Jim Belk Bard College
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Research

My primary research area is geometric group theory, the study of topological and geometric properties of infinite discrete groups, as well as the actions of infinite discrete groups on topological and geometric spaces. See Jon McCammond's geometric group theory page for an overview of the geometric group theory research community.

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.

Publications & Preprints

Here is a complete list of my papers and preprints. My collaborators include Francesco Matucci, Collin Bleak, Bradley Forrest, Brita Nucinkis, Conchita Martínez-Pérez, Matt Zaremsky, Stefan Witzel, Marco Varisco, Robert McGrail, Nabil Hossain, Sarah Koch, Kai-Uwe Bux, and my Ph.D. advisor Kenneth Brown.

Click on the   icon to download the corresponding manuscript.

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 F in Linear Time   (with N. Hossain, F. Matucci, and R. McGrail)
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.

Geometric Group Theory

Roughly speaking, geometric group theory is the study of finitely-generated groups as geometric objects. Given a finitely-generated group G, one can construct the Cayley graph of G. This graph has one vertex for each element of G, with edges corresponding to multiplication by generators. If we give each edge a length of one, then the resulting metric space can be thought of geometrically.

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.

Dynamical Systems

A dynamical system is any system who state changes with time according to a fixed rule. For example, any function from a topological space to itself can be thought of as a dynamical system, where moving forward in time corresponds to iterating the function.

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.

Quandles

Roughly speaking, a quandle is the algebraic structure obtained from a group by forgetting the multiplication and using conjugation as the main operation. These structures are intrinsically interesting, and have recently become important in knot theory as a new source of knot invariants.

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.

Voronoi Diagrams

A Voronoi diagram is a certain type of polygonal cell decomposition of the plane of interest in discrete geometry. Two summers ago, I participated in a summer research group with students concentrating on Voronoi diagrams and the dual Delaunay tessellations. This group included four students as well as myself, my wife Maria Belk, and Lauren Rose. Together, we proved an interesting graph-theoretic characterization of Delaunay tessellations, and we developed an understanding of all inscribable polyhedra with up to 12 vertices.