User:Jean Raimbault/sandbox

=Mathematicians= Tianyi Zheng, Amir Mohammadi

= Kazhdan's property (T)=


 * Groups with property (T) lead to good mixing properties in ergodic theory: again informally, a process which mixes slowly leaves some subsets almost invariant.
 * Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can efficiently approximate any given invertible matrix, in the sense that every matrix can be approximated, to a high degree of accuracy, by a finite product of matrices in the list or their inverses, so that the number of matrices needed is proportional to the number of significant digits in the approximation.

= Higher Teichmüller theory =

https://arxiv.org/pdf/1803.06870.pdf

https://www.ihes.fr/~kassel/Rio.pdf

= /Random walk =

= Full group =

= Bounded cohomology =

https://zbmath.org/?q=an%3A0516.53046

= Borel-Serre compactification =

= Cusp neighbourhood =

= Andreev's theorem =

= Orbit equivalence / Measure equivalence (group theory) =

Definition
Let $$\Gamma, \Lambda$$ be two discrete groups. They are said to be measure equivalent to each other if there exists a measure space $$(X, \mu)$$ and commuting $$\mu$$-preserving, essentially free actions $$\Gamma \curvearrowright (X, \mu) \curvearrowleft \Lambda$$ such that each action admits a measurable fundamental domain of finite measure.

This is an equivalence relation, though its transitivity is not obvious (it follows from the existence of fibred product in an appropriate category).

Examples of measurably equivalent groups
The fundamental example of two measure-equivalent groups is the following: if $$G$$ is a locally compact group and $$\Gamma, \Lambda$$ are lattices in $$G$$ they are measure equivalent. The coupling is given by the actions by multiplication on the left (for $$\Gamma$$) and on the right (for $$\Lambda$$) on $$G$$. For example, a nonabelian free group, a surface group and the modular group $$\mathrm{PSL}_2(\mathbb Z)$$ are measure equivalent to each other since each of them embeds as a lattice in $$\mathrm{PSL}_2(\mathbb R)$$.

In particular, any finite-index subgroup in a countable group $$\Gamma$$ is measure equivalent to it, so by transitivity if $$\Gamma'$$ has a finite-index subgroup isomorphic to a finite index subgroup of $$\Gamma$$ they are measure equivalent to each other.

The Ornstein--Weiss theorem discussed below implies in particular that any pair of countable amenable groups are measurably equivalent to each other.

Definition
Two measure-preserving actions $$\Gamma \curvearrowright (X, \mu)$$ and $$(Y, \nu) \curvearrowleft \Lambda$$ are said to be orbit equivalent to each other if there exists a measurable map $$\varphi : X \to Y$$ such that $$\varphi_*\mu = \nu$$ and $$\varphi$$ sends an orbit of $$\Gamma$$ to an orbit of $$\Lambda$$. That is, for $$\mu$$-almost every every $$x \in X$$ and for every $$\gamma \in \Gamma$$ there exists $$\lambda \in \Lambda$$ (depending on $$x$$ and $$\gamma$$) such that $$\varphi(\gamma x) = \lambda\varphi(x)$$, and the same property holds for an inverse (in the measurable category) of $$\varphi$$.

Stable orbit equivalence
To define stable orbit equivalence it is convenient to generalise the definiton of orbit equivalence to measured equivalence relations. A measurable equivalence relation $$\sim$$ on a Borel probability space $$(X, \mu)$$ is said to be measure-preserving if for any measurable transformation $$\phi$$ of $$X$$ such that $$x \sim \phi(x)$$ for $$\mu$$-almost all $$x \in X$$. The motivating example is that of the orbit relation of a measure-preserving group action. Two measure-preserving relations $$\sim_X,\, \sim_Y$$ on $$X, Y$$ are equivalent if there exists a measure-preserving isomorphism $$f : X \to Y$$ which sends almost every class of $$\sim_X$$ to a class of $$\sim_Y$$, i.e. $$x \sim_X x' \Rightarrow f(x) \sim_Y f(x')$$.

Two relations $$\sim_X,\, \sim_Y$$ on $$X, Y$$ are stably equivalent if there exists Borel subsets $$A \subset X, B\subset Y$$ such that $$A, B$$ interesect almost every class of $$\sim_X, \sim_Y$$, they have positive measure and the restrictions of $$\sim_X, \sim_Y$$ to $$A,B$$ (with the unique probability measure on $$A, B$$ which is a rescaling of the induced measure on $$X, Y$$) are equivalent (in the sense introduced in the preceding paragraph).

Two measure-preserving group actions are said to be stably orbit equivalent if their orbit equivalence relations are stably equivalent.

Stable orbit equivalence and measure equivalence
The relation between measure equivalence and orbit equivalence is then the following: two groups are measure equivalent to each other if and only if they have free probability measure preserving actions which are stably orbit equivalent to each other.

Examples and non-examples of orbit equivalent actions
Dye's theorem states that any two essentially free ergodic actions of the infinite cyclic groups $$\mathbb Z$$ are orbit equivalent to each other. The Ornstein--Weiss theorem discussed below extends this to actions of any pair of amenable groups. Even in the case of actions of $$\mathbb Z$$ this can be quite hard to see explicitely: for example, if $$\theta, \varphi \in \mathbb R$$ are irrational numbers such that $$\theta/\varphi$$ is also irrational then Dye's theorem states that there is a measurable bijection from the circle $$\mathbb R/\mathbb Z$$ to itself which sends almost every orbit of the rotation $$x \mapsto x + \theta$$ to an orbit of $$x \mapsto x + \varphi$$; but it is well--known that there is no measurable section for any of these rotations. Instead the bijection must be constructed in a non-effective way by using the Rokhlin lemma.

On the other hand every non-amenable group admits uncnuntably many orbit inequivalent free ergodic actions.

For some groups, for instance lattices in higher-rank semisimple Lie groups, the orbit equivalence class of their actions are very rigid.

Measure equivalence and amenability
An equivalence relation is said to be hyperfinite if it is an increasing union of equivalence relations with finite classes. D. Ornstein and B. Weiss proved that if a countable group $$\Gamma$$ is amenable then all its measure-preserving actions are hyperfinite. A theorem of H. Dye states that all probability measure preserving hyperfinite equivalence relations are generated by a single transformation and they are equivalent to each other. As a consequence of these two results infinite amenable groups form a single class for measure equivalence.

Invariants of measure equivalence
Some properties which are invariant under measure equivalence are:
 * Being amenable;
 * Having Kazhdan's property (T);
 * Having the Haagerup property;
 * Having a positive L2-Betti number in degree k (for any k);
 * Having cost equal to 1 (or >1).

Rigidity
As an analogue to quasi-isometric rigidity one can define measure rigidity for a group as follows: $$\Gamma$$ is measure rigid if any group which is measure equivalent to $$\Gamma$$ must be commensurable to $$\Gamma$$. Then A. Furman proved that lattices in higher-rank semisimple Lie groups are measure rigid.

Definitions
The cost of a Borel equivalence relation (with countable classes) on a Borel probability space $$(X, \mu)$$ is $$\inf_{\phi_1, \ldots, \phi_r} \sum_{i=1}^r \mu(A_i)$$ where $$\phi_i: A_i \to B_i$$ are partial isomorphism which generate the relation (that is, in almost all pairs of related points one can be mapped to the other by a finite composition of the $$\phi_i$$).

The cost of a probability measure preserving group action is the cost of its orbit equivalence relation (two points are related if a group element maps one to the other).

The cost of a countable group is the infimum of the cost over all its probability measure preserving, essentially free actions. A group is said to have fixed price if all of its essentially free actions have the same cost (which must equal the cost of the group itself).

Examples and properties
A finite group $$G$$ has fixed price, its cost equals $$1 - \frac 1{|G|}$$.

Any equivalence relation with infinite classes has cost at least 1. In particular the infinite cyclic group $$\mathbb Z$$ has fixed price 1. By the Ornstein--Weiss theorem it follows that any infinite amenable group has fixed price 1.

The free group of rank $$r$$ has fixed price equal to $$r$$. The fundamental group of a closed surface of genus $$g$$ has fixed price $$2g-1$$. More generally if $$\Gamma_1, \Gamma_2$$ are two fixed price groups and $$\Gamma_3$$ a common amenable subgroup (possibly finite) then the amalgamated product $$\Gamma_1 *_{\Gamma_3} \Gamma_2$$ has fixed price and cost equal to $$\mathrm{Cost}(\Gamma_1) + \mathrm{Cost}(\Gamma_2) - \mathrm{Cost}(\Gamma_3)$$.

The cost satisfies a form of multiplicativity: if $$\Gamma$$ contains a finite-index subgroup $$\Lambda$$ then $$\mathrm{Cost}(\Lambda) - 1 = [\Gamma:\Lambda](\mathrm{Cost}(\Gamma) - 1)$$.

If $$\Gamma$$ has Kazhdan's property (T) then its cost is equal to 1 (it is not known whether it must have fixed price).

The fundamental group of a 3-manifold has cost 1 (it is also not known whether it has fixed price).

Applications
The cost of a group is an isomorphism invariant of the von Neumann algebra associated to that group. As a consequence, the von Neumann algebras associated with free groups of different ranks are not isomorphic (this was first observed by other means in the work of Dan-Virgil Voiculescu).

It also allows to give examples of equivalence relations that cannot be generated by finitely many elements: the cost of the free group on countably many generators is infinite, so the orbit equivalence relation of any of its free actions gives such an example.

The cost has also applications to the rank gradient problem: if $$\Gamma \supset \Gamma_1 \supset \cdots \supset \Gamma_n \supset \cdots$$ are nested subgroups with $$\bigcap_{n \ge 1} \Gamma_n = \{1\}$$ then
 * $$\lim_{n \to+\infty} \frac{\mathrm{rank}(\Gamma_n) - 1}{[\Gamma:\Gamma_n]} = \mathrm{Cost}(\Gamma) - 1$$.

Career and distinctions
Guivarc'h is an emeritus professor at the university of Rennes. He was the recipient of the Paul Langevin award of the French Academy of Sciences and of the "prix fondé par l'Etat" in 2015. He was a participant in the "Marie Curie transfer of knowledge" program of the European union, in the project "HANAP".

Mathematical work
Most of Guivarc'h work concerns random walks on discrete and linear groups

= Goulnara Arzhantseva =

Goulnara Arjantseva, full name Goulnara Nurullovna Arzhantseva (Гульна́ра Нурулловна Аржа́нцева, * November 28 1973 in Perm Oblast, Soviet Union) is a Russian mathematician. She is a professor at the mathematics institute of University of Vienna and a co-direcor at the Erwin Schrödinger International Institute for Mathematical Physics. She is a specialist on geometric group theory and metric geometry.

Arzhantseva attended Kolomogorov school for young talents in physics and mathematics at Moscow State University where she later also studied in the mathematics department. She defended her Ph.D., on "Generic Properties of Finitely Presented Groups" under the direction of Aleksandr Olshansky there in 1998. Afterwards she occupied positions at the University of Geneva and the University of Neuchâtel. In October 2010 she was hired at the University of Vienna.

In 2010 she otained a Starting Independent Researcher Grant from the European Research Council.

Selected works

 * With V. Guba, M. Sapir: Metrics on diagram groups and uniform embeddings in a Hilbert space, Comment. Math. Helv. 81 (2006), no. 4, 911–929.
 * With C. Druţu, M. Sapir: Compression functions of uniform embeddings of groups into Hilbert and Banach spaces, J. Reine Angew. Math. 633 (2009), 213–235.
 * With E. Guentner, J. Špakula: Coarse non-amenability and coarse embeddings, Geom. Funct. Anal. 22 (2012), no. 1, 22–36.
 * With D. Osajda: Infinitely presented small cancellation groups have the Haagerup property, J. Topol. Anal. 7 (2015), no. 3, 389–406.

Weblinks

 * Personal webpage

= Zimmer Program = Zimmer's conjecture

Zimmer's conjecture is a statement in mathematics which forbids certain manifolds (a type of space) from admitting certain configuration of symmetries It was named after the mathematician Robert Zimmer.

Formally the conjecture states that certain groups, namely lattices in higher-rank Lie groups (the foremost example of which is $$\mathrm{SL}_n(\mathbb Z)$$ when $$n \ge 3$$) cannot act on manifolds of small dimension (where this is precisely expressed in terms of the Lie group).

The conjecture was widely open until 2017, when most cases were proven by Aaron Brown, Sebastián Hurtado-Salazar and David Fisher.

Types of actions
An action of a group $$\Gamma$$ on a manifold $$M$$ needs to be specified with a degree of regularity, and eventually an additional structure preserved. For example:
 * A smooth action is an action by smooth diffeomorphisms, that is a morphism $$\Gamma \to \mathrm{Diff}^\infty(M)$$; one can also ask for more or less differentiable actions, i.e. morphisms to $$\mathrm{Diff}^k(M)$$ where $$k$$ is an integer and $$\mathrm{Diff}^k(M)$$is the group of diffeomorphism of class $$C^k$$; one can even take $$k$$ real.
 * A volume-preserving action is defined as ollows: endow $$M$$ with a volume form $$\omega$$ (a top-degree, nowhere vanishing differential form) one can ask that a smooth (or at least differentiable) action preserves it; that is, if the action is given by a morphism $$\rho: \Gamma \to \mathrm{Diff}^\infty(M)$$ then the pullback \rho(\gamma)^*\omega is equal to $$\omega$$, for any $$\gamma \in \Gamma$$. The group of diffeomorphisms of $$M$$ preserving $$\omega$$ is usually denoted $$\mathrm{Diff}^\infty(M, \omega)$$ so a volume-preserving action is a morphism $$\Gamma \to \mathrm{Diff}^\infty(M, \omega)$$.

In all these cases the target group is infinite-dimensional Lie group.


 * If $$M$$ is a Riemannian manifold then one can ask that the action preserves the Riemannian metric, that is it is goven by a morphism to the isometry group $$\mathrm{Isom}(M)$$. This type of action

This is very different from the two types of action described previously since $$\mathrm{Isom}(M)$$ is a finite-dimensional Lie group; hence the study of isometric actions essentially reduces to the finite-dimensional representation theory of $$\Gamma$$.

Finally, one can ask for actions with the least degree of regularity.


 * A topological action is an action by homeomorphisms, that is a morphism $$\Gamma \to \mathrm{Homeo}(M)$$.

Statements
The exact statements of Zimmer's conjectures deals with a fixed group $$\Gamma$$ which is an irreducible lattice in a higher-rank semisimple Lie group $$G$$. They state that there is an integer $$h$$ depending only on $$G$$ (not on $$\Gamma$$) such that The interest of the conjectures lies in the fact that an explicit formula for $$h$$ is given; for instance, when $$G = \mathrm{SL}_n(\mathbb R)$$ the conjecture is that $$h = n$$; note that $$\mathrm{SL}_n(\mathbb R)$$ (hence all its subgroups) acts on the $$(n-1)$$-sphere since it has a linear representation on $$\mathbb R^n$$; moreover $$\mathrm{SL}_n(\mathbb Z)$$ acts on the $$n$$-torus, so both conjectures are sharp (of true) in this case.
 * If $$\dim(M) < h-1$$ then any smooth action of $$\Gamma$$ on $$M$$ has finite image;
 * If $$\dim(M) < h$$ the same is true of any volume-preserving action.

In general $$h$$ depends on the representation theory of $$G$$; let $$d$$ be the minimal dimension of a linear representation of $$G$$, let $$G_{\mathbb C}$$ be the complexification of $$G$$ and $$n$$ the smallest dimension (over $$\mathbb C$$) of a simple factor of $$G_{\mathbb C}$$; then the conjecture is stated with $$h = \min(n, d)$$.

The conjecture is motivated by the superrigidity theorem of Margulis.

topological Zimmer program https://arxiv.org/pdf/2002.01206.pdf

Generalizations
Generalizations include partial differential equations in several complex variables, or differential equations on complex manifolds. Also there are at least a couple of ways of studying complex difference equations: either study holomorphic functions which satisfy functional relations given by the difference equation or study discrete analogs of holomorphicity such as monodiffric functions. Also integral equations can be studied in the complex domain.

History
Some of the early contributors to the theory of complex differential equations include:
 * Pierre Boutroux
 * Paul Painlevé
 * Lazarus Fuchs
 * Henri Poincaré
 * David Hilbert
 * George David Birkhoff
 * Kōsaku Yosida
 * Hans Wittich
 * Charles Briot
 * Jean Claude Bouquet
 * Johannes Malmquist