From Self-Similar Structures to Self-Similar Groups

TL;DR

This paper investigates the conditions under which the limit spaces of contracting self-similar groups are self-similar structures, providing criteria, constructions, and examples to connect group actions with fractal-like spaces.

Contribution

It establishes necessary and sufficient conditions linking contracting groups and self-similar structures, including the p.c.f. case, and offers a construction method for such groups.

Findings

Characterization of when limit spaces admit self-similar structures

Conditions for self-similar structures to be p.c.f.

Construction of contracting groups from self-similar structures

Abstract

We explore the relationship between limit spaces of contracting self-similar groups and self-similar structures. We give the condition on a contracting group such that its limit space admits a self-similar structure, and also the condition such that this self-similar structure is p.c.f. We then give the necessary and sufficient condition on a p.c.f. self-similar structure such that there exists a contracting group whose limit space has an isomorphic self-similar structure; in this case, we provide a construction that produces such a contracting group. Finally, we illustrate our results with several examples.