James Belk

FOREST DIAGRAMS FOR ELEMENTS OF THOMPSON'S GROUP F

We introduce forest diagrams to represent elements of Thompsons group F. These diagrams relate to a certain action of F on the real line in the same way that tree diagrams relate to the standard action of F on the unit interval. Using forest diagrams, we give a conceptually simple length formula for elements of F with respect to the {x0,x1} generating set, and we discuss the construction of minimum-length words for positive elements. Finally, we use forest diagrams and the length formula to examine the structure of the Cayley graph of F.

Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V

Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group 2V is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of 2V on Cantor Space. Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem. To our knowledge, 2V furnishes the first concrete example of such a group, and gives an example of a direct undecidability result in the extended family of R. Thompson type groups.

A THOMPSON GROUP FOR THE BASILICA

We describe a Thompson-like group of homeomor- phisms of the Basilica Julia set. Each element of this group acts as a piecewise-linear homeomorphism of the unit circle that pre- serves 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 nitely generated. We also prove that the group is virtually simple.

Implementation of a solution to the conjugacy problem in Thompson's group F

We present an efficient implementation of the solution to the conjugacy problem in Thompsons group F. This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. We provide a description of our solution algorithm, including the data structure that represents strand diagrams and supports simplifications.

Röver's simple group is of type F ∞

We prove that Claas Rovers Thompson-Grigorchuk simple group V G has type F∞. The proof involves constructing two complexes on which V G acts: a simplicial complex analogous to the Stein complex for V , and a polysimplicial complex analogous to the Farley complex for V . We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.

The Word Problem for Finitely Presented Quandles is Undecidable

This work presents an algorithmic reduction of the word problem for recursively presented groups to the word problem for recursively presented quandles. The faithfulness of the reduction follows from the conjugation quandle construction on groups. It follows that the word problem for recursively presented quandles is not effectively computable, in general. This article also demonstrates that a recursively presented quandle can be encoded as a recursively presented rack. Hence the word problem for recursively presented racks is also not effectively computable.

CSPs and Connectedness: P/NP Dichotomy for Idempotent, Right Quasigroups

In the 1990s, Jeavons showed that every finite algebra corresponds to a class of constraint satisfaction problems. Vardi later conjectured that idempotent algebras exhibit P/NP dichotomy: Every non NP-complete algebra in this class must be tractable. Here we discuss how tractability corresponds to connectivity in Cayley graphs. In particular, we show that dichotomy in finite idempotent, right quasi groups follows from a very strong notion of connectivity. Moreover, P/NP membership is first-order axiomatizable in involutory quandles.

Deciding Conjugacy in Thompson's Group F in Linear Time

We present an efficient implementation of the solution to the conjugacy problem in Thompsons group F, a certain infinite group whose elements are piecewise-linear homeomorphisms of the unit interval. This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. We provide a comprehensive description of our solution algorithm, including the data structure that stores strand diagrams and methods to simplify them. We prove that our algorithm theoretically achieves a linear time bound in the size of the input, and we present a quadratic time working solution.