Notes of Andrei Zuk’s Cambridge lecture 19-01-2017

Random walks on random symmetric groups

Joint with Harald Helfgott and Akos Seres.

1. Mixing time

Related to expanders: the key word is mixing time.

Every finite simple group can be generated by 2 elements. This follows from the classication (no conceptual reason for that). How efficiently do these two elements generate ? This is what mixing time measures. We shall solve the case of the alternating group, with randomly chosen generators.

Definition 1 Fix a measure {\mu} on {G=\mathfrak{S}_n}. After {m} steps of random walk, the distribution of random element is {\mu^{\star m}}. Fix a resolution {\delta=1/e}. The mixing time is the least {m} such that

\displaystyle  \begin{array}{rcl}  \sum_{g\in G}|\mu^{\star m}-\frac{1}{|G|}|<\delta. \end{array}

Experience shows that mixing time behaves slightly differently from first eigenvalue {\lambda_1}: small scale changes in the graph can destroy expansion, whereas mixing time survives. In a sense, mixing time involves all eigenvalues, not just {\lambda_1}.

1.1. Expansion

Definable in terms of isoperimetric constants like Cheeger’s constant of a finite graph {X},

\displaystyle  \begin{array}{rcl}  h(X)=\min\{\frac{|\partial A}{|A|}\,;\,A\subset X,\,|A|\leq\frac{1}{2}|X|\}. \end{array}

Recall that a family of finite graphs is an expander if degree is bounded above and Cheeger constant is bounded away from 0.

The discrete Laplace operator is

\displaystyle  \begin{array}{rcl}  \Delta f(x)=f(x)-\frac{1}{\mathrm{deg}(x)}\sum_{y\sim x}f(y). \end{array}

Equivalently, a family of finite graphs is an expander iff degree is bounded above and {\lambda_1} is bounded away from 0.

1.2. Examples: Pinsker’s model

An infinite trivalent tree has {h(X)=1}. So regular trees are the best infinite expanders. However, finite trees do poorly.

To get expanders, the first successful source has been random graphs in the permutation model. Fix two copies {I} and {O} of {\{1,\ldots,n\}}. Pick permutations {\pi_1,\ldots,\pi_k\in\mathfrak{S}_n}. Connect {i\in I} to {\pi_j(i)} in {O}. Get a {k}-regular graph {X}, bipartite, with multiple edges. Different permutations give rise to different vertex-labelled graphs (but possibly many isomorphic graphs).

Theorem 2 (Pinsker) A symptotically almost surely, if {k\geq 3} and permutations are chosen at random, the Cheeger constant of {X} is {\geq 1/2}.

1.3. Proof

Introduce a wrong isoperimetric constant {h'}, where {|\partial A|}, {A\subset X} is replaced with {|\partial'A|=} number of vertices, {A\subset I}. Then {h(X)\geq h'(X)-1}.

Assume that {h'(X)\leq 3/2}. Then there exist {A\subset I} and {B\subset O} such that {\pi_i(A)} misses {B} for all {i}. There are at most

\displaystyle  \begin{array}{rcl}  \sum_{A\subset I,\,|A|\leq n/2}\sum_{B\subset O,\,3/2|A|}(|B|\cdots|B|(|B|-|A|+1)(n-|A|)!)^k \end{array}

such bad pairs, and this is {o((n!)^k)}.

1.4. Comparison with trees

If finite graph {X_n} is a quotient of {k}-regular tree {T_k}, then

\displaystyle  \begin{array}{rcl}  \limsup \lambda_1(X_n)\leq \lambda_0(T_k)=1-2\frac{\sqrt{k-1}}{k},\quad \limsup h(X_n)\leq h(T_k)=k-2. \end{array}

Bollobas improved this bound into {\limsup h(X_n)\leq \frac{1}{2}h(T_k)=\frac{k-2}{2}}.

The search of graphs achieving the asymptotic bound on {\lambda_1}, called Ramanujan graphs, happened to be successful.

1.5. Link with Kazhdan’s property

Theorem 3 (Margulis) Let {\Gamma} be a group finite property (T), generated by {S}. Let {\Gamma_n} be an infinite family of finite quotients of {\Gamma}. Then the family {Cay(\Gamma_n,S)} is an expander.

Conversely, if a finite 2-dimensional polyhedron has links {L} which satisfy {\lambda_1(L)>1/2}, then its fundamental group has property (T) (Zuk 1996).

2. Selberg’s conjecture

Consider congruence quotients of the hyperbolic plane, and their Riemannian Laplacians.

Theorem 4 (Selberg)

\displaystyle  \begin{array}{rcl}  \lambda_1(H^2/\Gamma(n))\geq\frac{3}{16}. \end{array}

Selberg conjectured that this bound could be improved to {1/4}. Also, {h(H^2)=1}, and one can wonder wether, asymptotically, congruence quotients have Cheeger constant tending to 1. This is not true.

Theorem 5 (Brooks-Zuk)

\displaystyle  \begin{array}{rcl}  \limsup h(H^2/\Gamma(n))\leq 0.46<\frac{3}{16}. \end{array}

Selberg’s theorem relies on the representation theory of {SL_2({\mathbb Z}/p{\mathbb Z})}. For the symmetric group, this use of representation theory has been unsuccessful yet.

3. The symmetric group

The following solves a conjecture by Diaconis.

Theorem 6 (Helfgott-Seress-Zuk) For a random choice of generators of {\mathfrak{S}_n}, the mixing time is at most {n^3\log n}.

This relies on studying the action of {\mathfrak{S}_n} on {k}-tuples of points. For each {k}, in analogy with Pinsker, one constructs a random graph with {2n^k} vertices, and shows that this produce expanders. Representation theory of {\mathfrak{S}_n} plays a role.

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Slides of Pansu’s Cambridge course, lectures 1 to 3, january 2017

Here are the slides of the first 3 lectures, jan. 18th, 25th and feb. 2nd, 2017.

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Notes of Stefan Vaes’ second Cambridge lecture 18-01-2017

Representation theory and cohomology for standard invariants

1. Tube algebra of a quasiregular inclusion

The historical reason for the word tube will not appear…

Recall that a finite index pair of {II_1} factors {T\subset S} is quasiregular if {S} is spanned by finite index {T}-submodules.

The main examples are SE-inclusions and cross-products {T\times\Gamma} attached to outer actions.

The definition has algebraic content: there is a canonical dense {*}-subalgebra {\zeta}, the linear span finite index {T}-submodules. In cross-product case, it is spanned by {Tu_g}, {g\in\Gamma}.

Theorem 1 (Popa-Schlyakhtenko-Vaes, after people did it in special cases) We construct a {*}-algebra {\mathcal{A}} such that its Hilbert representations are in {1-1} correspondence with {S}-bimodules

I explain this because it makes the tube algebra natural. Then I specialize to tensor categories.

1.1. Construction of {\mathcal{A}}

Data: {T\subset S} and a category {\mathcal{C}} of finite index {T}-bimodules.

As a vectorspace,

\displaystyle  \mathcal{A}=\bigoplus_{i,j\in Irr(\mathcal{C})}(i\zeta,\zeta j),

where {(i\zeta,\zeta j)} denotes the space of finite rank intertwiners from {L^2(S)\otimes_T H_j} to {H_i \otimes_T L^2(S)}. Finite rank means that it involves only finitely many of the finite index {T}-modules constituting {S}.

Notation. Given {\mathcal{F}} a subset of irreducibles, denote by {e_{\mathcal{F}}:L^2(S)\rightarrow L^2(S)} the projections onto the span of all of those in {\mathcal{F}}.

Then an intertwiner {V} is finite rank if {V=V(e_{\mathcal{F}}\otimes 1)=(1\otimes e_{\mathcal{F}}V} for some finite subset {\mathcal{F}} of irreducibles.

The composition of intertwiners is defined by

\displaystyle  \begin{array}{rcl}  VW=(1\otimes m)(V\otimes 1)(1\otimes W)(m^*\otimes 1), \end{array}

where {m:L^2(S)\otimes L^2(S)\rightarrow L^2(S)} is induced by multiplication.

There are canonical idempotents {p_i\in\mathcal{A}}, {i\in Irr(\mathcal{C})}, such that {p_i\mathcal{A}p_j=(i\zeta,\zeta j)}. The inclusion defines an element {\delta:L^2(T)\rightarrow L^2(S)} such that {p_i=(1\otimes \delta)(\delta^*\otimes 1)\in (i\zeta,\zeta i)}.

The adjoint is defined formally by {t:L^2(S)\rightarrow L^2(S)\otimes_T L^2(S)} as {t=m^*\delta}. In fact, only {(1\otimes e_{\mathcal{F}})t} makes sense. In reality, define, for {V\in (i\zeta,\zeta j)},

\displaystyle  \begin{array}{rcl}  V^\#=(t^*\otimes 1\otimes 1)(1\otimes V^*\otimes 1)(1\otimes 1\otimes t). \end{array}

This turns {\mathcal{A}} into a {*}-algebra.

1.2. Example of cross-product

Let {S=T\times \Gamma}. Then {\zeta=\bigoplus_{g\in \Gamma}g} is the algebraic direct sum of irreducible {T}-bimodules. {(g\zeta,\zeta h)=\bigoplus_{k,k'}(gk,k'h)}, where each summand is {{\mathbb C}} if {g=k'gk^{-1}}, 0 otherwise. Thus {\mathcal{A}} is built from the action of {\Gamma} on its set of conjugacy classes,

\displaystyle  \begin{array}{rcl}  \mathcal{A}=C_c(\Gamma)\times\Gamma. \end{array}

Also, the group algebra {{\mathbb C}\Gamma=p_e\mathcal{A}p_e} is merely a corner in {\mathcal{A}}.

Theorem 2 There is a {1-1} correspondence between Hilbert {S}-bimodules {H} such that {_T H_T} is a sum of objects of {\mathcal{C}}, and non-degenerate right Hilbert {\mathcal{A}}-modules {K}. The correspondence is

\displaystyle  \begin{array}{rcl}  \forall i\in Irr(\mathcal{C}),\quad Kp_i=(H,H_i). \end{array}

2. Construction of Ocneanu’s tube algebra

Data: a rigid {C^*}-tensor category {\mathcal{C}}.

As a vectorspace,

\displaystyle  \begin{array}{rcl}  \mathcal{A}=\bigoplus_{i,j\in Irr(\mathcal{C})}\bigoplus_{\alpha\in Irr(\mathcal{C})}(i\alpha,\alpha j), \end{array}

There are canonical idempotents {p_i\in\mathcal{A}}, {i\in Irr(\mathcal{C})}, such that

\displaystyle  p_i\mathcal{A}p_j=\bigoplus_{\alpha\in Irr(\mathcal{C})}(i\alpha,\alpha j),

and {p_E} such that

\displaystyle  \begin{array}{rcl}  p_E\mathcal{A}p_E=\bigoplus_{\alpha\in Irr(\mathcal{C})}{\mathbb C}, \end{array}

which I denote by {{\mathbb C}[Irr(\mathcal{C})]}. I call possitive the maps {\omega:{\mathbb C}[Irr(\mathcal{C})]\rightarrow{\mathbb C}} such that

\displaystyle  \begin{array}{rcl}  \forall V\in p_E \mathcal{A},\quad \omega(VV^\#)\geq 0. \end{array}

The multiplication of {V\in (i\alpha,\alpha j)} and {W\in (j\alpha,\alpha k)} is

\displaystyle  \begin{array}{rcl}  VW=\bigoplus_{\gamma\in Irr(\mathcal{C})}(\bigoplus_{X\in(\alpha\beta,\gamma)}(1\otimes X)(V\otimes 1)(1\otimes W)(X\otimes 1)). \end{array}

We need to play with three equivalent points of view: the tube algebra, bimodules, and unitary half braidings.

For SE-inclusions, the equivalence of tube algebra and Bimodules is due to Ghosh-C. Jones.

2.1. Unitary half-braiding

Data: an irreducible {X},and for each {\alpha\in\mathcal{C}}, a unitary morphism {\sigma_\alpha:\alpha X\rightarrow X\alpha} such that

\displaystyle  \begin{array}{rcl}  (1\otimes V)\sigma_\alpha=\sigma_\beta (V\otimes 1) \end{array}

if {V\in (\beta,\alpha)}, and

\displaystyle  \begin{array}{rcl}  \sigma_{\alpha\beta}=(\sigma_\alpha\otimes 1)(1\otimes \sigma_\beta). \end{array}

2.2. How to take tensor products of representations ?

OK in half-brading and bimodule pictures

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Notes of Stefan Vaes 1st Cambridge lecture 17-01-2017

Representation theory and cohomology for standard invariants

We study pairs {N\subset M} of {II_1} factors, with {dim_N M} finite.

There are a bunch of discrete invariants.

  • Standard invariants
  • Rigid {C^*} algebras
  • Planar algebras (Jones).

How can these invariants act on factors ? This is Popa’s theory.

We shall introduce representations on Hilbert spaces, and define {L^2} Betti numbers.

1. Overview: standard invariants

1.1. Jones tower

With a pair {N\subset M}, there comes a whole tower {N\subset M\subset M_1\subset\cdots}, with equal indices. {M_1=<M,e_0>}. {B_{ij}} is a grid of multimatrix algebras with {e_n\in B_{ij}} if {i<n<j}). Popa calls such a structure a {\lambda}-lattice.

1.2. Alternative view

View {M} as an {N-M}-bimodule. By tensor products over {N}, one gets more modules {M\otimes_N M\otimes_N M\cdots}. Then, as an {M-M}-bimodule, {M_1=M\otimes M}. Also, {End_{N-N}(M)\simeq N'\cap M_1}. We get a 2-category (of {N-M}-bimodules) with generator {_N M_M}.

1.3. Planar algebras

This is a diagrammatic way to write intertwiners {End_{M-M}(M\otimes_N M\otimes\cdots)}.

1.4. Popa’s classification

Theorem 1 (Popa 1992) The standard invariant is a complete invariant of the pair when {N\simeq M} is the hyperfinite {II_1} factor and the standard invariant is amenable.

This suggests to define amenability, property (T), Haagerup property,… for the standard invariants. This was done by Popa. In 2014, Popa and I gave an intrinsic definition for these notions fir {\lambda}-lattices and tensor categories. Very soon, Neshveyev-Yamashita defined unitary representations (the Drinfeld center). Immediately after, Ghosh-C. Jones showed that unitary representations coincide with ordinary representations of the tube algebra.

2. The symmetric-enveloping inclusion

Now we start again with a more synthetic view.

Start with {N\subset M}. Let {T=M\otimes M^{op}}. Let {S} be the unique {II_1} factor generated by {T} and a projection and such that

  • {<M\otimes 1,e>} are the toric construction for {N\otimes 1\subset M\otimes 1}.
  • Idem for {N^{op}}.

This automatically contains the entire Jones tower and also the Jones tunnel (a tower contained in {M}). Then

\displaystyle  \begin{array}{rcl}  M_m=(1\otimes M_{-n}^{op})' \cap S. \end{array}

Longo and Rehren, Morsodo have an alternative construction for {S}. Consider the category {\mathcal{C}} of all finite index {M}-bimodules appearing in {M^{\otimes_N^k}}. Every irredcible {\alpha\in Irr(\mathcal{C})} is realised by {_M(H_{\alpha})_M}. Define

\displaystyle  \begin{array}{rcl}  S_0 = \bigoplus_{\alpha\in Irr(\mathcal{C})}(H_\alpha\otimes \bar H_\alpha). \end{array}

This has an obvious product.

Theorem 2 There is an involution {*} and a positive functional {\tau},

\displaystyle  \begin{array}{rcl}  \tau(\xi_1\otimes\bar \xi_2)=\begin{cases} 0 & \text{ if }\alpha\not=\text{trivial}, \\ \text{trace on }M\otimes M^{op} & \text{otherwise}. \end{cases} \end{array}

Hence we get a pair of von Neumann algebras {T\subset S}. {\tau} is a trace when {N\subset M} is extremal (meaning that all bimodules in {\mathcal{C}} have equal left and right dimension.

3. Quasiregular inclusions

{T\subset S} is called quasiregular if {S} belongs to the closure of the span of finite index subbimodules.

Example 1. The symmetric-enveloping inclusion just described is.

Example 2. Crossed products {S=T\times \Gamma}.

The more general approach (by Popa, Shlyakhtenko and myself) goes as follows.

A “unitary respresentation” for {T\subset S} is a Hilbert {S}-bimodule {H} such that as a {T}-bimodule {_T H_T}, we have a direct sum of {T}-bimodules in a given category {\mathcal{C}}.

A “positive definite function” is a completely positive {T}-linear map {\phi:S\rightarrow S}. In the symmetric-enveloping example, such maps {\phi} are scalar on each {H_\alpha\otimes \bar H_\alpha}, therefore it boils down to a {{\mathbb C}}-valude fonction on {Irr(\mathcal{C})}. There in an obvious DNS construction that passes from positive definite functions to unitary representations.

4. Plan

In lecture 2, I will attach to {T\subset S} an {*}-algebra (like {{\mathbb C}\Gamma}).

In lecture 3, I will do cohomology, {L^2}-Betti numbers (like {{\mathbb C}\Gamma\subset L\Gamma}).

In lecture 4, I will give examples and computations.

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Gabriel Pallier’s notes of Alexandre Martin’s Cambridge lecture 13-01-2017

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Notes from talk 28 at NPCW01, Cambridge, January 13th 2017. A. Martin (Universität Wien), joint work with A. Genevois.

\paragraph{Reminders about graph products} Let {\Gamma} be a finite simplicial graph. For every {v \in V \Gamma}, let {G_v} be a group, and form the graph product

\displaystyle  G = \frac{\ast_{v \in V \Gamma} G_v}{\left\langle \left\langle \left\{ [g,g'] : g \in G_v, g' \in G_{v'}, (v,v') \in E \Gamma \right\} \right\rangle \right\rangle},

where {\langle \langle - \rangle \rangle} denotes taking the normal closure.

When {\Gamma} is discrete, this is a free product ; when {\Gamma} is complete, a direct product.

If for all {v \in V\Gamma}, {G_v = \mathbb{Z}} (resp. {G_v = \mathbb{Z}_2}) then {G} is a right angled Artin (resp. Coxeter) group.

\paragraph{Problem} Understand the structure and geometry of {\mathrm{Aut}(\Gamma)}. Today’s “baby” case : {\Gamma} is a long cycle (with length { n \geqslant 9}). The strategy is to construct a curve complex {Y}, and an adequate action {G \curvearrowright Y} such that {\mathrm{Aut}(G)} also acts on {Y}.

Recall that the Davis complex of a graph product is defined as follows:

  • {vertices are cosets {g G_{\Gamma'}} where {\Gamma'} is a clique of {\Gamma} and {G_{\Gamma'}} denotes the graph product over {\Gamma'}.}
  • {there is an edge from {g_{\Gamma''}} to {g_{\Gamma'}} whenever {\Gamma''} is a subclique of {\Gamma'} and {\vert \Gamma' \setminus \Gamma'' \vert = 1}.}
  • {the cubes in the graph are filled.}

This yields a {\mathrm{CAT}(0)} cube complex.

In our case, the Davis complex is the cubical subdivision of an {n}-gonal complex {X}. The stabilizers are as follows:

  • {Stabilizers of vertices are conjugated to {G_{v_i, v_{i+1}}}.}
  • {Stabilizers of edges are conjugated to {G_{v_i}}‘s.}
  • {Stabilizers of the whole polygon is trivial}

Strict fundamental domains are single polygons. Define a new complex {\Delta_X} from {X}, as follows: start form a graph whose vertices represent the wall-trees, and edges intersections of wall-trees. Then fill the embedded {n}-cycles.

{\Delta_X} is a {C'(1/4) -T(4)} complex.

Recall that {C'(1/4)} is the small cancellation condition whereas {T(4)} means that the links of vertices have girth at least {4}.

{G \curvearrowright \Delta_X} is weakly acylindrical. More precisely, write {v_T} for the vertex of {\Delta_X} induced by a wall-tree {T} of {X}; then, for any pait of vertices {v_T}, {v_{T'}} in {\Delta_X}

  • {If {d(v_T, v_{T'}) \geqslant 3} then {\mathrm{Stab}_{\Delta_X} (v_T) \cap \mathrm{Stab}_{\Delta_X}(v_{T'}) = \lbrace 1 \rbrace}.}
  • {If {d(v_T, v_{T'}) = 2 } then {\mathrm{Stab}_{\Delta_X} (v_T) \cap \mathrm{Stab}_{\Delta_X}(v_{T'})} is a subgroup in a conjugate of a {G_{v_i}} (Observe that those are not self-normalizing).}
  • {If {d(v_T, v_{T'}) = 1 } then {\mathrm{Stab}_{\Delta_X} (v_T) \cap \mathrm{Stab}_{\Delta_X}(v_{T'})} is conjugated to a {G_{v_i, v_{i+1}}} (Those are self-normalizing).}
  • {If {d(v_T, v_{T'}) = 0 } then {\mathrm{Stab}_{\Delta_X} (v_T) = \mathrm{Stab}_{\Delta_X}(v_{T'})} and are conjugated to a {G_{v_{i-1}, v_i, v_{i+1}}}.}

There exists an action {\mathrm{Aut}(G) \curvearrowright \Delta_X}.

Proof: First make {G} act on {\Delta_X^{(0)}} ; then observe by the weak acylindricity property that the edges are preserved (via a discussion on the distance between the images of a pair of adjacent vertices). \Box

The automorphism group of {G} admits the following decomposition:

\displaystyle  \mathrm{Aut}(G) = \mathrm{Inn}(G) \rtimes \mathrm{Out}(G) \simeq G \rtimes \left( \prod_{v \in V \Gamma} \mathrm{Aut}(G_v) \rtimes \underline{\mathrm{Aut}}(\Gamma) \right).

(Beware: Here {\underline{\mathrm{Aut}}(\Gamma)} denotes the automorphism group of {\Gamma} labelled with the isomorphism classes of the {G_v}‘s.)

Assume that all the {G_v} are finitely generated. Then {\mathrm{Aut}(G)} is an acylindrical group.

Assume {H \curvearrowright Y}, where {Y} is a finite-dimensional and irreducible {\mathrm{CAT}(0)} cube complex. Assume further that this action is essential, without fixed points at {\infty}, and that one can find hyperplanes {\widehat{ h_1}}, {\widehat{h_2} \subset Y} such that {\mathrm{Stab}(\widehat{h_1}) \cap \widehat{h_2}} is finite. Then {H} is either virtually cyclic or acylindrically hyperbolic.

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Gabriel Pallier’s notes of Chris Cashen’s Cambridge lecture 13-01-2017

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Notes from talk 29 at NPCW01, Cambridge, January 13th 2017. C. Cashen (Universität Wien), joint work with J. Mackay.

\paragraph{Goal} Construct “boundaries of hyperbolic groups” for non hyperbolic groups.

This builds on the following (Charney-Sultan) : let {X} be a {\mathrm{CAT}(0)} space. Define its contracting boundary {\partial_c X} as

\displaystyle  \partial_c X = \frac{\left\{ \text{contracting geodesic rays based at}\; o \right\}}{\text{bounded Hausdorff distance}}.

Pros and cons : This {\partial_c} is a QI invariant. However, for the Charney Sultan topology it is neither compact nor first countable (hence not metrizable) in general, and there is a loss of geometric intuition. On needs to find a better topology on this {\partial_c X}.

The topology of fellow-travelling quasi-geodesics on the contracting boundary of a f.g. group is a metrizable QI invariant.

\paragraph{Contractions} Let {X} be a proper, geodesic metric space. For {Z \subset X}, define {\pi_Z : X \rightarrow 2^Z} as

\displaystyle  x \mapsto \left\{ z \in Z : d(x,z) = d(x, Z) \right\}.

{\pi_Z(x)} is never empty. There is no bound on its diameter. Let {\rho} be a non-negative, non-decreasing sublinear function (i.e. {\lim f(r)/r = 0}).

Say {Z} is {\rho}-contracting if for any {x,y \in X},

\displaystyle  d(x,y) \leqslant d(x, Z) \implies \mathrm{diam}(\pi_Z(x) \cup \pi_Z(y)) \leqslant \rho (d(x,Z)).

{X} is hyperbolic. Then {Z} is contracting if and only if {Z} is quasi-convex.

{Z} is strongly contracting if it is {\rho}-contracting for some bounded {\rho}.

Let {\mu} be a function. {Z} is {\mu}-Morse if for all {L\geqslant 1}, {A \geqslant 0}, and for any {(L,A)} quasi-geodesic {\gamma} with endpoints on {Z}, {\gamma} lies in a {\mu(L,A)}-neighborhood of {Z}.

Given {\rho, L, A} as above, there exists {K,K'} and a {\rho}-contracting set {Z} such that for any continuous {(L,A)}-quasigeodesic ray {\gamma} starting on {Z}, either

  • {{\gamma} is trapped : the subset of {Z} that comes within distance {K} of {\gamma} is unbounded, and {\gamma} stays in a {K'} neighborhood of {Z}, or}
  • {{\gamma} escapes : there exists a last point on {Z} at distance {K} from {\gamma}, after which {\gamma} escapes “quickly”.}

In the second case, one can control the closest point on {Z} from {\gamma} by {J(t) = \mathrm{dist}(\gamma(t), Z)} after escape point : this is smaller than {\frac{4}{3} J + \mathrm{const}(\rho, L,A)}.

\paragraph{The FQ topology} Define the following topology: let {\eta} and {\varrho} be equivalence classes of contracting geodesic rays. Then {\eta} is close to {\varrho} if geodesics in {\eta} and {\varrho} closely follow travel for a long time.

Carney-Sultan’s idea is to control the contraction constants. Define {\partial_{c,\rho}^{FG} X = \left\{ \rho-\text{contracting rays} \right\} / \sim}, and then take the direct limit on sublinear {\rho}

\displaystyle  \partial_c^{DL} X = \lim_{\rightarrow} \partial_{c,\rho}^{FG} X.

{DL} is a QI invariant.

{\mathbb{Z}^2 \star \mathbb{Z} = \langle a,b,c \mid [a,b] \rangle}.

The preceding example illustrates a non-convergence problem ; there is need for a new topology. For any {\varrho \in \partial_c X}, denote by {\alpha^\varrho} a geodesic in {\varrho}.

For any {\varrho} there exists a sublinear {\rho^{\varrho}} so that all geodesics in {\varrho} are {\rho^{\varrho}}-contracting.

One can define a neighborhood system {\left\{ \mathcal{U}( \varrho, r ) \right\}_{\varrho \in \partial_c X, r \geqslant 1}} in a way such that a reasonable quasi geodesic in {\eta} cannot escape from {\alpha^\varrho} until after distance {r}. Denote by {FQ} (follow-travelling quasigeodesic) the topology on {\partial_c X} defined by : {U} is open if {\forall \varrho \in U, \exists r, \mathcal{U}(\varrho, r) \subset U}.

Let {X} be proper, geodesic metric space. Then {\partial_c^{FQ} X} is Hausdorff and regular. The {U(\varrho, n)} for {n \in \mathbb{N}} form a neighborhood basis.

If {G} is a finitely generated group, {\partial_c^{FQ} G} is well-defined (since this is a QI invariant).

Let {G} be a f.g. group. Then

  1. {{\partial_c G} is non-empty ; }
  2. {{\vert \partial_c G \vert = 2} iff {G} is virtually {\mathbb{Z}} ; }
  3. {If {\vert \partial_c G \vert = \infty}, then {G \curvearrowright \partial_c^{FQ} G} is minimal.}

Questions frrm the audience:

  1. {Are there examples where this invariant {FQ} is actually computable ?}
  2. {Is there a prefered metric on {\partial_c X} ?}

Answers:

  1. {Yes, some mixed free/direct products of {\mathbb{Z}}.}
  2. {Working on it…}

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Notes of Dominic Gruber’s Cambridge lecture 13-01-2017

Small cancellation theory over Burnside groups

Joint with Rémi Coulon.

1. A flexible tool for producing infinite periodic groups

Here are possible approaches.

1. Understand the proof that {B(S,n)} is infinite, and add a few relators to an infinite presentation. Olshanskii’s Tarski monster 1982. Hard.

2. Let {G} be hyperbolic and torsion free. Then {G} has infinite periodic quotients (Olshanskii 1991, Gromov-Delzant). Drawback: the exponent depends on {G}.

3. Our approach.

Theorem 1 (Coulon-Gruber) There exists {n_0} such that for all {n\geq n_0}, {n} odd, the following holds. Let {G=\langle S|R\rangle} be a {C'(\frac{1}{6})} presentation such that

  • {|S|\geq 2}, no relator of length {\leq 2}.
  • No relator is a proper power.
  • No third power is a subword of a relator.

Then {G/G^n} is infinite. Furthermore, no nonempty proper subword of a element of {R} represents the identity in {G/G^n}.

2. Examples of admissible data

In a group presentation, a piece is a common subword of two cyclic conjugates of relators. Say the presentation with cyclically reduced relators satisfies {C'(\lambda)} if whenever a piece {u} is a subword of a cyclic conjugate of a relator {r}, then {|u|<\lambda|r|}.

Examples.

  1. In the standard presentation of the genus 2 surface, all pieces have length 1.

  2. Let {S=\{a,b,t\}}. Let {x_i} be words in {a} and {b} of length {i}. Let {R} contain all words of the form {tx_{100N+1}tx^{100N+2}t\cdots tx^{100N+100}}. Then this satisfies {C'(\frac{1}{6})}.
  3. Let {S=\{a,b\}}. Let {x_\infty} be the Thue-Morse sequence. It contains no third power as subwords. Let {x_1=a}, {x_2=bb}, {x_3=aba}, … be the successive length {i} subwords. Apply previous construction, get a {C'(\frac{1}{6})} presentation. Our theorem provides an infinite periodic group.

    For {I\subset{\mathbb N}}, let {R(I)=\{r_i\,;\,i\in I\}}. This gives uncountably many different {n}-periodic groups.

3. Applications

1.Get {n}-periodic groups with coarsely embedded expander graphs.

2. Using the existence of {n}-periodic groups whose word problem is unsolvable, we get the following.

Theorem 2 Let {n\geq n_0} be not a prime. Let {(P)} be a property of groups such that

  • There exists a relatively finite presented {n}-periodic group {A} that has {(P)}.
  • There exists a relatively finite presented {n}-periodic group {B} such that any group containing {B} does not have {(P)}.

Then there does not exist an algorithm that takes a relatively finite presentation of an {n}-periodic group and decides wether it has {(P)} or not.

Here a relatively finite presentation means that one kills finitely many elements in {B(S,n)}.

Examples of suitable properties. Triviality, finiteness, being cyclic, abelian, nilpotent, solvable, amenable…

4. Periodic quotients of groups acting acylindrically on hyperbolic spaces

Let {X} be {\delta}-hyperbolic. Say {G} acts {(L,N,\epsilon)}-acylindrically on {X} if for all {x,y\in X} with {d(x,y)\geq L}, the number of elements of {G} moving each f {x} and {y} at most distance {\epsilon} away is at most {N}.

In 2013, Coulon showed that there exists {n_0} such that for all odd {n\geq n_0}, periodic quotients exist: let {G} act {(L,N,100\delta)}-acylindrically on a {\delta}-hyperbolic space {X} without elliptics. Then {G_n:=G/G^n} is infinite, and {G\mapsto G_n} is injective on a ball of radius 3 in the pseudo-metric induced by {X}.

The point is to understand the normal closure of high powers of all loxodromics simultaneously.

The method is small cancellation: {C'(\frac{1}{6})} implies that {Cay(G,S)} is hyperbolic and any cycle labelled by a relator is an isometrically embedded cycle graph. Small overlap implies a linear isometric inequality.

How does this extend to infinite presentations ?

Theorem 3 (Gruber-Sisto) Let {W\subset G} be a subset that contains all subwords of elements of {R}. Then {Cay(G,S\cup W)} is {\delta}-hyperbolic.

Apply this here. With our assumptions, {X} is nonelementary. Using the assumption that no {k+1}-st power is a subword of a relator, we show (Abbot and Hume did something similar) that the action of {G} on {X} is {(L,N,100\delta)}-acylindrical, for {N=N(k)}.

Why {k=2} ? If {\omega^k} is a subword of a relator, then the translation length of {\omega} on {X} is {\leq 1/k}.

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