Jim Belk

My research focuses primarily on the Thompson groups \(F\), \(T\), and \(V\) and their relatives, including groups of synchronous and asynchronous automata such as the Grigorchuk group and iterated monodromy groups. I am most interested in connections between these groups and other areas of mathematics, including automata and formal languages, dynamical systems, and fractal geometry.

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- 2016
**Hyperbolic Dynamics and Centralizers in the Brin-Thompson Group \(\boldsymbol{2V}\)**(with C. Martínez-Pérez, F. Matucci, and B. Nucinkis).

Manuscript in Preparation.- 2016
**Embedding Right-Angled Artin Groups into Brin-Thompson Groups**(with C. Bleak and F. Matucci).

Preprint (2016). arXiv:1602.08635.- 2016
**Rearrangement Groups of Fractals**(with B. Forrest).

Preprint (2016). arXiv:1510.03133.- 2016
**Some Undecidability Results for Asynchronous Transducers and the Brin-Thompson Group \(\boldsymbol{2V}\)**(with C. Bleak).

To appear in*Transactions of the American Mathematical Society*, arXiv:1405.0982.- 2016
**Röver's Simple Group is of Type \(\boldsymbol{F_\infty}\)**(with F. Matucci).

*Publicacions Matemàtiques*60.2 (2016), 501–552. doi:10.5565/PUBLMAT_60216_07.- 2015
**The Word Problem for Finitely Presented Quandles is Undecidable**(with R. McGrail).

In*Logic, Language, Information, and Computation*, pp. 1–13. Springer Berlin Heidelberg, 2015. doi:10.1007/978-3-662-47709-0_1.- 2015
**A Thompson Group for the Basilica**(with B. Forrest)

*Groups, Geometry, and Dynamics*9.4 (2015): 975–1000. doi:10.4171/GGD/333.- 2014
**Implementation of a Solution to the Conjugacy Problem in Thompson's Group \(\boldsymbol{F}\)**(with N. Hossain, F. Matucci, and R. McGrail)

*ACM Communications in Computer Algebra*47.3/4 (2014): 120–121. doi:10.1145/2576802.2576823.- 2014
**CSPs and Connectedness: P/NP Dichotomy for Idempotent, Right Quasigroups**(with B. Fish, S. Garber, R. McGrail, and J. Wood)

*Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2014 16th International Symposium on*, pp. 367–374. IEEE, 2014. doi:10.1109/SYNASC.2014.56.- 2014
**Conjugacy and Dynamics in Thompson's Groups**(with F. Matucci)

*Geometriae Dedicata*169.1 (2014): 239–261. doi:10.1007/s10711-013-9853-2.- 2013
**Deciding Conjugacy in Thompson's Group**(with N. Hossain, F. Matucci, and R. McGrail)*F*in Linear Time

*Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 15th International Symposium on*. IEEE, 2013. doi:10.1109/SYNASC.2013.19.- 2010
**Iterated Monodromy for a Two-Dimensional Map**(with S. Koch)

*In the Tradition of Ahlfors–Bers, V, 1–12, Contemp. Math.*, 510, AMS 2010. doi:10.1090/conm/510.- 2005
**Thompson's Group \(\boldsymbol{F}\) is Maximally Nonconvex**(with K. Bux).

*Geometric methods in group theory*, 131–146,*Contemp. Math.*, 372, AMS 2005. doi:10.1090/conm/372/06880- 2005
**Forest Diagrams for Elements of Thompson's Group \(\boldsymbol{F}\)**(with K. Brown).

Internat. J. Algebra Comput.**15**(2005), no. 5–6, 815–850. doi:10.1142/S021819670500261X- 2004
**Thompson's group \(\boldsymbol{F}\)**(Ph.D. thesis, Cornell University, supervised by K. Brown), arXiv:0708.3609.

My research on geometric group theory focuses on certain interesting examples of groups, especially the Thompson groups \(f\), \(T\), and \(V\). I am also interested in certain groups associated with Julia sets, including iterated monodromy groups of branched coverings, and certain Thompson-like groups that act on Julia sets. To a lesser extent, I am also interested in other well-known families of groups, including outer automorphism groups of free groups, mapping class groups, braid groups, Baumslag-Solitar groups, and so forth.

My past collaborators in geometric group theory have included my advisor Ken Brown as well as Kai-Uwe Bux and Francesco Matucci. More recently, I have been collaborating with Bradley Forrest on a number of topics related to geometric group theory and dynamical systems.

I am interested primarily in complex dynamics, which involves the iteration of polynomials and rational functions on the complex plane. The dynamics of such a function are related to the structure of its Julia set, which is usually a fractal.

My interest in dynamical systems stems from the iterated monodromy groups defined by Bartholdi and Nekrashevych. By associating a group to each rational function, these mathematicians were able to solve the *twisted rabbit problem*, which asked for an algorithm to identify a given period-three topological quadratic up to Thurston equivalence. This relation between complex dynamics and group theory is an exciting new development, and I have been collaborating with Sarah Koch to apply their methods to the classification of other branched coverings up to Thurston equivalence.

I was introduced to quandles by Bob McGrail, who has been investigating the constraint satisfaction problem over finite quandles. Together, Dr. McGrail and I have proven that the word problem for recursively presented quandles is not effectively computable. Together with Japheth Wood, we have also had some success in settling the constraint satisfaction problem for quandles and classifying the minimal types that appear.

I have also been investigating finite quandles on my own using the GAP computer algebra system. Using GAP, I have classified all finite connected quandles up to size 30, and I have calculated some of their homology and cohomology groups.