**Seminario teoría de grupos UAM-ICMAT **

**María Pe Pereira **

**(Universidad Complutense de Madrid) **

**"Curve monodromy, quasi-periodic diffeomorphism and tête-à-tête graphs" **

**Lunes 29/1/2018, 14:30, Aula 520, UAM **

**Resumen: I will report about a joint work with J. Fernández de Bobadilla and P. Portilla **

**which is also part of the PhD Thesis of the third author. **

**Norbert A’Campo defined tête-à-tête graphs and showed that if the monodromy **

**of a plane branch is periodic then it is a generalized Dehn twist along a tête-à-tête **

**graph. **

**We see that any periodic orientable diffeomorphisms of surfaces with non-empty **

**boundary is induced by a generalized Dehn twist along a tête-à-tête graph. In this **

**sense we generalize a result by Christian Graf. **

**We also study the more general case of quasi-periodic homeomorphisms of sur- **

**faces with boundary. Monodromy of an arbitrary plane curve is an example of **

**them. To codify this type of homeomorphisms we introduce the notion of mixed **

**tête-à-tête graph, improving a former version by A’Campo. We show that any **

**quasi periodic homeomorphism, subject to certain combinatorial restriction, can **

**be modelized with a mixed tête-à-tête graph. These restrictions are accomplished **

**by the monodromy of unibranch plane curves. **

**In this talk I will introduce the monodromy of plane branches, the quasi-periodic **

**automorpisms, the tête-à-tête graphs and the mentioned characterizations.**