We study the action of the group AutðF n Þ of automorphisms of a finitely generated free group on the degree 2 subcomplex of the spine of Auter space. Hatcher and Vogtmann showed that this subcomplex is simply connected, and we use the method described by K. S. Brown to deduce a new presentation of AutðF n Þ.

We construct rearrangement groups for edge replacement systems, an infinite class of groups that generalize Richard Thompson's groups F , T , and V . Rearrangement groups act by piecewise-defined homeomorphisms on many self-similar topological spaces, among them the Vicsek fractal and many Julia sets. We show that every rearrangement group acts properly on a locally finite CATp0q cubical complex, and we use this action to prove that certain rearrangement groups are of type F8.

Abstract. We describe a Thompson-like group of homeomorphisms of the Basilica Julia set. Each element of this group acts as a piecewise-linear homeomorphism of the unit circle that preserves the invariant lamination for the Basilica. We develop an analogue of tree pair diagrams for this group which we call arc pair diagrams, and we use these diagrams to prove that the group is finitely generated. We also prove that the group is virtually simple.

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The vitality of any modern society depends on the mathematical literacy of its citizens. No longer necessary for just a few key decision makers, mathematics is the most important skill that we can pass on to the future generation. Mathematics, the language of science, economics, engineering, and computer science, must be a skill of the people. In the 21st century, the covenant of public education mandates that colleges and universities be responsible for developing this skill.2 Unfortunately, the challenges within contemporary mathematics education are acute. The under-performance of American elementary, middle, and high school students on international standardized mathematics examinations has become a consistent element of the American educational landscape (Gonzales et al., 2008; Organisation for Economic Co-operation and Development [OECD], 2010). This deficiency can be seen at the collegiate level in the decreased mathematical readiness of incoming freshmen. In 2010, only 43% of all ACT-tested high school graduates met the Mathematics Readiness Benchmark (American College Testing Program, 2010). Lack of preparation in new college students translates into an increased need for remedial mathematics. At our institution, 28.2% of the incoming freshman class in 2010 required remedial placement for math skills. This percentage was up from 17.6% in 2009. These numbers are representative of an upward trend in remedially placed mathematics students at our institution (Forrest, et al., 2012).