Muhammed ULUDAG
Title: Mapping class groupoids and Thompson's groups
Abstract:
We aim to concoct a uniform treatment of mapping class groups and Thompson's groups and to generalise them.
As a by-product we obtain a description of the outer automorphism group of free groups as the isotropy group of a groupoid,
akin to the mapping class groupoid of Mosher and Penner.
We illustrate some arithmetic aspects of these groupoids at the end of our talk.
Title: Jimm, a fundamental involution
Abstract:
Dyer's outer automorphism of PGL(2,Z) induces an involution
of the real line, which behaves very much like a kind of modular
function. It has some striking properties: it preserves the set of
quadratic irrationals sending them to each other in a non-trivial way
and commutes with the Galois action on this set. It restricts to an
highly non-trivial involution of the set unit of norm +1 of quadratic
number fields. It conjugates the Gauss continued fraction map to the
so-called Fibonacci map. It preserves harmonic pairs of numbers
inducing a duality of Beatty partitions of N. It induces a subtle
symmetry of Lebesgue's measure on the unit interval.
On the other hand, it has jump discontinuities at rationals though its
derivative exists almost everywhere and vanishes almost everywhere. In
the talk, I plan to show how this involution arises from a special
automorphism of the infinite trivalent tree.