The goal is to give a short overview of topics in *symmetric transformations* on metric measure spaces. We will address the questions of when is the group of symmetries of a m.m. space *well-behaved*? And of what can be concluded concerning the induced geometry of a space with symmetries?

In the first part of the talk we study the existence of a differential structure on *symmetry groups* of metric measure spaces. For a *class *of m.m. spaces, we present a necessary and sufficient condition for the existence of such a structure. As a consequence, we recover classical results and provide new examples of spaces with smooth symmetry groups; such is the case of some spaces satisfying synthetic lower Ricci curvature bounds in the Lott-Sturm-Villani sense.

Motivated by these results, in the remainder we consider symmetric groups acting on m.m. spaces which satisfy a curvature-dimension type condition. We show that these conditions are preserved under quotient maps and glance into various novel applications. This second part is based on a collaboration together with Galaz-García, Kell, and Mondino.

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