[PDF] CAT(0) cubical complexes Exercice 1 (M). Prove that a CAT(0

View - Download CAT(0) cubical complexes Exercice 1 (M). Prove that a CAT(0

Previous PDF

Next PDF

CAT(0) cubical complexes

Exercice 1(M).Prove that a CAT(0) cubical complex whose links are all complete bi-partite graphs is a product of two trees. Exercice 2(M).Show that a product of two trees does not embed in R

3, even locally.

Exercice 3(E).Square a surface group.

Exercice 4(E).Show that the triple space of a group is empty if and only if the group is elementary. Exercice 5(M).Show that if a groupGacts cocompactly on a CAT(0) cube complexX, then every hyperplane is acted on cocompactly by its stabilizer. Exercice 6(E/O).Draw the CAT(0) cube complex associated to all partitions ofnpoints, wheren= 1;2;3 and 4. Can you give a descrip- tion for the general case? Exercice 7(M).LetLbe a discrete collection of lines which has nitely many parallelism classes (for example, think of the plane tri- angluated by unit isosceles triangles). Consider the setS=R2L. What is the cube complex associated to this space with walls (S;L)? Exercice 8(M/H).LetLbe a transverse collection ofnlines inH2, withn >1. Then, there exists a numberR=R(L)>0 such any line intersecting all the lines inL, intersects the ball of radiusRabout the origin. Exercice 9(M).The Coxeter groupPGL2(Z) acts by isometries on the upper half-plane preserving a hyperbolic tessellation by isometric triangles, each with one ideal vertex as illustrated on gure 1. This gives the plane a wall space structure, where the walls are the mirrors for the re ections. Describe the associated CAT(0) cube complex. Exercice 10(M/H).Show that a codimension one subgroupHin a discrete groupGgivesGa structure of space with walls. Exercice 11(M/H).LetXbe a CAT(0) cube complex,g2Aut(X) andhbe a half-space. (1) Ifgskewersh, thengis hyperbolic and any axis forgcrosses ^h. (2) Ifgis hyperbolic and the axis ofgcrossesh, then for some n2Z, we have thatgnskewersh. 1 2 The

Coxeter

complex for the group PGL CZ embedded inthe hyperbolic plane upper half plane modelFigure 1.The Coxeter complex for the groupPGL2(Z) embedded in the upper half plane model of the hyperbolic space Exercice 12(H).LetXbe a CAT(0) cube complex,g2Aut(X), let gbe a hyperbolic isometry ofXand lethbe a halfspace. Then one of the following holds. (1) Some power ofgskewersh. (2) Some power ofg ipsh. (3) Some power ofgstabilizes^h. The rst of the above possiblities is when the axis ofgmeets^hand the last is when the axis forglies in a bounded neighborhood of^h. Exercice 13(O).LetGact properly and cocompactly on a product of two trees, doesGcontainF2Z? What aboutZ2Z? E=easy, M=medium, most of those are taken from Sageev's course and require to apply the theory, H=hard, O=open,quotesdbs_dbs29.pdfusesText_35

[PDF] Réseaux de Petri

[PDF] Examen professionnel Informatique, système d 'information

[PDF] Graphes exercices et correction

[PDF] Correction examen théorie des jeux 2009-2010 - Ceremade

[PDF] Exercice n° HU 0601 - Corrigé

[PDF] 10

[PDF] 14

[PDF] Examen d analyse personnel technique (ANT) - carrieres gouv

[PDF] Examen d 'habileté ? comprendre les lois et règlements (CLRB)

[PDF] Informations admissions 2016-2017 - Gymnase français de Bienne

[PDF] Examen d analyse personnel technique (ANT) - carrieres gouv

[PDF] Examen d 'aptitude au travail de bureau (APTB) - carrieres gouv

[PDF] Atomistique

[PDF] Électrostatique et électrocinétique 1re et 2e années - 2ème édition

[PDF] Examen d 'admission aux études d 'ingénieur civil Université