**Seminario Teoría de Grupos**

**CURVE MONODROMY, QUASI-PERIODIC DIFFEOMORPHISMS AND TETE-A-TETE GRAPHS.**

**MARÍA PE PEREIRA**

**UCM**

**Jueves 18 de enero, 14:30 hr.**

**Aula 520, módulo 17**

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

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

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

**of a plane branch is periodic then it is a generalized Dehn twist along a tête-a-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-a-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-a-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-a-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-a-tête graphs and the mentioned characterizations.**