## Nicolas Matte Bon [1]

Given a group, we are interested in understanding and classifying its

actions on one-dimensional manifolds, that its representations into

the groups of homeomorphisms or diffeomorphisms of an interval or the

circle. In this talk we will address this problem for a class of

groups arising via an action on intervals of a special type, called

locally moving. A well studied example in this class is the Thompson

group. We will see that if G is a locally moving group of

homeomorphisms of a real interval, then every action of G on an

interval by diffeomorphisms (of class C^1) is semiconjugate to the

natural defining action of G. In contrast such a group can admit a

much richer space of actions on intervals by homeomorphisms, and we

will investigate the structure of such actions. This is joint work

with Joaquín Brum, Cristóbal Rivas and Michele Triestino.