Jim Belk University of St Andrews
Home | CV | Research | Talks | Teaching | Writing

Research

My research lies in the intersection of group theory, dynamical systems, and topology. 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. I have also done some significant work in complex dynamics, which studies the iteration of holomorphic functions in one or more complex variables.

Within geometric group theory my research focuses on discrete groups of homeomorphisms such as the Thompson groups \(F\), \(T\), and \(V\) and the Grigorchuk group. By studying the dynamics of such homeomorphisms, we can learn about algebraic properties of the group such as subgroup structure, isomorphisms, and conjugacy. I am also quite interested in boundaries of hyperbolic groups and the dynamics of such a group acting on its boundary. All of this research touches on symbolic dynamics and automata theory as well as fractal geometry. I am also an expert in using geometric topology and CAT(0) geometry to build complexes and analyze the finiteness properties of groups of homeomorphisms.

Within complex dynamics my research focuses on using methods from geometric group theory and geometric topology to answer questions about holomorphic functions. My recent preprint with Justin Lanier, Dan Margalit, and Becca Winarski offers a purely geometric algorithm for recognizing the Thurston class of a topological polynomial, and more recently I have been working with Dan Margalit and Becca Winarski to develop a geometric proof of Thurston's theorem for topological polynomials. All of our methods are built on techniques developed for understanding mapping class groups and the Nielsen–Thurston classification.

Publications & Preprints

Here is a complete list of my papers and preprints. My collaborators include Collin Bleak, Kai-Uwe Bux, Peter Cameron, Bradley Forrest, Nabil Hossain, James Hyde, Sarah Koch, Justin Lanier, Dan Margalit, Francesco Matucci, Robert McGrail, Shayo Olukoya, Martyn Quick, Rachel Skipper, Rebecca Winarski, Matthew Zaremsky and my Ph.D. advisor Kenneth Brown.

Click on the   icon to download the corresponding manuscript.

Stabilizers in Higman–Thompson Groups
with James Hyde and Francesco Matucci.
Preprint (2021). arXiv:2104.05572.
Conjugator Length in Thompson's Groups
with Francesco Matucci.
Preprint (2021). arXiv:2101.10316.
Embeddings of Q into Some Finitely Presented Groups
with James Hyde and Francesco Matucci.
Preprint (2020). Accepted to Bulletin of the American Mathematical Society (pending revisions). arXiv:2005.02036.
Twisted Brin–Thompson Groups
with Matthew C. B. Zaremsky.
Preprint (2020). Accepted to Geometry & Topology. arXiv:2001.04579.
Recognizing Topological Polynomials by Lifting Trees
with Justin Lanier, Dan Margalit, and Rebecca R. Winarski.
Preprint (2019). arXiv:1906.07680.
Rational Embeddings of Hyperbolic Groups
with Collin Bleak and Francesco Matucci.
Preprint (2017). Accepted to Journal of Combinatorial Algebra. arXiv:1711.08369.
2020
On the Asynchronous Rational Group
with James Hyde and Francesco Matucci.
Groups, Geometry, and Dynamics 13.4: 1271–1284.
2020
Embedding Right-Angled Artin Groups into Brin-Thompson Groups
with Collin Bleak and Francesco Matucci.
Math. Proceedings of the Cambridge Philosophical Society 169.2: 225–229.
2019
Rearrangement Groups of Fractals
with Bradley Forrest.
Transactions of the American Mathematical Society 372.7: 4509–4552.
2017
Some Undecidability Results for Asynchronous Transducers and the Brin-Thompson Group \(\boldsymbol{2V}\)
with Collin Bleak.
Transactions of the American Mathematical Society 369.5: 3157–3172.
2016
Röver's Simple Group is of Type \(\boldsymbol{F_\infty}\)
with Francesco Matucci.
Publicacions Matemàtiques 60.2: 501–552.
2015
The Word Problem for Finitely Presented Quandles is Undecidable
with Robert McGrail.
In Logic, Language, Information, and Computation, pp. 1–13. Springer.
2015
A Thompson Group for the Basilica
with Bradley Forrest.
Groups, Geometry, and Dynamics 9.4: 975–1000.
2014
Implementation of a Solution to the Conjugacy Problem in Thompson's Group \(\boldsymbol{F}\)
with Nabil Hossain, Francesco Matucci, and Robert McGrail.
ACM Communications in Computer Algebra 47.3/4: 120–121.
2014
CSPs and Connectedness: P/NP Dichotomy for Idempotent, Right Quasigroups
with Benjamin Fish, Solomon Garber, Robert McGrail, and Japheth Wood.
Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2014 16th International Symposium on, pp. 367–374. IEEE.
2014
Conjugacy and Dynamics in Thompson's Groups
with Francesco Matucci.
Geometriae Dedicata 169.1 (2014): 239–261.
2013
Deciding Conjugacy in Thompson's Group F in Linear Time
with Nabil Hossain, Francesco Matucci, and Robert McGrail.
Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 15th International Symposium on. IEEE.
2010
Iterated Monodromy for a Two-Dimensional Map
with Sarah Koch.
In the Tradition of Ahlfors–Bers, V, 1–12, Contemp. Math., 510, AMS.
2005
Thompson's Group \(\boldsymbol{F}\) is Maximally Nonconvex
with Kai-Uwe Bux.
Geometric methods in group theory, 131–146, Contemp. Math., 372, AMS.
2005
Forest Diagrams for Elements of Thompson's Group \(\boldsymbol{F}\)
with Kenneth Brown.
International Journal of Algebra and Computation 15, no. 5–6, 815–850.
2004
Thompson's group \(\boldsymbol{F}\)
Ph.D. thesis, Cornell University, supervised by Kenneth Brown.