}}+t^2 g_{\nu}+dt^2. \end{array}$

Resulting space is coarse equivalent to ${O_{\Gamma}(Y)}$.

Example. Let ${Y={\mathbb R}/{\mathbb Z}}$ and ${\Gamma}$ cyclic group generated by translation ${y\mapsto y+\alpha}$. Then suspension is torus ${({\mathbb R}/{\mathbb Z})^2}$ with linear foliation of slope ${\alpha}$.

Question. Let ${\Gamma={\mathbb Z}}$ acting by matrix ${\begin{pmatrix} 2 &1 \\ 1 & 1 \end{pmatrix}}$ on the 2-torus. Compare ${O_\Gamma(T^2)}$ with warping along linear foliation defined by one eigenspace.

Question. Let ${\Gamma}$ act on its profinite completion ${\hat\Gamma}$ (or on its pro-${p}$ completion). Show ${O_{\Gamma}\hat\Gamma}$ is coarsely equivalent to the box space.

G. Arzhantseva points out Damian Sawicki’s paper on warped cones and profinite completions. It contains many examples and results parallel to those of Roe below.

4. Theorems

4.1. Monstruosity

Yu introduced property A as a coarse metric generalization of Haagerup property. It is designed to produce coarse Hilbert embeddings.

Definition 3 (Yu) Say space ${X}$ has property A if there exists a sequence of families of probability measures ${x\mapsto f_n(x)}$ on ${X}$ such that

1. Uniformly bounded support: ${\mathrm{supp}(f_n(x))\subset B(x,r_n)}$.
2. Spreading: ${\|f_{n}(x)-f_n(x')\|_1 \leq\epsilon_n(s)\rightarrow 0}$ uniformly as ${d(x,x')\leq s}$.

Next is one of a hundred equivalent definitions of amenability of actions. All actions of amenable groups are amenable, but there are more. For instance, the action of a hyperbolic group on its ideal boundary is amenable. Morally (this is exactly true only for rank one Lie groups), this means that stabilizers are amenable.

Definition 4 Say ${\Gamma}$ action on compact space ${Y}$ is amenable if there exists a family of finite support probability measures ${y\mapsto\mu_{n}(y)}$ on ${\Gamma}$ such that ${\forall \gamma\in\Gamma}$,

$\displaystyle \begin{array}{rcl} \sup_{y\in Y}\|\gamma\cdot\mu_{n}(y)-\mu_{n}(\gamma y)\|_1 \rightarrow 0 \quad \textrm{as }n\rightarrow\infty. \end{array}$

Note that ${\Gamma}$, as a metric space, has property A iff it admits an amenable action on some compact space.

Roe 2005. If action of ${\Gamma}$ on ${Y}$ is amenable, then ${O_\Gamma(Y)}$ has Yu’s property A. In particular, it has a coarse embedding into a Hilbert space.

Roe 2005. Let ${G}$ be a compact Lie group and ${\Gamma a dense subgroup. Then

$\displaystyle \begin{array}{rcl} O_\Gamma (G) \textrm{ has property }A \quad &\Rightarrow&\quad \Gamma \textrm{ is amenable}.\\ O_\Gamma (G) \textrm{ has coarse Hilbert embedding} \quad &\Rightarrow&\quad \Gamma \textrm{ has Haagerup property}.\\ \end{array}$

These results indicate that warped cones can be monstruous (if ${\Gamma}$ is Kazhdan, for instance). Roe even announced a warped cone counterexample to coarse Baum-Connes conjecture, but nothing appeared until

Drutu-Nowak 2015. Certain warped cones have ghost projections in their Roe algebras.

4.2. A new source of expanders

Let ${S}$ be a generating set for ${\Gamma}$. Say ${\Gamma}$ action is expanding in measure if ${\mu(SA)\geq (1+\epsilon)\mu(A)}$ for all subsets ${A}$ with ${\mu(A)\leq 1/2}$.

Vigolo 2016. Let ${\Gamma}$ preserve a probability measure on ${Y}$. Then levels of ${O_\Gamma (Y)}$ constitute a coarse expander iff ${\Gamma}$ action on ${L^2(Y)}$ is expanding in measure.

Question. Investigate further properties of these new expanders.

Question. Compare Vigolo’s construction to classical constructions of expanders, like Margulis type (${\Gamma}$ acting on ${\hat\Gamma}$) or zig-zag products (perhaps requires generalizing warped cones to finitely generated equivalence relations).

5. Actions with spectral gaps

It turns out that measure expansion on ${Y}$ is equivalent to spectral gap on ${L^2(Y)}$.

Definition 5 Let ${\Gamma}$ act linearly isometrically on some Banach space ${B}$. Say action has a spectral gap if if there exists ${\epsilon>0}$ and a generating set ${S}$ such that ${\forall v\in B}$, ${|v|=1}$, ${\exists s\in S}$ such that ${|sv-v|\geq\epsilon}$.

For instance, Kazhdan groups have a uniform spectral gap in any unitary representation. On the other hand, irrational rotations of the circle never have spectral gaps on ${L^2(\textrm{circle})}$. Nevertheless, even free groups can have spectral gaps in specific representations.

Therefore we want to understand a few more examples of dense subgroups ${\Gamma with a spectral gap.

The adequate context is that of a symmetric probability measure ${\mu}$ on ${G}$ (think of ${\Gamma}$ as generated by the support of ${\mu}$). One always assumes that the support of ${\mu}$ generates a dense subgroup. A spectral gap on ${L^2(G)}$ means that the spectral radius of the averaging operator

$\displaystyle \begin{array}{rcl} \int_{G}L_g \,d\mu(g) \end{array}$

is ${<1}$.

Benoist-de Saxcé 2016 (generalizing Bourgain-Gamburd). Let ${G}$ be a compact simple Lie group. ${\mu}$ has a spectral gap iff ${\mu}$ is almost Diophantine in the following sense: ${\mu}$ does not put much mass on tubular neighborhoods of connected subgroups. Specifically, ${\exists c}$ and ${C}$ such that ${\forall n}$, for every proper closed subgroup ${H,

$\displaystyle \begin{array}{rcl} \mu\{HB(e^{-Cn})\}\leq e^{-cn}. \end{array}$

Benoist-de Saxcé 2016. Assume that ${\forall g\in\mathrm{supp}(\mu)}$, ${Ad_g}$ has algebraic entries (in a fixed basis of ${\mathfrak{g}}$). Then ${\mu}$ is almost Diophantine.

We see that the structure of ${\Gamma}$ does not show up (is ${\Gamma}$ free, most of the time ?), the essential role is played by the structure of ${G}$.

The main tool is de Saxce’s Product Theorem (elaboration on results of Bourgain-Gamburd): if a subset ${A\subset G}$ near identity does not grow much under tripling ${A\mapsto AAA}$, then ${A}$ is Hausdorff close to a proper closed subgroup. Size is measured by covering number ${N(A,\delta)}$.

## Notes of Eric Swenson’s Cambridge lecture 2-2-2017

Infinite torsion subgroups of ${CAT(0)}$ groups

Question. Let ${G}$ act geometrically on a proper ${CAT(0)}$ space. Can ${G}$ have an infinite torsion subgroup ?

Known. For cube complexes, Wise and Sageev show that a torsion group fixes a point (eventually at infinity). Tits alternative. So answer is no for these.

Coulon and Guirardel have an example of an infinite torsion group acting properly on an infinite dimensional ${CAT(0)}$ cube complex.

Today, as a kind of therapy, I collect all I know on this irritating open question.

1. ${CAT(0)}$ geometry

In ${CAT(0)}$ spaces, angles are well defined as limits of Euclidean comparison angles at small scales. Angle is always at most equal to the Euclidean comparison angle.

The visual boundary consists of equivalence classes of unit speed geodesic rays. Shadows of small balls seen from far away define a topology. If space ${X}$ is proper (need not be finite dimensional), ${X\cup\partial X}$ is compact and finite dimensional.

Angles in large biangles converge and define the angle metric on ${\partial X}$, that defines a finer topology. Let ${d_T}$ denote the corresponding path metric, known as the Tits metric. Angle and Tits coincide below level ${\pi}$. We denote by ${TX}$ the visual boundary equipped with Tits metric’s topology.

2. Conical limit points

If ${H, let ${\Lambda H}$, the limit set, denote the set of limit points of an orbit. A ray ${\alpha\in\Lambda H}$ is conical if some tubular neighborhood of ${\alpha}$ contains an unbounded subset of some orbit.

Theorem 1 If ${H has a conical limit point, then there is no bound on the orders of elements of ${H}$.

Indeed, there are numbers ${p_n\rightarrow\infty}$ and elements ${h_n\in H}$ such that ${h_n\alpha(p_n)\in \bar B(x_0,N)}$. Define rays

$\displaystyle \begin{array}{rcl} \alpha_n(t)=h_n(\alpha(t+p_n)),\quad t\in [-p_n,+\infty). \end{array}$

Up to extracting a subsequence, one can assume that ${\alpha_n}$ converge to a geodesic line ${L}$ uniformly on compact subsets. For ${j\gg i\gg 0}$, let ${g=h_i^{-1}h_j}$, and consider points ${\alpha(p_i)}$ (very close to ${g(\alpha(p_j))}$), ${g(\alpha(p_i))}$ (very close to ${g^2(\alpha(p_j))}$), and ${g^2(\alpha(p_i))}$ (very close to ${g^3(\alpha(p_j))}$). Hence the angle ${<_{g(\alpha(p_i))}(\alpha(p_i),g^2(\alpha(p_i)))}$ is almost ${\pi}$. Say ${g}$ has order ${n}$. Then iterates ${g^k(\alpha(p_i))}$ form the vertices of an ${n}$-gon with equal angles. In a ${CAT(0)}$ space, the sum of the angles of an ${n}$-gon is at most ${(n-2)\pi}$, this forces ${n}$ to be large.

Corollary 2 Every ${CAT(0)}$ group has a hyperbolic element.

Corollary 3 If ${H a ${CAT(0)}$ group is an infinite torsion group, then ${H}$ has no conical limit points.

3. Further results

Theorem 4 Let ${H acting geometrically on ${CAT(0)}$ space ${X}$. If

$\displaystyle \begin{array}{rcl} diam_T \Lambda H > 2\pi, \end{array}$

then ${H}$ contains a hyperbolic isometry.

Theorem 5 Let ${H acting geometrically on ${CAT(0)}$ space ${X}$. If ${H}$ is infinite torsion group, and if ${dim(X)}$ is minimal among such spaces, then ${\Lambda H}$ is infinite. Furthermore,

$\displaystyle \begin{array}{rcl} radius_T \Lambda H > \frac{\pi}{2}, \end{array}$

and ${H}$ does not fix a point in ${\partial X}$.

## Notes of Claus Koestler’s Cambridge lecture 26-01-2017

An elementary approach to unitary representations of Thompson’s group ${F}$

${F=\langle X_n \,|\,X_k X_n=X_{n+1}X_k,\,0\leq k. Unknown wether amenable or not. Does not contain free subgroups.

1. Sources for unitary representations of ${F}$

Characters!

Theorem 1 (Gohm-Kostler, Dudko-Madzrodski) Extreme points of the set of characters and in one to one correspondance with points of the 2-torus (characters of the abelianisation ${{\mathbb Z}^2}$) plus ${\{0,0\}}$ (left regular representation).

1. ${F}$ is non-alenable.
2. ${C^*_r(F)}$ has a unique normalized trace.

Not so productive.

1.2. Subfactor approach

Theorem 3 (Jones 2014) TFAE Any subfactor yields a unitary representation of ${F}$.

1.3. Probabilistic approach

Theorem 4 (Gohm, Evans, Bhat, Wills, C. Jones) Every non-commutative stationary Markov chain yields a unitary representation of ${F}$.

1.4. Graphical picture

for the monoid ${F^+}$ (Belk 2004).

Represent generators ${X_0}$ and ${X_1}$ as diagrams: collection of edges joining an ${n+1}$-point set to an ${n}$-point set, with two edges joining at 0 (resp. at 1). Such diagram can be composed. Get a category whose objects are finite sets ${([n])_{n\in{\mathbb N}}}$ and morphisms are finite binary forests.

In the semicosimplicial category ${\Delta_S}$, same objects, morphisms are increasing functions. They satisfy again the relations of ${F}$.

There is a covariant functor from ${\Delta_S}$ to the category of NonCommutativePS. With Evans et de Finetti, we related coface identities to spreadability and non-commutative Bernoulli shift.

2. From Markov chains to representations

Start with construction on infinite sets, kind of limit of the previous one: partial shifts.

Pass to Hilbert spaces: partial shifts on sequences of vectors in a fixed Hilbert space ${D}$. Let ${U}$ and ${V}$ be unitaries ${D\rightarrow D\oplus D}$. Inserting them in partial shifts ${S_k}$, get unitaries ${U_k}$.

Theorem 5 The unitaries ${T_k:=U_{k+1}S_{k+1}}$ satisfy Thompson ${F}$‘s relations.

We call this the standard form representation of ${F}$.

Theorem 6 Let ${\pi:F\rightarrow\mathcal{H}}$ be a unitary representation such that

1. No fixed vectors.
2. ${\mathcal{H}}$ is generated by fixed vectors of generators ${X_n}$.

Then ${\pi}$ has standard form up to modifications of ${S_k}$‘s.

2.1. Outlook

Does the V. Jones’ example fit into this framework ?

Study the ${C^*}$ and von Neumann algebras of ${F}$.

2.2. Question

What if you embed ${F}$ into diffeos of the circle and compose with unitary representations coming from conformal field theory ?

## Notes of Ian Leary’s Cambridge lecture 26-01-2017

Generalizing Bestvina-Brady groups using branched covers

Joint with Ignat Soroko and Robert Kropholler.

Initial motivation: prove that there exist uncountably many groups of type ${FP}$.

1. Finiteness properties

Recall that ${G}$ is type ${F}$ if it has a finite ${K(\pi,1)}$, i.e. it acts freely, cellularly and cocompactly on a contractible CW complex.

Say ${G}$ is type ${FH}$ if it acts freely, cellularly and cocompactly on an acyclic CW complex (same homology as a point).

Say ${G}$ is type ${FH}$ if it acts freely, cellularly and cocompactly on an acyclic CW complex (same integral homology as a point).

Say ${G}$ is type ${FL}$ if ${{\mathbb Z}}$ has a resolution of finite length by finitely generated free ${{\mathbb Z} G}$-modules.

Say ${G}$ is type ${FP}$ if ${{\mathbb Z}}$ has a resolution of finite length by finitely generated projective ${{\mathbb Z} G}$-modules.

All these properties imply that ${G}$ is torsion free.

There is no known group that is ${FP}$ but not ${FL}$ or not ${FH}$ over ${{\mathbb Z}}$. If one replaces ${{\mathbb Z}}$ with ${{\mathbb Q}}$, one finds examples showing that ${FP({\mathbb Q})\not= FL({\mathbb Q})\not=FH({\mathbb Q})}$.

Bestvina-Brady associate a group ${BB_L}$ to a flag complex ${L}$, the kernel of the obvious map of the Artin group ${A_L}$ to ${{\mathbb Z}}$. Then

${BB_L}$ is finitely generated iff ${L}$ is connected.

${BB_L}$ is finitely presented iff ${L}$ is 1-connected.

${BB_L}$ is type ${F}$ iff ${L}$ is a point.

${BB_L}$ is ${FH}$ iff ${BB_L}$ is ${FP}$ iff ${L}$ is acyclic.

There are countably such groups. All embed in groups of type ${F}$.

3. New examples

3.1. Properties

Start with a connected finite flag complex ${L}$, with universal cover ${\tilde L}$, and a subset ${S\subset{\mathbb Z}}$. I build a finitely generated group ${G_L(S)}$ with following properties.

If ${S\subset T}$, there is an epimorphism ${G_L(S)\rightarrow G_L(T)\rightarrow 1}$.

For fixed ${L}$, the following are equivalent:

1. ${\forall S\subset{\mathbb Z}}$, ${G_L(S)}$ is ${FP}$.
2. ${\forall S\subset{\mathbb Z}}$, ${G_L(S)}$ is ${FH}$.
3. ${L}$ and ${\tilde L}$ are both acyclic.

If ${L}$ is not simply connected,

• There exist uncountably many isomorphism types of ${G_L(S)}$ as ${S}$ varies. With Ignat Soroko and Robert Kropholler, we even show that there are uncountably many quasi-isometry types.
• ${G_L(S)}$ is finitely presented iff ${S}$ is finite,
• ${G_L(S)}$ embeds in a finitely prresented group iff ${S}$ is recursively enumerable.

3.2. Corollaries

Using a trick of Mike Davies, we get

Theorem 1 For all ${n\geq 4}$, there exist an aspherical ${n}$-manifold ${M^n}$ with uncountably many regular acyclic covers with non-isomorphic (and propably non quasi-isometric) groups of deck transformations.

Theorem 2 Every countable group embeds in a group of type ${FP_2}$.

Question. If you believe in Gromov’s assertion (that a statement balid for all countable groups is either false or obvious), find a straightforward proof of this.

3.3. Construction

Let ${T_L}$ be the Salvetti complex. View it as a subcomplex of the torus (product of circles, one for each vertex). ${T_L}$ is a ${K(A_L,1)}$.

Addition defines a map ${T_L\rightarrow T_p}$ which, in homotopy, is Bestvina-Brady’s morphism. Lift it to induced ${{\mathbb Z}}$-cover, get a real function ${h}$ on ${\hat T_L}$. ${BB_L}$ is the fundamental group of ${\hat T_L}$. Each vertex link in this complex is the sphericalisation ${S(L)}$: replace vertices by simplices and edges by joins of such simplices.

Theorem 3 Fer every subset ${S}$ of integers, there is a ${CAT(0)}$ cube complex ${X_L^{(S)}}$ which is a regular branched covering of ${\hat T_L}$, branched only at vertices, whose links are isomorphic to

1. ${S(L)}$ for vertices in ${h^{-1}(S)}$,
2. ${S(\tilde L)}$ for vertices not in ${h^{-1}(S)}$,

We define ${G_L(S)}$ as the fundamental group of ${X_L^{(S)}}$.

Recall that in a ${CAT(0)}$ space, the link of a vertex is a strong deformation retract of its complement, hense fundamental groups of links. If ${L}$ has no local cut points, ${S(\tilde L)=\widetilde{S(L)}}$. If ${L}$ has cut-points, embed ${L}$ into ${L\times I}$ to get rid of them.

To distinguish these groups, we use Bowditch’s length spectrum.

## Notes of Dima Shlyakhtenko’s Cambridge lecture 25-01-2017

Homology ${L^2}$-Betti numbers for subfactors and quasiregular inclusions

Joint with Sorin Popa and Stefan Vaes.

There were several competing definitions, and they turn out to be equivalent, relief.

1. ${L^2}$ Betti numbers

Associative algebras ${A}$ (and bimodules ${V}$) have a Hochschild cohomology defined as follows. Cochains are maps ${A^{\otimes k}\rightarrow V}$. Coboundary is such that 1-cocycles are derivations. If ${A}$ is augmented (e.g. a group algebra), right ${A}$-modules can be turned into bimodules by using the augmentation ${\epsilon}$ as a left action.

When ${A={\mathbb C}\Gamma}$ is a group algebra, ${\Gamma=\pi_1(X)}$ and ${\tilde X}$ is contractible, then ${H^\cdot({\mathbb C}\Gamma,{\mathbb C})=H^\cdot(X)}$. Another choice of coefficients is ${\ell^2(\Gamma)}$, (or its completion ${L(\Gamma)}$). Then ${H^\cdot(\Gamma,\ell^2(\Gamma))}$ are modules over ${L(\Gamma)}$, hence have a dimension, the Betti number

$\displaystyle \begin{array}{rcl} \beta_k^{(2)}(\Gamma)=\mathrm{dim}_{L(\Gamma)}H^k(\Gamma,\ell^2(\Gamma)). \end{array}$

${L^2}$-Betti numbers of infinite groups are additive under free products and multiplicative under direct products.

A factor has some grouplike symmetry. To embody it, we have two choices, ${C^*}$-tensor categories and ${M\otimes M^{op}}$ included in Popa’s symmetric enveloping algebra.

2. The ${C^*}$-tensor categories point of view

Let ${\mathcal{C}}$ be a tensor category. When ${\alpha}$, ${\beta}$ are irreducible objects, we denote by ${(\alpha,\beta)}$ the set of intertwiners.

The tube algebra is defined, as a vectorspacen by

$\displaystyle Tube(\mathcal{C})=\bigoplus_{i,i,\alpha\in Irr(\mathcal{C})}(i\alpha,\alpha j).$

The multiplication ${\mathcal{A}_{ij}\times\mathcal{A}_{jk}\rightarrow\mathcal{A}_{ik}}$ is

$\displaystyle \begin{array}{rcl} VW=(V\otimes 1)(1\otimes W). \end{array}$

Define the idempotent ${p_i=id_{ie,ei)}\in\mathcal{A}_{ii}}$, where ${e}$ is the trivial representation. They generate a subalgebra ${\mathcal{B}}$. The augmentation ${\epsilon:\mathcal{A}\rightarrow{\mathbb C}}$ is defined by

$\displaystyle \begin{array}{rcl} \epsilon(p_i)=0,~(i\not=e),\quad \epsilon(id_\alpha)=d(\alpha). \end{array}$

We think of as a representation of ${\mathcal{A}}$ as a “representation” of ${\mathcal{C}}$.

The cohomology of ${\mathcal{A}}$ and a right ${\mathcal{A}}$-module ${K}$ is defined as follows. Cochains are morphisms ${p_e\mathcal{A}^{\otimes_\mathcal{B}(n+1)}}$ to ${K}$. The coboundary is given by the same formula as Hochschild.

Fot ${n=1}$, ${H^1}$ corresponds to derivations ${D}$ ${D(VW)=D(V)W+\epsilon(V)D(W)}$, modulo inner derivations ${D_\xi(V)=\epsilon(V)\xi-\xi V}$.

Possible choices for ${K}$: ${\epsilon}$ yields cohomology ${L^2(\mathcal{A},Trace)}$ leads to ${L^2}$ Betti numbers.

3. The quasiregular inclusion point of view

Let ${T\subset S}$ be a quasiregular inclusion. Define the tube algebra by a similar formula for the category ${_T S_T}$. It is Morita equivalent to the preceding tube algebra.

Cochains involve tensor powers of ${S}$ over ${T}$.

In case of trivial coefficients, i.e. take module ${S}$. Then get ${H_k(T\subset S,S)=0}$ if ${T}$ has finite index in ${S}$. This had been observed by Jones. More genrally, ${H_\cdot(\mathcal{A}_{\mathcal{C}},\epsilon)=0}$ if ${\mathcal{C}}$ has finite depth.

For calculations, one replaces cocycles ${Z}$ and coboundaries ${B}$ with smaller subspaces.

For instance, for amenable categories (in a Folner sense), Betti numbers vanishes. For finite index inclusions, ${\beta_0^{(2)}=1/[S:T]}$. Additivity under free products and multiplicativity under direct products od categories holds.

3.1. Graphical picture for trivial coefficients

On a 2-sphere with punctures, draw cochains as bodies with as many legs as factors in a tensor product. Then coboundary is alternating sum of picture obtained by removing each puncture at a time.

4. Computations

Mystery: what is ${H_k(T\subset T;}$ trivial coefficients ${)}$ ?

Kyed, Raum, Vaes, Valvekans have a recent result.

One may conjecture that for the category of quantum groups, one recovers Betti numbers of quantum groups. The study of fusion rings seems to indicate that it could be wrong.

One recovers Gaboriau’s Betti numbers for group ${\Gamma}$ acting on space ${X}$, ${T=L^{\infty}(X)}$ but ${S}$ is not quite ${L^{\infty}(X)\times\Gamma}$.