Notes of Mario Bonk’s lecture

Dynamics and quasiconformal geometry

1. Motivation : Cannon’s conjecture

${G}$ Gromov hyperbolic group. When is the ideal boundary a topological 2-sphere ? It is the case when ${G}$ is virtually the fundamental group of a compact hyperbolic 3-manifold.

Version 1: Are there other examples ?

Alternative formulation

Version 2: Show that the ideal boundary ${\partial G}$, equipped with a visual metric, is quasi-symmetric to the standard 2-sphere.

1.1. Visual metrics

Recall that the visual metric ${d(a,b)=e^{-\epsilon (a,b)_p}}$, where the Gromov product ${(a,b)_p}$ is (up to a bounded additive error) the distance of base point ${p}$ to the geodesic joing ${a}$ to ${b}$. Changing the parameter ${\epsilon}$ changes ${d}$ to a snowflake equivalent metric (i.e. a power of it). The word comes from the fact that the von Koch snowflake curve is snowflake equivalent to the real line.

1.2. Quasi-symmetry

Quasi-symmetric means that ratios ${\frac{d(x,y)}{d(x,z)}}$ in one metric are controlled by similar ratios in the other. Equivalently, balls in one metric are pinched between concentric balls in the other, with a bounded ratio of radii. This has to do with quasi-conformality (an infinitesimal version of quasi-symmetry): for Euclidean domains, quasi-symmetry is equivalent to qausi-conformality.

Proposition 1 Under mild assumptions, Gromov hyperbolic metric spaces are quasi-isometric iff their ideal boundaries are quasi-symmetric.

A Gromov hyperbolic group ${G}$ acts on its ideal boundary by quasi-symmetries, in a uniform manner. To make this quantitative, one must inteoduce the notion of quasi-Möbius map (replace ratios ${\frac{d(x,y)}{d(x,z)}}$ with cross-ratios of 4-tuples of points).

Version 2 implies Version 1. Indeed, if ${\partial G}$ is quasi-symmetric to the round sphere, ${G}$ acts by uniformly quasi-Möbius on the round sphere. A result of Sullivan and Tukia implies that the action is by Möbius transformations.

2. The quasi-symmetric uniformization problem

When is a metric space ${X}$ is quasi-symmetric to a standard space ${X_0}$ ?

This is relevant for the

2.1. Kapovich-Kleiner conjecture

Let ${G}$ be a Gromov hyperbolic group whose ideal boundary is homeomorphic to a Sierpinsky carpet (start with a square, cut in 9 pieces, remove central square, iterate in each of the 8 remaining squares). Does ${G}$ arise from a standard situation in hyperbolic geometry ?

This is equivalent to showing that ${\partial G}$ is quassymmetric to a round carpet (remove disjoint geometric circle from a circle, until no interior is left).

Cannon conjecture implies Kapovich-Kleiner conjecture.

This is also relevant to

2.2. Other problems in semi-group dynamics

From a branched covering ${f}$, one defines a Gromov hyperbolic graph ${G_f}$ (this a rather long story, I willnot give details).

Theorem 2 (Bonk-Meyer, Haissinsky-Pilgrim) Let ${f:S^2\rightarrow S^2}$ be a postcritically finite expanding branched covering. Then ${f}$ is conjugate to a rational map iff ${\partial G_f}$ is quasi-symmetric to the round sphere.

Sometimes it is true, sometimes it is not. The visual metrics arising from branched covering maybe non quasi-symmetric to the standard sphere.

2.3. Example: the snow sphere

Start with the boundary of the cube. Subdivide each face in 9 squares, build a small cube on the middle square, and iterate. This produces a metric which is not a snowflake of the standard sphere (it has rectifiable curves).

Question. Is it quasi-symmetric to the standard sphere ?

At first sight, one would bet that the answer is no. If it were, all squares in the construction should remain uniformly round. Flattening the first stage of the construction to a cube is easy, but doing this at all stages accumulates distorsion. Nevertheless,

Theorem 3 (Meyer) The snow sphere is quasi-symmetric to the round sphere.

3. What is known ?

A lot of positive results for dimensions 0 and 1. No positive results in higher dimensions ${n\geq 3}$. Semmes writes that all the naive facts one could think of turn out to be wrong. So interesting things happen in dimension 2.

3.1. Low dimensional results

Theorem 4 (Tukia-Väisälä) A metric on the circle is quasi-symmetri to the standard metric iff

1. It is doubling.
2. It has bounded turning: every arc ${a}$ with endpoints ${x}$ and ${y}$ has diameter${(a)\leq C\,d(x,y)}$.

Similar result for Cantor sets.

3.2. Results in dimension 2

Theorem 5 (Bonk-Kleiner) Let ${S}$ be a metric sphere homeomorphic to the 2-sphere. Assume that

1. ${S}$ is linearly locally connected (this is a necessary condition).
2. ${S}$ is Ahlfors 2-regular (this is not at all necessary).

Then ${S}$ is quasi-symmetric to the standard sphere.

Note that visual metrics of hyperbolic groups are Ahlfors-regular (Coornaert).

Theorem 6 (Bonk-Kleiner) Let ${G}$ be Gromov hyperbolic group whose ideal boundary is homeomorphic to the 2-sphere. Assume that the conformal dimension is attained as a minimum. Then ${\partial G}$ is quasi-symmetric to the standard sphere.

Recall that the conformal dimension is the infimal dimension of Alhfors regular metric spaces quasi-symmetric to ${\partial G}$.

We strongly use 2 dimensions, but an intermediate step applies in all dimensions: conformal dimension attained implies metric is Löwner. Note that there are examples (due to Bourdon and Pajot) of groups whose conformal dimension is not attained. These examples have boundaries which are not 2-spheres.

Notes of Zoltan Balogh’s lecture nr 2

Horizontal convexity in the Heisenberg group

Joint with Andrea Calogero and Alexandru Kristály.

1. Alexandrov’s theorem in Euclidean space

I think of a convex function as a function which stays above its supporting affine function at every point. The set of slopes (covectors) of these affine functions at ${x_0}$ is called the subdifferential ${\partial u(x_0)}$ of ${u}$ at ${x_0}$. The union of all these sets over the domain is denoted by ${\nabla u(\Omega)}$. Its measure ${\mathcal{L}^n(\nabla u(Omega))}$ is sometimes called the Monge-Ampère measure of ${u}$.

Theorem 1 (Alexandrov) There is a dimension dependant constant ${C}$ with the following effect. Let ${\Omega}$ be an open, bounded convex domain in ${{\mathbb R}^n}$. Let ${u}$ be a convex function on the closure of ${\Omega}$, which vanishes on the boundary. Then, for all ${x_0\in\Omega}$,

$\displaystyle \begin{array}{rcl} |u(x_0)|^n\leq C\,d(x_0,\partial \Omega).\mathrm{diameter}(\Omega)^{n-1}.\mathcal{L}^n(\nabla u(Omega)). \end{array}$

This is used in PDE (Caffarelli,…).

2. Convexity in Heisenberg group

Since left translations ar affine is exponential coordinates, Heisenberg group carries an affine structure. Therefore convex domains will simply be Euclidean convex.

Definition 2 (Several competing groups) Say a function on a convex domain ${\Omega}$ of Heisenberg group is ${H}$-convex if its restriction to every horizontal line of ${\Omega}$ is convex.

The subdifferential of ${u}$ at ${x_0}$ is a subset of ${{\mathbb R}^{2n}}$.

Note that there are H-convex functions in ${\mathbb{H}^n}$ which are very irregular (e.g. Weierstrass) in the vertical direction.

2.1. Results

We define a horizontal slicing diameter : this is the maximal diameter of the intersection of ${\Omega}$ with horizontal planes ${H_x}$, ${x\in\mathbb{H}^n}$. We also define a horizontal slicing Monge-Ampère measure

$\displaystyle \begin{array}{rcl} \mathcal{L}_{HS}^{2n}(\nabla_H u(\Omega))=\sup_{x\in\Omega}\mathcal{L}^{2n}(\nabla_H u (H_x \cap\Omega)). \end{array}$

Theorem 3 There is a dimension dependant constant ${C}$ with the following effect. Let ${\Omega}$ be an open, bounded convex domain in ${\mathbb{H}^n}$. Let ${u}$ be a convex function on the closure of ${\Omega}$, which vanishes on the boundary. Then, for all ${x_0\in\Omega}$,

$\displaystyle \begin{array}{rcl} |u(x_0)|^{2n}\leq C\,d(x_0,\partial \Omega).\mathrm{diameter}_{HS}(\Omega)^{2n-1}.\mathcal{L}_{HS}^{2n}(\nabla u(\Omega)). \end{array}$

This improves earlier results by Garofalo et al. where the distance to the boundary appeared with a negative power.

3. Proof

3.1. Back to the Euclidean case

Lemma 4 (Comparison principle) Let ${u}$, ${v}$ be continuous functions on the closure of ${\Omega}$. Assume that ${u\leq v}$. Then

$\displaystyle \begin{array}{rcl} \nabla v(Omega)\subset \nabla u(\Omega). \end{array}$

Indeed, any supporting hyperplane of the graph of ${u}$, when raised, will touch the graph of ${v}$.

Alexandrov compares the graph of ${u}$ with the cone on ${\partial \Omega}$ with vertex at ${(x_0,u(x_0))}$. Its subdifferential is concentrated at the vertex. Let ${x}$ be the nearest point in the boundary. In the subdifferential ${\partial v(x_0)}$, there is a covector ${p_1}$ of size ${\sim|u(x_0)|/d(x,x_0)}$. All othe covectors ${p}$ in ${\partial v(x_0)}$ satisfy

$\displaystyle \begin{array}{rcl} |p|\geq \frac{|u(x_0)|}{\mathrm{diameter}(\Omega)}. \end{array}$

3.2. Failure of comparison principle in Heisenberg group

There exists functions ${u,v}$ on a cyclinder ${\Omega}$, which are equal n the boundary and ${u\leq v}$, but ${\nabla_H v(\Omega)\not\subset \nabla_H u(\Omega)}$.

Indeed, set ${v(x,y,t)=t}$. Check that ${\partial _H v(x,y,t)=(2y,-2x)}$, so that ${\nabla_H v(\Omega)=B(0,2)}$ contains the origin. Modify ${v}$ in an annulus,

$\displaystyle \begin{array}{rcl} u(x,y,t)=t-(1-t^2)g(x,y) \end{array}$

where ${g}$ has support in an annulus. Assume that ${0\in \nabla_H u(\Omega)}$. Then ${0\in \partial_H u(q)}$, and ${u}$ achieves its minimum on ${H_q}$ at point ${q}$. One can achieve that this never happens.

3.3. Comparison for convex functions

What saves us is that comparison holds for convex functions.

Theorem 5 Let ${\Omega}$ be a convex domain in ${\mathbb{H}^n}$, let ${u,v}$ be convex functions on ${\Omega}$ that are equal on the boundary. Assume that for some ${x_0\in\Omega}$, there exists ${p\in\partial_H v(x_0)}$ such that, for al ${x\in H_{x_0}\cap\Omega}$ different from ${x}$,

$\displaystyle \begin{array}{rcl} v(x) > v(x_0)+p.(\pi(x)-\pi(x_0)). \end{array}$

Then ${p\in \partial_H u(x_0)}$.

The proof uses degree theory for set valued maps. For simplicity, let us assume that ${u}$ is smooth, and ${x_0=0}$. Let ${U=H_{x_0}\cap \Omega}$ projected to ${{\mathbb R}^{2n}}$. We view ${\partial_H u}$ as a mapping of ${U}$ to ${{\mathbb R}^{2n}}$. To show that ${p}$ belongs to its image, it suffices to show that the degree of ${\partial_H u}$ on ${U}$ at ${p}$ is non zero. We check that this is the case when ${\partial_H u}$ is replaced with ${\partial_H v}$. Then a linear homotopy allows to conclude. Indeed, assume by contradiction that the homotopy hits ${p}$ along ${\partial U}$, i.e. there exists a point ${x\in\partial\Omega\cap H_{x_0}}$ and ${t\in[0,1]}$ such that

$\displaystyle \begin{array}{rcl} t\partial_H u(x)+(1-t)\partial_H v(x)=\partial_H v(x_0). \end{array}$

Along the horizontal line from ${x_0}$ to ${x}$,

$\displaystyle \begin{array}{rcl} u(x_0) \geq u(x)+\partial_H u(x).(\pi(x_0)-\pi(x)),\quad v(x_0) \geq v(x)+\partial_H v(x).(\pi(x_0)-\pi(x)). \end{array}$

Take the convex combination of these two inequations, get and inequality that contradicts the assumption ${v(x) > v(x_0)+p.(\pi(x)-\pi(x_0))}$.

Computation of the index for ${v}$.

3.4. End of the proof

One gets

$\displaystyle \begin{array}{rcl} |u(x_0)|^{2n}leq C\,d(x_0,H_{x_0}\cap\partial \Omega).\mathrm{diameter}_{HS}(\Omega)^{2n-1}.\mathcal{L}_{HS}^{2n}(\nabla u(Omega)). \end{array}$

There remains to replace ${d(x_0,H_{x_0}\cap\partial \Omega)}$ with ${d(x_0,\partial \Omega)}$. This relies of an Harnack inequality, which allows to replace ${x_0}$ with a nearby point where the horizontal plane is tilted and hits the boundary at a distance comparable to the distance of ${x_0}$ to the boundary.

Next sessions : April 4th and May 14th.

Notes of Francois Vigneron’s lecture

Multifractal analysis on the Heisenberg group

Joint with Stéphane Seuret.

1. Motivation

1.1. Sources of multifractal analysis

Multifractal analysis is a toolbox for data analysis (textures, financial or experimental data, diophantine approimation,… see the program of our seminar at UPEC). It introduces classification parameters based on absence of regularity. Why the Heisenberg group ? People from image analysis ask about non isotropic textures.

1.2. Historic examples

Weierstrass’ example of a function ${W_h}$ which is ${C^h}$-Hölder but nowhere differentiable (for every ${0.

Riemann function which is differentiable at infinitely many rational points (but not all of them) and nowhere else,

$\displaystyle \begin{array}{rcl} \sum_n n^{-2}\sin(2\pi n^2 x). \end{array}$

Its multifractal spectrum was not computed before 1996 (S. Jaffard).

Both examples are self similar. A picture of Weierstrass’ function suggests that irregularity is spread all over. A picture of Riemann’s function looks very different : spikes, points with different left and right derivatives, differentiability point, all over.

Definition 1 The pointwise regularity exponent ${h_f(x_0)}$ of ${f}$ at ${x_0}$ is the largest ${\alpha>0}$ such that ${f}$ is, up to a polynomial, ${O(|x-x_0|)^\alpha)}$.

The multifractal spectrum is the function

$\displaystyle \begin{array}{rcl} d_f(h)=\mathrm{dim}_{\mathrm{Hausdorff}}(\{x\,;\, h_f (x)=h\}). \end{array}$

Example 1 Weierstrass function ${W_h}$ is monofractal : every point has pointwise regularity exponent ${h}$.

The multifractal spectrum of Riemann’s function is made of a segment joining ${h=1/2}$ and ${h=3/2}$ plus a point at ${h=3/2}$.

2. Heisenberg group

The strong anisotropy turns out not the make a big change for the specific questions we solve the existing techniques adapt rather easily. This is why we can state theorems

2.1. Wavelets

One can construct smooth, exponentially decaying functions ${\psi_{j,k}^{\epsilon}}$ concentrated at ${2^{-j}\circ k}$, where ${k\in {\mathbb Z}^3}$ (which is a subgroup), with vanishing moments, which form a basis of ${L^2}$.

2.2. Results

Theorem 2 (Global Hölder regularity) Let ${s=k+\sigma}$. A function belongs to ${C^s}$ iff its ${k}$-th horizontal derivatives are ${C^\sigma}$. Also iff its wavelet coefficients

Theorem 3 (Pointwise Hölder regularity) Let ${s=k+\sigma}$. If a function ${f}$ is ${C^s}$ at ${x_0}$, then

$\displaystyle \begin{array}{rcl} 2^{js}|d_{j,k}^\epsilon(f)|\leq C(1+2^j d(x_{j,k},x_0))^s . \end{array}$

Conversely, if this holds, then ${f}$ is ${C^t}$ at ${x_0}$ for all ${t.

2.3. Generic spectrum in H\” older and Besov classes

Theorem 4 Monofractal functions (at ${s}$) form a dense ${G_\delta}$ subset of ${C^s}$.

Definition 5 ${f\in B_{p,q}^s}$ if

$\displaystyle \begin{array}{rcl} \|2^{j(s-Q/p)}|d_{j,k}^\epsilon(f)|\|_{\ell^p(k)}\in \ell^{q}(j) . \end{array}$

Note that ${B_{p,q}^s \subset C^{s-Q/p}}$ for ${s>Q/p}$ and ${s\notin Q/p +{\mathbb N}}$. Here, ${Q=4}$.

Theorem 6 For a dense ${G_\delta}$ subset of ${B_{p,q}^s}$, the spectrum is a segment between ${(s-Q/p,0)}$ and ${(s,Q)}$. For all functions of ${B_{p,q}^s}$, the spectrum is below this segment.

Indeed, for the standard example of a Besov function (expressed in wavelet expansion), the pointwise Hölder exponent at ${x_0}$ is related to the dyadic approximation rate of ${x_0}$. Dimensions of isoapproximable sets can be computed. It is a special case of a very general result by Beresnevich, Dickinson and Velani (2006).

Harmonic functions of minimal growth

With Amir, Benjamini, Duminil-Copin, Meyerovich, Yadin.

1. Motivation

Gromov’s polynomial growth theorem. Kleiner’s new proof that uses Lipschitz harmonic functions.

Any group has non trivial Lipschitz harmonic functions. Polynomial growth implies that they form a finite dimensional space (inspired by a similar result by Colding and Minicozzi on Riemannian manifolds).

Question. Conversely, does finite dimensionality of the space of Lipschitz harmonic functions imply polynomial growth ?

1.1. Examples

What about Grigorchuk group ? Assume ${V=\{Lipschitz harmonic functions\}}$ is finite dimensional. This is a torsion group. Its image in the linear group ${GL(V)}$ is finite. This easily leads to a contradiction. We conclude that, for Grigorchuk group, ${V}$ is infinite dimensional.

What about lamplighter group ? I construct an infinite dimensional space of positive Lipschitz harmonic functions on ${{\mathbb Z}_2 \wr {\mathbb Z}}$. Key : function must be big only in a small part of the unit sphere around each point. Let ${T_r}$ be the stopping time when the lamplighter reaches ${\pm r}$. Consider

$\displaystyle \begin{array}{rcl} h(g,r)=\mathop{\mathbb P}^g(X_{T_r}\textrm{ is a configuration with all negative lamps off}). \end{array}$

This is essentially the probability that the walk of the lamplighter stays in ${{\mathbb Z}_+}$. So this is ${O(1/r)}$, and ${rh(g,r)}$ converges as ${r}$ tends to infinity. There is one such function for each ${g\in G}$ and they are linearly independant.

The lamplighter group ${G_1}$ has no sublinear harmonic functions.

$\displaystyle \begin{array}{rcl} h(g)=\mathop{\mathbb E}_g(h(X_{T_r})). \end{array}$

Sublinearity implies that ${\mathop{\mathbb E}_g(h(X_{T_r})1_{X_{T_r}=r})}$ goes to 0. On the other hand,

$\displaystyle \mathop{\mathbb E}_g(h(X_{T_r})1_{X_{T_r}=0})=h(e),$

showing that ${h(g)=h(e)}$.

2. Result

Theorem 1 Assume ${V=\{Lipschitz harmonic functions\}}$ is finite dimensional. Then ${G}$ has no sub-linear harmonic functions.

2.1. Proof

Assume ${G}$ is amenable. Let ${H}$ be the image in ${GL(V}$ mod constants${)}$. By Tits, we know that ${H}$ is virtually solvable. By Malcev, ${H}$ is virually uniformly triangularizable. Furthermore, ${H}$ preserves a norm (Lipschitz constant of harmonic functions), so it is virtually abelian. Thus ${G}$ has a finite index subgroup ${A}$ such that, for ${h\in V}$ in a suitable basis, ${gh=\lambda h+c}$. Taking commutators shows that ${\lambda=1}$. If ${h}$ is sub-linear, then ${c=0}$. Argument ends with the fact that virtually abelian groups have no sublinear harmonic functions.

2.2. More

Conjecture: ${G}$ has no sublinear harmonic functions if and only if ${G}$ is diffusive, i.e. simple random walk escapes at speed ${\sqrt{n}}$.

Notes of Bertrand Deroin’s lecture

Random walks on left-orderable groups

I view random walks as a tool to study left-orderable groups.

1. Left-orderable groups

This means a group with a left-invariant order.

Torsion in an obstruction. Not easy to find firther obstructions, and indeed, many classes of groups are left-orderable.

• torsion free nilpotent groups (Malcev),
• free groups (Magnus),
• surface groups,
• virtually, every 3-manifold group (Agol).

Conjecture: If ${G}$ is a lattice in a simple Lie group of real rank ${\geq 2}$, then ${G}$ is not left-orderable.

Theorem 1 (Witte) True for finite index subgroups of ${SL(n,{\mathbb Z})}$, ${n\geq 3}$.

Conjecture (Linnell): Let ${G}$ be a finitely generated left-orderable group. Then either ${G}$ contains ${F_2}$, or ${G}$ surjects onto ${{\mathbb Z}}$.

Theorem 2 (Witte) Let ${G}$ be a finitely generated amenable left-orderable group. Then ${G}$ surjects onto ${{\mathbb Z}}$.

1.1. Actions on the real line

Proposition 3 Countable left-orderable groups coincide with subgroups of Homeo${^+({\mathbb R})}$.

Idea of proof: enumerate elements of ${G}$ and map them to ${{\mathbb R}}$ in an order preserving manner. Left translations act on a subset of ${{\mathbb R}}$ and extend by continuity to homeos.

2. Actions on the circle

Theorem 4 (Ghys) If ${G}$ is a lattice in a simple Lie group of real rank ${\geq 2}$, then every actions of ${G}$ on the circle by homeomorphisms has a finite orbit.

Theorem 5 (Ghys) If ${G\subset Homeo^+(S^1)}$ has no finite orbit, then either ${G}$ contains ${F_2}$, or ${G}$ surjects onto ${{\mathbb Z}}$.

2.1. Quasi-periodic actions on the real line

Can one upgrade left-orderable (action on ${S^1}$ with a fixed point) to action on ${S^1}$ without finite orbits ? One can do a step in this direction.

Definition 6 ${\rho:G\rightarrow Homeo^+({\mathbb R})}$ is quasi-periodic if its conjugates by translations form a relatively compact subset in ${Hom(G,Homeo^+({\mathbb R}))}$.

A source of such actions is the following situation. Let ${X}$ be a compact space with a free action of ${{\mathbb R}}$ and a ${G}$-action mapping ${{\mathbb R}}$-orbits to ${{\mathbb R}}$-orbits. Conversely, every quasi-periodic action on ${{\mathbb R}}$ arises in this way.

Let ${\mu}$ be a symmetric probability distribution on ${G}$ whose finite support generates ${G}$. Assume ${G\subset Homeo^+({\mathbb R})}$. This defines a Markov process on ${{\mathbb R}}$, where ${p(x,y)=\mu\{g\in G\,;\,gx=y\}}$.

Theorem 7 (Deroin-Kleptsyn-Navas-Parwani) Assume that orbits are dense. Then there exists a conjugation (unique up postcomposition by affine transformations) such that the Deriennic property holds: ${\forall x\in{\mathbb R}}$, the average displacement ${\mathop{\mathbb E}_{x,\mu} (gx)=x}$. Moreover, this action is quasi-periodic.

3.1. Proof

Uses recurrence. There exists a compact interval ${K\subset {\mathbb R}}$ such that for all ${x\in{\mathbb R}}$, almost surely, the trajectory ${g_n\ldots g_1 x}$ hits ${K}$ infinitely often.

From recurrence, we get existence of a stationary Radon measure on ${{\mathbb R}}$ (Ornstein-Weiss).

Notes of Itai Benjamini’s 2014 lecture

Invariant random structures

1. First passage percolation

Multiply length of edges of the 2-grid by 1 or 10 with equal probabilities. Known : there is an asymptotic shape. What is it ?

UIPT : the limit is not deterministic.

Stationary first passage percolation. On trees (Dekking and Host 1991).

Theorem 1 (Benjamini-Tessera) FPP on vertex transitive graphs of polynomial groth almost surely Gromov-Hausdorff converges after rescaling to a Carnot group.

Convergence to a deterministic metric space works on other groups ? Lamplighter ? Let ${X}$, ${Y}$ be independant realizations. Does ${d_X(o,v_n)/d_Y(o,v_n)}$ tend to 0 ? Bound ${Var(d_X(o,v_n))}$ ?

Side question: say a graph is ${C}$-roughly transitive if there is a ${C}$-quasi-isometry mapping every point to every other point. Does this imply space is quasi-isometric to a homogeneous space ?

2. Partitions

Partition a graph into infinitely many roughly connected infinite subgraphs, each one touching finitely many others. Which Cayley graphs admit invariant random partitions of that sort (i.e. a measure on partitions, invariant under automorphisms) ?

${{\mathbb Z}^2}$ admits one. Exercise: Regular tree does not.

Theorem 2 (Benjamini-Tessera) Positive first ${L^2}$-Betti number implies no such invariant random partitions.

Conjecture: Lamplighter does not admit IRP.

2.1. Iterations

Given a random partition, can you random partition again ? Try ${{\mathbb Z}\times}$ tree.

3. Invariant majority dynamics

For odd degree graph, each vertex changes its mind according to majority of its neighbours.

For finite graphs, the process stabilizes in finite time.

Theorem 3 (Benjamini-Tamuz) Start with invariant random opinions. Then almost surely opinions stabilize in finite time (every second time).

4. Harmonic measure

Given a set ${S}$, start random walk at some vertex and stop when it hits ${S}$, yielding harmonic measure on ${S}$.

Let ${G_n}$ be finite, connected, vertextransitive graphs with bounded degree and large diameters,

$\displaystyle \begin{array}{rcl} |G_n|=o(diam(G_n)^d). \end{array}$

Then, for any set ${S_n}$, harmonic measure is supported on a set of size ${o(|G_n|)}$.

4.1. Beyond polynomial growth

On expanders, for sets of at most half the size, when averaging over starting vertices, harmonic measure is supported on a set of size proportional to size of the set (with Yadin).

On lamplighter ${{\mathbb Z}_2 \wr {\mathbb Z}_n}$, there is a set of size ${|G|/4}$ for which harmonic measure is supported on a set of size proportional to size of the set, and other sets where harmonic measure is supported on a small subset.

Notes of Bartosz Trojan’s lecture

Heat kernel on affine buildings

Consider finite support probability distributions ${p(v,\cdot)}$ on an affinebuiding which are spherical : depends only on distance.

Plays the role of heat kernel on symmetric spaces. For such a kernel, one has rather sharp asymptotic estimates (Anker-Ji), used to dtermined the Martin boundary. There remains open questions about the asymptotic behaviour of Green function.

1. Case of ${{\mathbb Z}^r}$

Theorem 1 (Lawler-Limic)

$\displaystyle \begin{array}{rcl} p_n(v)\sim (det nB_s)^{-1/2}e^{-n\phi(\delta)}, \end{array}$

in some ball ${|v|<\leq\rho n}$. Here ${B_s =D^2 \log\kappa(s)}$, ${\kappa}$ is the Fourier transform of probability distribution ${p}$, ${\phi(\delta)=\delta\cdot s-\kappa(s)}$, ${s}$ is the unique solution of ${\Delta s=...}$.

${s}$ is real analytic on the interior of the convex hull ${M}$ of the support of ${p}$, but blows up at the boundary. Here is how.

Theorem 2

$\displaystyle \begin{array}{rcl} \frac{e^{s\cdot v}}{\kappa(s)}\geq c\,dist(\delta,\partial M)^\eta. \end{array}$

2. Generalization to affine buildings

We have analogous estimates, with extra terms: distance to boundary of Weyl chamber.

Eralier results on on-diagonal behaviour (local limit theorem) by Sawyer, Gerl, Woess (trees), Lindbauer, Voit, Cartwright, Tolli and finally Parkinson. Off-diagonal behaviour by Lalley (trees), Anker, Schapira, Trojan for ${\tilde{A}_n}$, general case here.

2.1. Spherical analysis

There is a spherical inversion formula reconstructing the probability distribution in terms of its Gelfand Fourier transform (goes back to Macdonald 1971 in special cases). Macdonald polynomial arise as multiplicative finctions on the algebra of finitely supported invariant operators.