## Notes of Richard Schwartz’ third Cambridge lecture 17-05-2017

Iterated barycentric subdivisions and steerable semi-groups

In two dimensions, there are many different affinely natural procedures on simplices: the barycentric subdivision (defined by coning and induction on dimension), yielding 6 triangles; the truncation of corners, yielding 4 triangles. The second is better behaved for numerics (triangles do not get thin, they stay similar to each other, in fact only 2 shapes are encountered).

Question. What shapes of simplices arise in an iterated subdivision scheme?

Diaconis and McMullen show that almost every triangle produced gets thin.

1. The space of shapes

I am aiming at a different question. Shapes form a topological space. Let ${X^n}$ be the space of labelled ${n}$-simplices mod scaling. Say we normalize so that volume stays equal to 1. Then ${X^n}$ is a principal homogeneous space of ${Sl(n,{\mathbb R})}$. An affinely natural subdivision process amounts to a subdivision of the standard simplex.

Question. Does the iteration scheme produce a dense set of shapes?

The answer is positive in 2 dimensions (Barany-Beardon-Carne 1990). I gave a positive answer in 3 and 4 dimensions, I will explain my solution. My guess is that answer should be negative for ${n}$ large enough.

The question is equivalent to the density of the semi-group generated by the matrices that map the standard simplex to each of the labelled tiles.

Lemma: the subgroup ${S}$ generated by the subdivision is either discrete or dense.

The proof is not very enlightening. It follows that the subgroup ${\langle S\rangle}$ generated by the subdivision is dense.

Lemma: If ${S}$ contains a bounded infinite walk, then the semi-group ${S}$ is dense in ${\langle S\rangle}$. Indeed, a long bounded walk contains segments ${g_1\cdots g_k}$ which are close to 1, hence an expression of inverses as elements of the semi-group ${S: g_1^{-1}=g_2 \dots g_n}$ up to small error.

1.1. ${n=2}$

If you label right, then all generators are infinite order elliptics.

1.2. ${n=3}$

A computer search revals some infinite order elliptic elements. They represent a positive fraction of words of length 3.

Experiments suggest existence of an infinite walk up to ${n=10}$.

2. Proof

Look at horoballs in ${X}$. Fix an origin ${o\in X}$.

Definition 1 A horoball is special if it contains the origin ${o}$ in its boundary.

A set ${W\subset X}$ is a strong wheel if every special horoball contains a point of ${W}$ in its interior.

A semi-group ${S}$ is steerable if the orbit ${So}$ contains a steering wheel.

Lemma: If ${S}$ is steerable, then ${S}$ has a bounded infinite walk.

The idea is that if a walk goes far away from ${o}$, use a special horoball containing it in its boundary, pick a point there, start again.

Therefore, the point is to exhibit a steering wheel.

Lemma: Consider the Hadamard map ${(X,o)\rightarrow H^N}$, ${N=\frac{n(n+1)}{2}-1}$ (geodesic polar coordinates) preserving min sectional curvature. This is distance non-decreasing. This allows to carry the problem to 9-dimensional hyperbolic space.

The wheel is a finite set of 144 points on the boundary of some ball ${B}$ centered at ${o}$. To certify that they form a steering wheel, apply the following criterion: each of them points towards the center of a special horoball. If the convex hull of these centers contains ${B}$, then the horoballs cover ${\partial B}$, hence the wheel.

3. Questions

What makes you think the answer should be negative in high dimensions? Because ${n!}$ is too small a number of horospheres to cover a sphere of dimension ${N=\frac{n(n+1)}{2}-1}$. Experiments indicate that this phenomenon could start at ${n=9}$.

The semi-group would be uniformly discrete?

## Notes of Viktor Schroeder’s first informal Cambridge lecture 16-05-2017

Moebius structures on boundaries, I

This is an informal series of 3 lectures. I start with boundaries of hyperbolic groups. I will continue with Furstenberg boundaries of higher rank symmetric spaces (joint work with my student Beyrer).

1. Moebius structures

4 distinct points in a set can be split into pairs in 3 ways, whence an epimorphism ${\mathfrak{S}_4\rightarrow\mathfrak{S}_3}$, with kernel

$\displaystyle \begin{array}{rcl} \{1,(12)(34),(13)(24),(14)(23)\}. \end{array}$

Say a quadruple is regular if all points are distinct, and admissible if no 3 of them coincide.

1.1. Semi-metrics

A semi-metric on a set is a function ${X\times X\rightarrow[0,+\infty]}$ which is

• symmetric,
• positive on distinct pairs,
• at most one point can have infinite distance to an other point. For this point, distance to all points is infinity.

1.2. Cross-ratios

The Moebius structure of a semi-metric can be defined in 3 equivalent ways. Given a quadruple ${(x_1,x_2,x_3,x_4)\in X}$, let ${\rho_{ij}=\rho(x_i,x_j)}$ and define cross-ratio

$\displaystyle \begin{array}{rcl} \frac{\rho_{12}\rho_{34}}{\rho_{14}\rho_{23}}. \end{array}$

The resulting 6 numbers (after permutations) can be organized in different ways.

1. View

$\displaystyle \begin{array}{rcl} crt_\rho:Reg_4\rightarrow \Sigma=\{(a:b:c)\in{\mathbb R} P^2\,;\,a,b,c>0\}. \end{array}$

This extends to ${Adm_4\rightarrow \hat\Sigma=\Sigma}$ union 3 points. Under permutation, ${crt_\rho}$ changes via ${\mathfrak{S}_3}$ as above.

2. View

$\displaystyle \begin{array}{rcl} \mathbb{X}:Reg_4\times\Theta\rightarrow{\mathbb R}_+, \end{array}$

where ${\Theta}$ is the 3-point set of splittings in pairs. The product of the 3 functions equals 1. Under permutation, ${\mathbb{X}}$ changes via taking inverse and ${\mathfrak{S}_3}$ acting on ${\Theta}$.

3. Alternatively, one may replace values by their logarithms.

1.3. Sub-Moebius structures

Definition 1 A sub-Moebius structure on a set ${X}$ is a function ${crt:Ad_4\rightarrow\hat\Sigma}$ satisfying

1. Normalization. ${crt(x,x,y,z)=(0:1:1)}$.
2. Symmetry. ${crt(\pi(q))=\phi(\pi(crt(q)))}$.

1.4. The cocycle condition

Not all sub-Moebius structures arise from semi-metrics. Those arising from semi-metrics satisfy an extra equation, the cocycle condition:

$\displaystyle \begin{array}{rcl} crt(\alpha,x,y,\beta)crt(\alpha,y,z,\beta)=crt(\alpha,x,z,\beta). \end{array}$

Theorem 2 (Buyalo) A sub-Moebius structure arises from a semi-metric if and only if it satisfies the cocycle condition.

Indeed, set

$\displaystyle \begin{array}{rcl} \rho_{\alpha,\beta,\omega}(x,y)=\frac{\mathbb{X}(\alpha,x,y,\beta)}{\mathbb{X}(\alpha,x,\omega,\beta)\mathbb{X}(\alpha,\omega,y,\beta)}. \end{array}$

This is a semi-metric. Different choices of ${\alpha,\beta,\omega}$ define the same sub-Moebius structure.

1.5. Moebius equivalent semi-metrics

Say two semi-metrics are Moebius equivalent if they define the same sub-Moebius structure. Here are constructions of semi-metrics Moebius equivalent to a given one ${\rho}$.

• Multiplication with a constant ${\lambda}$. ${\lambda\rho}$.
• Involution. Given ${\omega\in X}$,

$\displaystyle \begin{array}{rcl} \rho_\omega(x,y)=\frac{\rho(x,y)}{\rho(x,\omega)\rho(\omega,y)}. \end{array}$

• Multiplication with a positive function ${\lambda}$.

$\displaystyle \begin{array}{rcl} \rho_\lambda(x,y)=\lambda(x)\lambda(y)\rho(x,y). \end{array}$

2. Boundaries

Let ${X}$ be a ${CAT(-1)}$ metric space. Fix origin ${o\in X}$. Then (Bourdon) the semi-metric

$\displaystyle \begin{array}{rcl} \rho_o(x,y)=e^{-(x|y)_o} \end{array}$

is a metric. Changing ${o}$ gives a Moebius equivalent metric.

2.1. The Ptolemaic inequality

Moebius structures on boundaries of ${CAT(-1)}$ spaces satisfy an extra inequality, which we call Ptolemaic inequality,

$\displaystyle \begin{array}{rcl} \rho_{12}\rho_{34}\leq \rho_{23}\rho_{14}+\rho_{13}\rho_{24}. \end{array}$

This means that ${crt}$ takes its values in the triangle ${\Delta\subset\hat\Sigma}$ with vertices at the extra points.

${CAT(0)}$ metrics are Ptolemaic. The sphere with the chordal metric (i.e. isometric to a subset of Euclidean space) is Ptolemaic. The sphere in its Riemannian metric is not Ptolemaic.

Thus triangle inequality does not imply Ptolemaic. Conversely, Ptolemaic inequality does not imply triangle inequality for all metrics in the class. However, if a sub-metric Ptolemaic Moebius structure has a point at infinity, then it satisfies triangle inequality. Also the Moebius class contains bounded sub-metrics which are metrics.

2.2. Hyperbolic groups

Mineyev has constructed metrics on hyperbolic groups whose visual distances define a Ptolemaic Moebius structure. More on this next time.

## Notes of Yves Cornulier’s Cambridge lecture 16-05-2017

Commensurating actions of groups of birational transformations

Joint work with Serge Cantat.

1. Birational geometry

This is more than a century old. Let ${X,Y}$ be irreducible varieties over ${{\mathbb C}}$. A rational map ${X\rightarrow Y}$ is a regular (given by the ratio of two polynomials) function between Zariski dense open subsets, modulo coincidence on a dense open subset. A birational map ${X\rightarrow Y}$ is a regular function between Zariski dense open subsets which is an isomorphism of varieties, again modulo coincidence on a dense open subset.

Example. The map ${{\mathbb C}^2\rightarrow P^1\times P^1}$, ${(x,y)\mapsto ((x:1);(y:1))}$ is birational. Therefore ${P^1\times P^1}$ and ${P^2}$ are birationally equivalent. Therefore a rational map comes in several different models. One often needs to consider all of them simultaneously.

A set of rational functions on ${{\mathbb C}^d}$ forms a birational map to ${{\mathbb C}^d}$ iff it is algebraically free over ${{\mathbb C}}$ and generates the field ${{\mathbb C}(t_1,\ldots,t_d)}$.

Notation. ${Bir(X)}$ is the group of birational equivalences of ${X}$, i.e. the automorphism group of the field ${{\mathbb C}(X)}$ of rational functions on ${X}$.

Example. It turns out that ${Bir({\mathbb C})=Bir(P^1)=Aut(P^1)=PGl(2,{\mathbb C})}$. On the other hand, ${Bir(P^2)}$ contains ${Aut(P^2)=PGl(3,{\mathbb C})}$ but also ${Aut(P^1\times P^1)=PGl(2,{\mathbb C})\times PGl(2,{\mathbb C})}$. However, ${Bir(P^2)}$ contains an abelian group of infinite rank, the set of maps ${(x,y)\mapsto (x+P(y),y)}$, where ${P\in{\mathbb C}[t]}$.

The difficulty of ${Bir}$ is that it comes with quasi-actions which are not quite actions. Some results have been obtained in dimension 2, nearly nothing is known in higher dimensions. For instance, one does not know which finite groups occur in ${Bir(P^3)}$, and it is still possible (but unlikely) that every finite group occurs in ${Bir(P^4)}$.

2. Commensurating actions

2.1. Commensurated sets

Say a group ${G}$ acting on a set ${E}$. Say a subset ${A\subset E}$ is commensurated if ${\forall g\in G}$, ${\ell_A(g):=|A\Delta gA|<\infty}$ (Stallings, Dunwoody,…). Say that ${A}$ is transfixed if there exists a ${G}$-invariant subset ${A'}$ such that ${|A\Delta A'|<\infty}$. It turns out that it is equivalent to ${\ell_A}$ being bounded (proved by several authors in the 1960’s, with good bounds finally obtained by W. Neumann).

Example. Let ${G}$ act on a Schreier graph ${G/H}$. Then a subset is commensurated iff its boundary is finite. A subset is transfixed iff it is finite or its complement is finite. In particular, there exists a commensurated, non-transfixed subset iff the Schreier graph has at least 2 ends.

Example. Let ${G}$ act on set ${E}$, and ${A\subset E}$. Let ${\ell_A^2(E)=1_A+\ell^2(E)\subset{\mathbb R}^E}$ is an affine Hilbert space. It is ${G}$ invariant iff ${A}$ is commensurated. This provides an affine isometric action of ${G}$.

Example. Cubulation. Define a graph whose vertices are subset ${B\subset E}$ such that ${|B\Delta A|<\infty}$. Put an edge between ${B}$ and ${B\cup\{b\}}$. This a median graph, therefore an action on a ${CAT(0)}$ cube complex arises from it. Conversely, every action of ${G}$ on a ${CAT(0)}$ cube complex induces a commensurating action on the set ${E}$ of half-spaces, with ${A}$ being the subset of half-spaces contaning a fixed vertex. This gives huge cube complexes or actions, far from being optimal.

2.2. Property FW

Definition 1 Say ${G}$ has property FW if for every commensurating action of ${G}$ on ${(E,A)}$, ${A}$ is transfixed.

Say ${G}$ has property FW relative to subgroup ${H}$ if for every commensurating action of ${G}$ on ${(E,A)}$, ${A}$ is transfixed by ${H}$.

Example. For every cyclic distorted subgroup ${H, ${G}$ has property FW relative to ${H}$ (Haglund).

This can be used to prove that ${Sl(d,{\mathbb Z})}$, ${d\geq 3}$, or ${Sl(d,{\mathbb Z}[\sqrt{2}])}$, ${d\geq 2}$, has property FW (using bounded generation by exponentially distorted unipotents).

Property (T) implies property FW (use ${\ell^2}$ action).

Proposition 2 or any group ${G}$, the following are equivalent:

1. ${G}$ has property FW.
2. Every isometric action of ${G}$ on a ${CAT(0)}$ cube complex has a fixed point.
3. Every isometric action of ${G}$ on a median graph has a finite orbit.
4. If ${G}$ is finitely generated, every Schreier graph of ${G}$ has at most 1 end.
5. For all actions of ${G}$ on sets ${E}$, ${H^1(G,{\mathbb Z}^{(E)})=0}$ (functions with finite support).

3. Birational groups

Fix a projective variety ${X}$. Let ${Hyp(X)}$ be the set of irreducible hypersurfaces of ${X}$. Every subgroup ${G acts on a set ${E}$ which commensurates ${A=Hyp(X)}$ as follows. The crucial point is the following. If ${X}$ is smooth (in fact, normal is sufficient) and ${f:Y\rightarrow X}$ is a birational morphism, inverse images of hypersurfaces are well defined: among the several subvarieties whose union is ${f^{-1}(H)}$, there is exactly one which maps onto ${H}$. Therefore one can define ${E=\widetilde{Hyp}(X)}$ as the inverse limit of ${Hyp(Y)}$ over all birational morphisms ${Y\rightarrow X}$. Birational self-maps of ${X}$ act on ${E}$ and the subset ${A=Hyp(X)}$ is commensurated.

If ${G}$ acts by automorphism of a Zariski open set of ${X}$, then ${G}$ transfixes ${A}$. This is also true for pseudo-automorphisms, i.e. isomorphisms between proper Zariski open subsets of ${X}$.

Theorem 3 Let ${G\subset Bir(X)}$. Then ${G}$ transfixes ${Hyp(X)}$ iff the ${G}$ action on ${X}$ is conjugate to an action by pseudo-automorphisms.

Example. The monomial action of ${Sl(n,{\mathbb Z})}$ on ${{\mathbb C}^{n^2}}$ raises each coordinate to a power given by a matrix coefficient. These are pseudo-automorphisms.

Theorem 4 Let ${X}$ have dimension 2. Let ${G\subset Bir(X)}$ have property FW. Then the action of ${G}$ on ${X}$ is conjugate to an action by automorphisms on some variety ${Y}$, with a short list of exceptions.

This fails in higher dimensions (see the monomial action). There is a corresponding statement for groups with relative property FW.

## Notes of Michael Davis’ Cambridge lecture 16-05-2017

Action dimension of a group

1. Dimensions

Let ${G}$ be a group. Assume ${G}$ has a finite ${K(G,1)}$. The geometric dimension of ${G}$ is the minimum dimension of a complex homotopic to ${K(G,1)}$. The action dimension of ${G}$ is the minimum dimension of a manifold with boundary homotopic to ${K(G,1)}$. Embedding complexes in manifolds gives

$\displaystyle \begin{array}{rcl} gdim\leq actdim\leq 2 gdim. \end{array}$

1.1. Obstructor dimension

Bestvina-Kapovich-Kleiner introduced obstructor dimension. Say a finite CW complex ${K}$ is an ${m}$-obstructor if van Kampen’s classical for embedding of ${K}$ in ${{\mathbb R}^m}$ does not vanish.

Example: a non-planar graph is a 2-obstructor.

They say that ${K\subset\partial G}$ if there exists a coarse embedding of the Euclidean cone over ${K}$ to ${EG}$. This holds for instance if ${G}$ has a boundary (e.g. hyperbolic or ${CAT(0)}$). Therefore they define obstructor dimension as ${2+}$ max of ${m}$ such that ${K\subset\partial G}$.

Theorem 1 (Bestvina-Kapovich-Kleiner 2001)

$\displaystyle \begin{array}{rcl} actdim\geq obdim. \end{array}$

Equality often holds.

Example. ${G=F_2\times\cdots\times F_2}$ (${d+1}$ factors) has ${actdim=2d+2}$. If ${d=1}$, ${gdim=2}$, ${obdim=4}$.

Example. ${G}$ a non-uniform lattice.

Example. ${G=}$ mapping class group.

2. Results on action dimension

Avramidi-Davis-Okun-Schreve: RAAG’s.

Le-Davis-Huang: general Artin groups.

Le-Schreve: simple complexes of groups. This means a functor from a poset to the category of groups.

Today, ${Q}$ will be the poset of simplices (including the empty simplex) of a simplicial complex ${L}$.

2.1. Gluing

Form the disjoint union of products ${K(G_\sigma,1)\times Cone(link(\sigma))}$ and identify according to inclusions. Denote result by ${BG(L)}$.

Example: the Artin complex of a Coxeter system is constructed from the nerve ${L}$ whose simplices correspond to subsets of vertices generating finite Coxeter subgroups. Call the corresponding Artin groups ${A_\sigma}$ spherical Artin subgroups.

Example: the graph product complex. Let ${L}$ be a flag complex. Then ${L^1}$ is a simplicial graph. The graph procduct is the free product of vertex groups ${G_v}$ modded out by ${[G_s,G_A]=1}$ each time ${\{s,A\}\in L^1}$. This has a finite ${K(G,1)}$ iff ${L}$ is a flag complex.

2.2. Results

I compute the action dimension of the families of examples above.

Theorem 2 Let ${L}$ be the nerve of a Coxeter system, let ${d}$ be its dimension. Suppose the corresponding Artin group ${A}$ has a finite ${K(A,1)}$. Then

1. ${H_d(L,{\mathbb Z}/2{\mathbb Z})\not=0}$ implies ${actdim(A)=obdim(A)=2gdim(A)}$.
2. (Le’s thesis) If ${L}$ embeds in a contractible complex of the same dimension (EDCE), then ${actdim(A)\leq 2d+1}$. (Due to C. Gordon for ${d=1}$).
3. For RAAG’s, if ${H_d(L,{\mathbb Z}/2{\mathbb Z})=0}$, then ${actdim(A)\leq 2d+1}$.

Theorem 3 Let ${L}$ be a simplicial complex, ${G}$ its grpah product. Let ${M_v}$ be an aspherical manifold of minimal dimension which is a ${K(G_v,1)}$. Let ${m_v=dim(M_v)}$ if ${M_v}$ has non-empty boundary, ${=dim(M_v)+1}$ if ${M_v}$ is closed. Set ${m_\sigma=\sum_{v\in\sigma}m_v}$. Then

1. ${actdim(G)\leq\max\{m_\sigma\}}$. Furthermore, if no ${M_v}$ is closed, equality holds.
2. If all ${M_v}$ are closed, and if ${L}$ is EDCE), then ${actdim(G)\leq\max\{m_\sigma\}-1}$.

3. Proofs

First perform suitable gluings, and then find obstructions.

1. Gluing. Glue together manifolds along codim 0 subsets of their boundaries. Eventually multiply smaller dimensional manifolds with disks in order to raise all pieces to the same dimension. For instance, glue surfaces to a 2-disk along intervals of their boundaries. Call this disk a dual disk.

2. Obstructors. For RAAGs, the ${K(A,1)}$ is a join of tori, ${EA}$ contains a union of Euclidean spaces. Its visual boundary ${OL}$ is called the octahedralization of ${L}$ (vertices are doubled). We show that this is a ${2d+1}$-obstructor if its top homology does not vanish.

For graph products, in a similar way we produce a join of spheres which is an obstructor.

4. Questions

Swenson: what about these exotic contractible manifolds whose boundaries are not spheres? We are thinking of this. Ultimately, I think we shall have to exclude them.

## Gabriel Pallier’s notes of Peter Kropholler’s Cambridge lecture 12-05-2017

A random walk around soluble groups theory

Notes by Gabriel Pallier.

1. Minimax groups

Recall that for any prime number ${p}$, the Pruefer group ${C_{p^\infty}}$ may be defined as ${\mathbb{Z}[1/p] / \mathbb{Z}}$. Pruefer group are also called quasi-cyclic, as any proper subgroup of a Pruefer group is finite and cyclic.

Definition 1 A polyminimax group ${G}$ is one with a composition series

$\displaystyle 1 = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G,$

where ${G_{i+1}/G_i}$ is either cyclic or quasicyclic.

This defines a class of groups, intermediate between polycyclic and soluble.

Example to be thought of : the additive group of the ring ${\mathbb{Z} \left[ \frac{1}{4+7i} \right]}$ is isomorphic to a direct product ${C_{5^{\infty}} \times C_{13^{\infty}}}$. As such, it is an abelian, non finitely-generated polyminimax group.

Theorem 2 (Kropholler 1984) If ${G}$ is a finitely generated soluble group, then

• {Either, ${G}$ is polyminimax, or}
• {${G}$ has a lamplighter section ${C_p \, \mathcal{o} \, \mathbb{Z}}$ for some prime ${p}$.}

Classical known results : if ${G}$ is polycyclic, then

1. {Hall 1960 : ${\mathbb{Z} G}$ is Noetherian.}
2. {Hall-Roseblade 1973 : irreducible ${\mathbb{Z} G}$-modules are finite.}

Theorem 3 (Jacoboni, 2016) Let ${G}$ be a finitely generated metabelian group. Assume that ${G}$ has Krull dimension greater or equal to ${2}$. Then ${G}$ has a section isomorphic to ${\mathbb{Z} \, \mathcal{o} \, \mathbb{Z}}$ or ${F / (F'^p) F''}$, where ${F}$ denotes the free group on two generators.

The assumption on the Krull dimension of ${G}$ may be expressed as follows : any sequence

$\displaystyle A \rightarrow G \rightarrow Q\rightarrow 1,$

with abelian ${A}$ and ${Q}$ makes ${A}$ a Noetherian module over the finitely generated commutative ring ${\mathbb{Z} Q}$ (indeed ${Q}$ acts on ${A}$ via the conjugation action of ${G}$ on ${A}$). Then it is sufficient that ${A}$ have Krull dimension at leat ${2}$.

For polyminimax groups, the return probability ${p(2n)}$ (or ${p(n)}$ if one authorizes to pause at some times) is bounded below by

$\displaystyle p(n) \geq e^{-n^{1/3}}.$

This was claimed by Pittet and Saloff-Coste in 2004, however with a mistake in the proof. Jacoboni proves under the same assumption as in theorem 3 (${G}$ metabelian, finitely generated with Krull dimension ${\geq 2}$), a reverse inequality

$\displaystyle p(n) \leq \exp ( -n^{1/3} (\log n)^{2/3}).$

2. Finitely Generated groups with no ${\mathbb{Z} \, \mathcal{o} \, \mathbb{Z}}$ sections

Let ${G}$ have finite Hirsch length, i.e.

$\displaystyle h(G) := \sum_{i \geq 0} \dim_{\mathbb{Q}} \left( G^{(i)} / G^{(i+1)} \otimes \mathbb{Q} \right) < +\infty.$

(Observe that only the ${\mathbb{Z}}$ sections increment the Hirsch length).

Theorem 4 (Kropholler-Jacoboni, 2016) If ${G}$ is soluble, finitely generated, without ${\mathbb{Z} \, \mathcal{o} \, \mathbb{Z}}$ section and ${h(G) = + \infty}$, then ${G}$ admits a quotient ${\overline{G}}$ as follows:

$\displaystyle A \rightarrow \overline{G} \rightarrow Q\rightarrow 1,$

with ${h(Q) < + \infty}$, ${Q}$ non polyminimax, ${A}$ a torsion-free abelian, ${h(A) = + \infty}$, ${A}$ decomposing as

$\displaystyle A = \bigoplus_{n \in \mathbb{Z}} A_n,$

${A_n \triangleleft \overline{G}}$, ${h(A_n) < + \infty}$.

Corollary 5 If ${G}$ is finitely generated, with a well-defined Krull dimension, then either ${G}$ has a ${\mathbb{Z} \, \mathcal{o} \, \mathbb{Z}}$ section, or ${h(G) < + \infty}$.

3. Open problem

Let ${G}$ be polyminimax and ${M}$ a simple ${\mathbb{Z} G}$-module. Is ${M}$ elementary abelian ? Here Hall’s method would work only provided that ${\mathbb{Z} G}$ is Noetherian. A necessary condition for this is that ${G}$ be amenable, since Bartholdi (2016) produces injection ${\mathbb{Z} G^n \hookrightarrow \mathbb{Z} G^{n-1}}$ for non amenable ${G}$.

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## Notes of Emmanuel Breuillard’s Cambridge workshop lecture 12-05-2017

How to quickly generate a nice hyperbolic element?

Joint work with Koji Fujiwara, done here in Cambridge during this program. Although we started thinking of it long ago.

Let ${G}$ be a group, ${S}$ a finite set. How large must ${n}$ be in order that knowledge of ${S^n}$ implies knowledge of ${\Gamma:=\langle S \rangle}$?

If ${\Gamma}$ contains nice elements, do they already occur in ${S^n}$?

If ${\Gamma}$ has fast growth, does there exist ${g\in S}$ whose powers grow as fast?

1. Hilbert 5th problem

In Gleason’s 1952 proof that NSS groups are Lie groups, the following quantity arises: for ${U}$ a neighborhood of identity,

$\displaystyle \begin{array}{rcl} n_U(S)=\inf\{n\,;\,S^n\not\subset U\}. \end{array}$

Gleason’s Lemma states that there exists ${g\in S}$ such that

$\displaystyle \begin{array}{rcl} n_U(g)\leq C\,n_U(S). \end{array}$

Yamabe extended this into a trapping lemma.

2. Groups acting on trees

Serre’s lemma: given tree isometries ${a}$ and ${b}$ without common fixed point, at least one of ${a}$, ${b}$ or ${ab}$ has no fixed point at all.

It follows that finitely generated torsion groups have a global fixed point on trees.

An analogous result holds in ${Sl(2,{\mathbb C})}$: if ${a}$, ${b}$ and ${[a,b]}$ are elliptic, then ${ab}$ is hyperbolic.

3. Escaping elliptic elements

In a Euclidean isometry group, if all elements are elliptic, does ${\Gamma}$ have a global fixed points? No! Bass gave a counterexample in ${{\mathbb R}^4={\mathbb C}^2}$. He takes two complex affine motions whose linear parts generate elements of ${SU(2)}$ none of whose has eigenvalue 1.

However, it is the case in irreducible symmetric spaces: there exists a global fixed point in the space or on its boundary. The proof is algebraic, and non-quantitative. In Euclidean buildings, Anne Parreau shows that there must be a fixed point in the building itself.

Does there exists ${n}$ such that if ${\Gamma}$ contains a non-elliptic element, then there is one already in ${S^n}$?

It turns out that answer is no for symmetric spaces but yes for Bruhat-Tits buildings.

3.1. Infinite order

Non-elliptic and infinite order behave differently.

Theorem 1 (Effective Schur’s Lemma) There exists ${n=n(d)}$ such that if ${\gamma\subset Gl(d,{\mathbb C})}$ generates an infinite subgroup, then ${S^n}$ contains an element of infinite order.

The proof uses equidistribution of Galois orbits. ${n(d)}$ must tend to infinity with ${d}$ (Bartholdi-Cornulier, de la Harpe).

3.2. Counterexample

Start with elements in ${K}$ generating a free subgroup such that no element but identity has eigenvalue 1 (this exists by Baire and Borel/Larsen’s theorem on density of images of word maps). Perturb them a bit in ${G}$.

If elliptic elements are not Zariski-dense, a bound exists. It follows from Eskin-Mozes-Oh’s escape from subvarieties theorem.

4. Joint spectral radius

Rota and Strang 1960: for a finite set ${S}$ of matrices, joint spectral radius is

$\displaystyle \begin{array}{rcl} R(S)=\lim_{n\rightarrow\infty}(\max_{g\in S^n}|g|)^{1/n} \end{array}$

It is important in applied maps (wavelets…).

Berger-Wang: can replace norms with eigenvalues.

Note that ${R(S)=1}$ iff ${\Gamma}$ is made of elliptic elements.

Bochi 2002: the Berger-Wang theorem holds in finite time. There are ${k}$ and ${c}$ depending on dimension only such that

$\displaystyle \begin{array}{rcl} R(S)\geq \sup_{n\leq n(d)}(\max_{g\in S^n}|\lambda(g)|)^{1/n}\geq c\, R(S). \end{array}$

The proof is algebraic.

5. Geometric point of view

I want to give a geometric proof of some of the above results.

On a metric space, a set of isometries has a joint minimal displacement

$\displaystyle \begin{array}{rcl} L(S):=\inf_{x\in X}\max_{x\in S}d(x,sx). \end{array}$

and an asymptotic minimal displacement

$\displaystyle \begin{array}{rcl} \ell(S):=\lim \frac{1}{n}L(S^n). \end{array}$

When ${S}$ has only one element, ${\ell(g)}$ equals translation length.

Set

$\displaystyle \begin{array}{rcl} \lambda(S)=\max_{s\in S}\lambda(s), \quad\textrm{and}\quad \lambda(S)=\frac{1}{k}\max_{s\in S^k}\ell(s). \end{array}$

Berger-Wang generalizes.

Question. When does equality ${\lambda_\infty(S)=\ell(S)}$ hold?

For affine isometric actions on Hilbert spaces, vanishing of ${L}$ and ${\ell}$ correspond to vanishing of reduced cohomology,…

5.1. Results

Equality ${\lambda_\infty(S)=\ell(S)}$ holds for buildings, symmetric spaces and hyperbolic spaces.

Theorem 2 (Breuillard-Fujiwara) For buildings, there is ${k=O(dim(X))}$ such that equality ${\lambda_k(S)=\ell(S)}$ holds.

For symmetric spaces there are ${k,C=O(dim(X))}$ such that inequalities ${\ell(S)-C\leq\lambda_k(S)\leq\ell(S)}$ holds.

For ${\delta}$-hyperbolic metric spaces, there is ${C=C(\delta)}$ such that ${\ell(S)-C\leq\lambda_2(S)\leq\ell(S)}$.

Last statement generalizes Serre’s lemma. Our proof is a quasification of Serre’s. The second generalizes Bochi’s. Our proof is not fully geometric, it still uses some linear algebra.

It follows that if ${L(S)>0}$, there is an element of length ${k}$ which is responsible for that.

5.2. Proof

I stress how useful Helly’s theorem is: in a ${d}$-dimensional ${CAT(0)}$ space, if convex subsets have non-empty ${d+1}$-wise intersections, then the intersection of the whole family is non-empty.

There are even more general versions. There is a ${\delta}$-hyperbolic version.

It implies that ${L(S)=\sup\{L(S')\,;\,S'\subset S,\,|S'|\leq d+1\}}$.

5.3. Application to uniform exponential growth

Corollary: for a ${\delta}$-hyperbolic space, either ${L(S)\leq C\,\delta}$ (i.e. ${S}$ almost fixes a point) or two elements generating a free semi-group can be found in ${S^3}$.

Note that the second case may never occur (e.g. Burnside groups).

This improves on Besson-Courtois-Gallot 2011. They obtained exponential growth for pinched Riemannian manifolds, but could not find a free semi-group, due to possible elliptics.

We recover their result because almost fixed points are ruled out by Margulis Lemma.

In hyperbolic spaces, a version of Margulis lemma follows for the structure theorem for approximate groups. Therefore we can state:

Corollary: for a ${\delta}$-hyperbolic space in which every ball of radius ${2\delta}$ is covered by ${K}$ balls of radius ${\delta}$, a free sub-semigroup is generated by elements of length ${\leq N(K)}$ independent on ${\delta}$.

Hope to cover mapping class groups.

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## Notes of Danny Calegari’s Cambridge lecture 12-05-2017

Laminations and external angles for similarity pairs

Joint with Alden Walker.

Ending laminations have an analogue inholomorphic dynamics, impression laminations (aka Thurston lamination).

Similarity pairs is a theory intermediate between holomorphic dynamics and Kleinian groups. I explain the analogue of an ending lamination in this theory.

Example: uniformizing the complement of a locally connected Julia set gives rise to a surjective map of the circle to its boundary. Connecting by geodesics in hyperbolic planes tuples of points mapped to the same point gives rise to a lamination of hyperbolic plane.

1. Similarity pairs

Example of a similarity pair. Two affine maps of ${{\mathbb C}}$ with the same stretch ${c\in{\mathbb C}}$, ${|c|<1}$, but distinct fixed points. E.g. ${f(z)=cz+1}$ and ${g(z)=cz-1}$. Let ${G}$ be the semi-group they generate. It has a limit set ${\Lambda}$, a compact set which is forward invariant under ${G}$.

Easy to draw: start with a large disk, take its image by the ${n}$-ball of ${G}$ (a union of little disks), for ${n}$ large (indeed, ${\Lambda}$ is the intersection of these as ${n}$ tends to infinity).

1.1. Connectivity

The Barnsley-Harrington Mandelbrot set is the set of parameters ${c}$ for which the ${\Lambda}$ is connected. Equivalently if ${f\Lambda\cap g\Lambda\not=\emptyset}$. Equivalently, for which ${c}$ is a root of a power series with coefficients in ${\{-1,0,1\}}$. Equivalently, for which ${c}$ is a limit of roots of polynomials with coefficients in ${\{-1,0,1\}}$.

How do roots arise ? Start with a point ${\alpha\in{\mathbb C}}$ and apply a positive word in ${f}$ and ${g}$. Get

$\displaystyle \begin{array}{rcl} 1\pm c\pm c^2 \pm c^3+\cdots+\pm c^{n-1}+c^n \alpha. \end{array}$

This converges to ${F(c)}$, ${F}$ a ${\pm1}$ series. If ${f\Lambda\cap g\Lambda\not=\emptyset}$, a ${\{-1,0,1\}}$-series vanishes on ${c}$.

A computer picture shows that ${M}$ looks like an annulus with outer boundary the unit circle and a hairy inner boundary with two whiskers: segments of the real axis. Calegari-Koch-Walker: the interior of ${M}$ is dense in ${M}$ away from the real axis. This helps drawing certified pictures. Play with our software! ${M}$ turns out to have infinitely many holes. We conjecture that holes accumulate on every point of the frontier of ${M}$ (away from the real axis). We also conjecture that algebraic points of the frontier are dense there.

Questions.

1. How do cutpoints of ${\Lambda}$ depend on ${c}$ ?
2. Give a simple topological model for the action of ${G}$ on ${\Lambda}$.

1.2. Uniformization

When ${c\in M}$, uniformize the complement of ${\Lambda}$, yeilds a parametrization of ${\partial \Lambda}$. For ${c\in\partial M}$, the dynamics on ${\partial\Lambda}$ can be partially lifted to the circle. It can be merged into a single piecewise continuous map ${H:S^1\rightarrow S^1}$ that preserve the dynamical lamination.

Theorem 1 (Calegari-Walker) A pair of points of ${S^1}$ corresponds to a cut-point of ${\Lambda}$ iff all iterates of ${H}$ are defined on it.

This leads to a criterion for the existence of cut-points, in terms of the dynamical lamination. This is algorithmic: a directed graph is inductively produced, cut-points correspond to infinite directed paths in it. Walk on this graph amounts to dynamics on the set of cut-points.

The action of of ${H}$ on ${S^1}$ is conjugate to a (discontinuous) piecewise linear action.

1.3. Dendrites

When the stretch of ${H}$ is equal to 2, ${\Lambda}$ is a dendrite, ${H}$ is quasiconformally conjugate to degree 2 rational maps studied by Bandt, Solomyak.

2. Questions

What Hausdorff dimension? Easy: cut into two pieces, apply dilation by ${c}$.

What if ${c}$ is algebraic? Analogue to Misiurewicz points.

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