## Senior Projects Spring 2015

#### Title

Homeomorphism Groups of Fractals

Spring 2015

Mathematics

James Belk

#### Abstract/Artist's Statement

Fractals are geometric objects that often arise from the study of dynamical systems. Besides their beautiful structures, they have unusual geometric properties that mathematicians are interested in. People have studied the homeomorphism groups of various fractals. In 1965, Thompson introduced the Thompson groups $F \subseteq T \subseteq V$, which are groups of piecewise-linear homeomorphisms on the unit interval, the unit circle, and the Cantor set, respectively. Louwsma has shown that the homeomorphism group of the Sierpinski gasket is $D_3$. More recently, Belk and Forrest investigated a group of piecewise-linear homeomorphisms on the Basilica, which is the Julia set associated with the quadratic polynomial~$z^2-1$. Weinrich-Burd and Smith, respectively, have studied the Julia sets for the maps~$\phi(z) = z^{-2}-1$ and~$\psi(z) = z^2+i$, and presented Thompson-like groups acting on these Julia sets.

In this project, we study the Julia set associated with the rational function $f(z)=(z^2+1)/(z^2-1)$. We construct a fractal $E_4$ that has the same geometric structure as the Julia set, and show that the homeomorphism group of $E_4$ is $D_4 \times \mathbb Z/2$. We construct another fractal $E_3$ with the same local structure as $E_4$. We prove that the homeomorphism group of $E_3$ is finitely generated, and show a finite presentation for this group. Furthermore, we show that this group contains an index-$2$ Kleinian subgroup. Finally, we give a geometric presentation of this group, and describe the limit set of this group acting on the Riemann sphere. The limit set appears to be homeomorphic to the $E_3$ fractal.

On-Campus only