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We develop a Thompson-like group for the Julia set associated to the polynomial z^2 + i, in a similar manner as the Thompson group for the Basilica developed by Belk and Forrest. This Julia set is a dendrite, and we include a discussion of the geometric branching structure that arises as a result. We prove that this group is finitely generated and give explicit generators for it. Also, we show that the commutator subgroup has infinite index by way of showing that the group has an infinite cyclic quotient. Additionally, we find that this group contains isomorphic copies of the modular group, Thompson's group F, and two diagram groups of interest . Finally, we develop a scheme for generalizing this group to act on other dendrite Julia sets, and conjecture that, in those cases, the groups found will have properties similar to this one.
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Smith, William Timothy, "Thompson-Like Groups for Dendrite Julia Sets" (2013). Senior Projects Spring 2013. 266.