Self-similar quantum groups

We introduce the notion of self-similarity for compact quantum groups. For a finite set $X$, we introduce a $C^{\ast }$-algebra ${\mathbb{A}}_X$, which is the quantum automorphism group of the infinite homogeneous rooted tree $X^{\ast }$. Self-similar quantum groups are then certain quantum subgroups of ${\mathbb{A}}_X$. Our main class of examples are called finitely-constrained self-similar quantum groups, and we find a class of these examples that can be described as quantum wreath products by subgroups of the quantum permutation group.