Thomas Koberda

photo

Associate Professor of Mathematics

E-mail: firstname dot lastname at gmail dot com

Office: Kerchof 304

Mailing Address:
Department of Mathematics
University of Virginia
P. O. Box 400137
Charlottesville, VA 22904-4137 USA

Here is my CV: pdf

Teaching

In the spring semester of 2020 I will teach Math 3315 (Advanced Calculus and Linear Algebra). All course materials/syllabi can be found on the relevant collab sites.

I taught a topics course on right-angled Artin groups at Yale in the spring of 2013. Here are some course notes. Someday I will clean these up and turn them into a book.

Conferences

I co-organize the Virginia Topology Conference, which we hope will be annual. Here is the 2016 page. The 2017 dates are November 10-12, and here is a link.

Research

My research is best described as geometric group theory. I am especially interested in mapping class groups of surfaces, right-angled Artin groups, and one-dimensional dynamics. I am also interested in the geometry and topology of 3-manifolds, and have recently become interested in the topology of arrangements.

Here are my research papers, in the order that they were written. In an effort to spam the arXiv as little as possible, the pdf versions on the UVa server (when available) are more up-to-date than the arXiv versions.

45. "Shapes of hyperbolic triangles and once-punctured torus groups." Joint with Sang-hyun Kim, Jaejeong Lee, Ken'ichi Ohshika, and Ser Peow Tan, with an appendix by Xinghua Gao. (Submitted). pdf

44. "Expanders and right-angled Artin groups." Joint with Ramón Flores and Delaram Kahrobaei. (Submitted). pdf

43. "Model theory of the curve graph." Joint with Valentina Disarlo and Javier de la Nuez González. (Preprint). pdf

42. "An algebraic characterization of k-colorability." Joint with Ramón Flores and Delaram Kahrobaei. (Submitted). pdf

41. "Small C1 actions of semidirect products on compact manifolds." Joint with Christian Bonatti, Sang-hyun Kim, and Michele Triestino. (To appear in Algebr. Geom. Topol.). pdf

40. "Integrability of moduli and regularity of Denjoy counterexamples." Joint with Sang-hyun Kim. (To appear in Discr. Contin. Dynam. Syst.) pdf

39. "A cryptographic application of the Thurston norm." Joint with Ramón Flores and Delaram Kahrobaei. (International Journal of Computer Mathematics: Computer Systems Theory 5 (2020), no. 1, 15-24.) pdf

38. "Commensurators of thin normal subgroups." Joint with Mahan Mj. (Submitted) pdf

37. "Algebraic Structure of Diffeomorphism Groups of One-Manifolds." Joint with Jaewon Chang and Sang-hyun Kim. (Submitted) pdf

36. "An effective Lie-Kolchin theorem for quasi-unipotent matrices." Joint with Feng Luo and Hongbin Sun. (Linear Algebra Appl. 581 (2019) 304-323) pdf

35. "Non-freeness of groups generated by two parabolic elements with small rational parameters." Joint with Sang-hyun Kim. (To appear in Michigan Math. Journal) pdf Ancillary file relnum-v2

34. "Commutators, commensurators, and PSL2(Z)." Joint with Mahan Mj. (Submitted) pdf

33. "Algorithmic Problems in right-angled Artin groups: Complexity and Applications." Joint with Ramón Flores and Delaram Kahrobaei. (J. Algebra 519 (2019), 111-129) pdf

32. "Diffeomorphism groups of critical regularity." Joint with Sang-hyun Kim. (To appear in Invent. Math.) pdf

31. "What is... an acylindrical group action?" (Not. Amer. Math. Soc. 65 (2018), no. 1, 31-34.) pdf

30. "Free products and the algebraic structure of diffeomorphism groups." Joint with Sang-hyun Kim. (J. Topol. 11 (2018), 1053-1075) pdf

29. "Actions of right-angled Artin groups in low dimensions." (Handbook of Group Actions (Vol. V) ALM 48, Ch. 6, pp. 227-260) pdf

28. "Two-chains and square roots of Thompson's group F." Joint with Yash Lodha. (To appear in Ergodic Theory Dynam. Systems) pdf

27. "Representations of surface groups with finite mapping class group orbits." Joint with Indranil Biwas, Mahan Mj, and Ramanujan Santharoubane. (New York J. Math. 24 (2018), 241-250) pdf NYJM link

26. "Irreducibility of quantum representations of mapping class groups with boundary." Joint with Ramanujan Santharoubane. (Quantum Topology 9 (2018), no. 4, 633-641) pdf

25. "Chain groups of homeomorphisms of the interval." Joint with Sang-hyun Kim and Yash Lodha. (Ann. Sci. de l'ENS (4) 52 (2019), no. 4, 797-820) Older, extended version Shorter, accepted version

24. "Flexibility of group actions on the circle." Joint with Sang-hyun Kim and Mahan Mj. (Lecture Notes in Mathematics, 2231. Springer, Cham, 2019) pdf

23. "Residually finite rationally p groups." Joint with Alexander I. Suciu. (Comm. Contemp. Math. 22 (2020), no. 3, 1950016, 44 pp.) pdf

22. "A representation theoretic characterization of simple closed curves on a surface." Joint with Ramanujan Santharoubane. (Math. Res. Lett. 25, No. 5 (2018), 1485-1496) pdf

21. "Unsmoothable group actions on compact one-manifolds." Joint with Hyungryul Baik and Sang-hyun Kim. (J. Eur. Math. Soc. 21 (2019), no. 8, 2333-2353) pdf

20. "Quotients of surface groups and homology of finite covers via quantum representations." Joint with Ramanujan Santharoubane. (Invent. Math. 206 (2016), 269-292) pdf

19. "The geometry of purely loxodromic subgroups of right-angled Artin groups." Joint with Johanna Mangahas and Samuel J. Taylor. (Trans. Amer. Math. Soc. 369 (2017), no. 11, 8179-8208) pdf

18. "Injective maps between flip graphs." Joint with Javier Aramayona and Hugo Parlier. (Ann. Inst. Fourier 65 (2015), no. 5, 2037-2055) pdf

17. "Right-angled Artin subgroups of the C∞ diffeomorphism group of the real line." Joint with Hyungryul Baik and Sang-hyun Kim. (Israel J. Math. 213 (2016), no. 1, 175-182) pdf

16. "Anti-trees and right-angled Artin subgroups of planar braid groups." Joint with Sang-hyun Kim. (Geom. Topol. 19 (2015), no. 6, 3289-3306) pdf

15. "An effective algebraic detection of the Nielsen-Thurston classification of mapping classes." Joint with Johanna Mangahas. (J. Top. Analysis 7 (2015), no. 1, 1-21) pdf

14. "Right-angled Artin groups and finite subgraphs of curve complexes." Joint with Sang-hyun Kim. (Osaka J. Math. 53 (2016), 705-716) pdf

13. "The geometry of the curve graph of a right-angled Artin group." Joint with Sang-hyun Kim. (Int. J. Algebra Comput. 24 (2014), no. 2, 121-169) pdf

12. "An obstruction to embedding right-angled Artin groups in mapping class groups." Joint with Sang-hyun Kim. (Int. Math. Res. Not. IMRN 2014, no. 14, 3912-3918) pdf

11. "Alexander varieties and largeness for finitely presented groups." (J. Homotopy Rel. Structures 9 (2014), no. 2, 513-531) pdf

10. "Embedability between right-angled Artin groups." Joint with Sang-hyun Kim. (Geom. Topol. 17 (2013), no. 1, 493-530) pdf Appendix

9. "Ping-pong lemmas with applications to geometry and topology." (Geometry, topology and dynamics of character varieties, 139-158, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 23, World Sci. Publ., Hackensack, NJ, 2012) I cannot post this paper on my website due to the copyright agreement.

8. "Faithful actions of automorphisms on the space of orderings of a group." (New York J. Math. 17 (2011) 783-798) pdf

7. "Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups." (Geom. Funct. Anal. 22 (2012), no. 6, 1541-1590) pdf

6. "Entropy of automorphisms, homology and the intrinsic polynomial structure of nilpotent groups." (In the tradition of Ahlfors-Bers. VI, 87-99, Contemp. Math., 590, Amer. Math. Soc., Providence, RI, 2013) pdf

5. "Residual properties of fibered and hyperbolic 3-manifolds." (Topology Appl. 160 (2013), no. 7, 875-886) pdf

4. "Effective indices of subgroups in one-relator groups with free quotients." (Comm. Alg. 39 (2011), no. 11, 4336-4341) arXiv link

3. "Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification." (Geom. Dedicata 156 (2012), 13-30) Geometriae Dedicata link

2. "On ergodic transformations that are both weakly mixing and uniformly rigid." Joint with J. James, K. Lindsey, C.E. Silva and P. Speh. (New York Journal of Mathematics 15 (2009) 393-403) NYJM link

1. "Measurable Sensitivity." Joint with J. James, K. Lindsey, C.E. Silva and P. Speh. (Proceedings of the American Mathematical Society 136 (2008) 3549-3559) Proceedings link

Non-original work and writings unsuitable for publication

1. "Class field theory and the MKR dictionary for knots." This was my minor thesis at Harvard. I am not a number theorist and I expect that there are many mistakes and inaccurate statements contained therein. It is expository and does not contain many proofs. pdf

2. "Notes on recent work of Dani Wise." These are some recent expository notes which contain a new theorem as well. This is a new version, fixing two minor errors in the first version. The corrections are: 1. one needs a relative version of the main theorem to establish virtual fibering of cusped hyperbolic 3-manifolds, and 2. graph groups are residually finite rationally solvable, but it is not because they are residually torsion-free nilpotent. pdf

3. "Representations of Galois Groups on the Homology of Surfaces." Joint with A. Silberstein. arXiv link

4. "On some of the residual properties of finitely generated nilpotent groups." pdf, arXiv link

5. "Teichmüller polynomials, Alexander polynomials and finite covers of surfaces." (Eternal preprint) pdf