### Mathematics Colloquium

# Hubert Flenner

### (Ruhr-Universität Bochum)

## "Flexible varieties and infinite transitivity of automorphism groups"

** Date: ** |
Mon, April 16, 2012 |

** Time: ** |
17:15 |

** Place: ** |
Research II Lecture Hall |

**Abstract:** Given an affine algebraic variety $X$ of dimension $n\ge 2$, we let SAut$(X)$ denote the special automorphism group of $X$ i.e., the subgroup of the full automorphism group Aut$(X)$ generated by all one-parameter unipotent subgroups. We show that if SAut$(X)$ is transitive on the smooth locus Reg$(X)$ then it is infinitely transitive on Reg$(X)$, i.e. $n$-transitive for every $n$. In turn, the transitivity is equivalent to the flexibility of $X$. The latter means that for every smooth point $x\in X$ the tangent space $T_xX$ is spanned by the velocity vectors at $x$ of one-parameter unipotent subgroups of Aut$(X)$. We provide also various modifications and applications.