Jim Belk
University of St Andrews

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.

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.