The space of marked groups and analytic properties

The space of k-marked groups was introduced by Grigorchuk. It is the space of all k-generated groups with marking (the ordered k-generators); it is endowed with a compact metrizable topology (the Cayley topology, in other words, the Chabauty topology with respect to F_k). We will discuss some behavior of analytic properties in terms of this topology, partly in connection to coarse geometry of the (infinite) disjoint union of finite marked groups.