Abstract
Iterated monodromy groups of postcritically finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the IMG of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpiński carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose IMG has exponential growth.
| Original language | English |
|---|---|
| Pages (from-to) | 1489-1518 |
| Number of pages | 30 |
| Journal | Proceedings of the London Mathematical Society |
| Volume | 116 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - 4 Jun 2018 |
| Externally published | Yes |
Funding
Received 3 July 2017; published online 2 February 2018. 2010 Mathematics Subject Classification 37F10, 37F25 (primary), 20E08 (secondary). M. Hlushchanka is supported by the Studienstiftung des Deutschen Volkes and is grateful for its continuing support. D. Meyer has been supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (project no. 271983). C 2018 The Author(s). The Proceedings of the London Mathematical Society is copyright C London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.