WORKSHOP
Journées de Géométrie Hyperbolique
Fribourg, 15 - 16 October 2015
PROGRAMME
All lectures will take place in Room 0.108, Physiology Building PER 9
The coffee breaks take place in the Mensa
The public thesis presentation of Matthieu Jacquemet will take place in Pérolles 2, Lecture Hall A230
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THURSDAY |
|
FRIDAY |
| 10:15 |
Doctoral exam Jacquemet (not public) |
10:10-11:00 |
Tumarkin
Cluster algebras and reflection groups |
| |
|
Coffee break |
Mensa |
|
|
11:30-12:20 |
Granier
A discrete subgroup of PU(2,1) whose limit
set is homeomorphic to the Menger curve |
| lunch |
Brasserie le Commerce |
lunch |
Brasserie le Commerce |
| 14:10-15:00 |
Ratcliffe
Salem numbers and arithmetic hyperbolic groups |
|
|
| Coffee break |
Mensa |
14:30-15:20 |
Parker
Complex hyperbolic triangle groups |
| 15:30-16:20 |
Modification: Im Hof
The quest for orthoschemes |
|
|
| 16:30-17:20
| Guglielmetti
Commensurability of arithmetic hyperbolic Coxeter groups |
|
|
| |
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17:15-18:15 |
Public thesis presentation: Jacquemet |
| 19:00 |
Conference Dinner
Restaurant Aigle Noir |
18:15
| Apéro, Pérolles 2 |
Download the pdf version of the programme with schedule and abstracts by clicking here
Talks - titles and abstracts
Vincent EMERY (Bern) :
CANCELLED
Quasi-arithmetic hyperbolic lattices and volume
I will introduce the notion of quasi-arithmetic lattice (due to Vinberg)
and discuss the volume of hyperbolic quotients by such lattices. The
discussion will be illustrated by an explicit computation for
a quasi-arithmetic Coxeter group in dimension 5.
Jordane GRANIER (Fribourg) :
A discrete subgroup of PU(2,1) whose limit set is homeomorphic to the Menger curve
The limit set of a discrete subgroup of isometries of the hyperbolic space (real or complex) is defined as the set of accumulation points of an orbit. A result by M. Kapovich and B. Kleiner classifies the spaces of topological dimension 1 that can appear as limit sets of convex cocompact subgroups of Isom(H^n): these are the circle, the Sierpinski carpet and the Menger curve. The only explicit examples known of groups with a limit set homeomorphic to the Menger curve are subgroups of PO(n,1) constructed by M. Bourdon. In this talk, I will describe how to construct a new example in the isometry group of the complex hyperbolic plane PU(2,1).
Rafael GUGLIELMETTI (Fribourg) :
Commensurability of arithmetic hyperbolic Coxeter groups
The classification of hyperbolic Coxeter groups up to commensurability is a difficult problem where numerous tools can be used but only partial answers are known. However, when we restrict ourselves to arithmetic groups, we can associate to each commensurability class a complete set of invariants. Moreover, when the dimension of the space is even, the invariant consists only of the ramification set of a single quaternion algebra, which is defined over Q if the group is non-compact and over a number field if the group is compact.
In this talk, I will explain the construction of these invariants and give a few examples.
John PARKER (Durham) :
Complex hyperbolic triangle groups
It is well known that the group generated by reflections in the sides
of a hyperbolic triangle is rigid, even when embedded in the isometry
group of higher dimensional hyperbolic space. However it is possible
to deform such a group when it is embedded in the isometry group of
complex hyperbolic space. In his ICM talk, Rich Schwartz gave a series
of conjectures about such groups. In particular, he conjectured that
discreteness of these representations is controlled by a particular
element. In this talk I will give a survey of the topic and then
discuss certain cases where Schwartz's conjecture is true. This is
joint work with Jieyan Wang and Baohua Xie and with Pierre Will.
John RATCLIFFE (Vanderbilt) :
Salem numbers and arithmetic hyperbolic groups
This talk is about a direct relationship between a certain class of algebraic integers called Salem numbers and translation lengths
of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field.
Pavel TUMARKIN (Durham) :
Cluster algebras and reflection groups
Cluster algebras are defined inductively via repeatedly applied
operation of mutation. During the last decade it turned out that the
mutation rule appears in various contexts. We use linear reflection
groups to build a geometric model for acyclic cluster algebras, where
“partial” reflections play the role of mutations. Hyperbolic examples
show up as partial cases of the general picture.
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