From Self-Similar Structures to Self-Similar Groups

2010/11/08 by Daniel J. Kelleher, Kelleher, Daniel J., Benjamin A. Steinhurst +3
Mathematics · #20E08 (Primary) 20F65 (Secondary) #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20E08 #msc:20F65

paper · pdf · doi:10.48550/arxiv.1011.1817

24 pages, 11 figures

arxiv created 2010/11/19 · arxiv updated 2010/11/23

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.

Related

• Self-similar groups and topological dimension
• Boone--Higman Embeddings for Contracting Self-Similar Groups
• Graph-directed systems and self-similar measures on limit spaces of self-similar groups
• Self-similarity and limit spaces of substitution tiling semigroups
• Self-similar products of groups