Work in Progress

  1. Hyperbolic Dynamics and Centralizers in the Brin-Thompson Group 2V, with J. Belk, C. Martinez-Perez, B. Nucinkis, in preparation
  2. Intersection growth in nilpotent groups, with I.Biringer, K.Bou-Rabee, M.Kassabov, in preparation
  3. Free subgroups of Thompson's group V and ping-pong, with C. Bleak, E. Bieniecka, in progress
  4. Groups with polynomial Intersection growth, M. Kassabov, I. Snopce, in progress

Research Papers

  1. Extensions of automorphisms of self-similar groups, with P. V. Silva, submitted (arXiv)
      • In this work we study automorphisms of synchronous self-similar groups, the existence of extensions to automorphisms of the full group of automorphisms of the infinite rooted tree on which these groups act on. When they do exist, we obtain conditions for the continuity of such extensions with respect to the depth metric, but we also construct examples of groups where such extensions do not exist. We study the case of the lamplighter group L_k = Z_k \wr Z and show that all of its automorphisms admit a continuous extension and determine necessary and sufficient conditions for when the automorphism group of L_k is finitely generated.
  2. Rational embeddings of hyperbolic groups, with J.Belk, C.Bleak, submitted (arXiv)
      • We prove that a large class of Gromov hyperbolic groups G, including all torsion-free hyperbolic groups, embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskii. The proof involves assigning a system of binary addresses to points in the Gromov boundary of G, and proving that elements of G act on these addresses by transducers. These addresses derive from a certain self-similar tree of subsets of G, whose boundary is naturally homeomorphic to the horofunction boundary of G.
  3. On the asynchronous rational group, with J.Belk, J.Hyde, submitted (arXiv)
      • We prove that the asynchronous rational group defined by Grigorchuk, Nekrashevych, and Sushchanskii is simple and not finitely generated. Our proofs also apply to certain subgroups of the rational group, such as the group of all rational bilipschitz homeomorphisms.
  4. An Invitation to Inverse Group Theory, with J. Araujo, C. Casolo, P. Cameron, accepted by the Israel Journal of Mathematics (arXiv)
      • An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are: (1) If a finite group has an integral, then it has a finite integral. (2) A precise characterization of the set of natural numbers n for which every group of order n is integrable: these are the cubefree numbers n which do not have prime divisors p and q with q dividing p - 1. (3) An abelian group of order n has an integral of order at most n^{1+o(1)}, but may fail to have an integral of order bounded by cn for constant c. (4) A finite group can be integrated n times (in the class of finite groups) if and only if it is the central product of an abelian group and a perfect group. There are many other results on such topics as centreless groups, groups with composition length 2, and infinite groups. We also include a number of open problems.
  5. Embedding Right-Angled Artin Groups into Brin-Thompson Groups, with J.Belk, C.Bleak, accepted by the Mathematical Proceedings of the Cambridge Philosophical Society (arXiv)
      • We prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group nV. It follows that many other groups can be embedded into some nV (e.g., any finite extension of any of Haglund and Wise's special groups), and that various decision problems involving subgroups of nV are unsolvable.
  6. Presentations of generalisations of Thompson's group V , with C.Martinez-Perez, B.Nucinkis, Pacific Journal of Mathematics 296(2), 2018, 371-403 (arXiv and journal article)
      • We consider generalisations of Thompson's group V, denoted by Vr(Σ), which also include the groups of Higman, Stein and Brin. It was shown by the authors in (Forum Math. 28 (2016), no. 5, 909-921) that under some mild conditions these groups and centralisers of their finite subgroups are of type F∞. Under more general conditions we show that the groups Vr(Σ) are finitely generated and, under the mild conditions mentioned above, we see that they are finitely presented and give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and then apply a general procedure to shorten this presentation. In the appendix, we give a proof of this general shortening procedure.
  7. Intersection growth in groups, with I.Biringer, K.Bou-Rabee, M.Kassabov, Transactions of the American Mathematical Society 369(12), 2017, 8343-8367 (arXiv and journal article)
      • The intersection growth of a group G is the asymptotic behavior of the index of the intersection of all subgroups of G with index at mostn, and measures the Hausdorff dimension of G in profinite metrics. We study intersection growth in free groups and special linear groups and relate intersection growth to quantifying residual finiteness.
  8. On groups with slow intersection growth, with M.Kassabov, Proceedings of the Edinburgh Mathematical Society 60(2) 2017, 387-390 (arXiv and journal article)
      • Intersection growth concerns the asymptotic behavior of the index of the intersection of all subgroups of a group that have index at most n. In this note we show that the intersection growth of some groups may not be a nicely behaved function by showing the following seeming contradictory results: (a) for any group G the intersection growth function i_G(n) is super linear infinitely often; and (b) for any increasing function f there exists a group G such that i_G below f infinitely often.
  9. Embeddings into Thompson's group V and coCF groups, with C.Bleak, M.Neunhoffer, Journal of the London Mathematical Society (2) 94 (2016), no. 2, 583-597 (arXiv and journal article)
      • Lehnert and Schweitzer show in (Bull. Lond. Math. Soc. 39(2), 235--241, 2007) that R. Thompson's group V is a co-context-free (coCF) group, thus implying that all of its finitely generated subgroups are also coCF groups. Also, Lehnert shows in his thesis that V embeds inside the coCF group QAut(T_{2,c}), which is a group of particular bijections on the vertices of an infinite binary 2-edge-colored tree, and he conjectures that QAut(T_{2,c}) is a universal coCF group. We show that QAut(T_{2,c}) embeds into V, and thus obtain a new form for Lehnert's conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson's group V. In particular we classify precisely which Baumslag-Solitar groups embed into V.
  10. The Conjugacy Problem in Extensions of Thompson's group F, with J.Burillo, E.Ventura, Israel Journal of Mathematics 216 (2016), no. 1, 15-59 (arXiv and journal article)
      • We consider the twisted conjugacy problem and the orbit decidability problem for certain actions on Thompson's group F. We show that the former is solvable unconditionally and that the latter is solvable under an additional assumption. By using general criteria introduced by Bogopolski, Martino and Ventura in (Trans. Amer. Math. Soc. 362(4):2003--2036, 2010), we construct an extension of F where the conjugacy problem is unsolvable and, under an additional assumption, we construct extensions of F with solvable conjugacy problem. As a byproduct of our techniques, we give a new proof of a result of Bleak-Fel'shtyn-Goncalves in (Pacific J. Math. 238 (2008), no. 1, 1--6) which shows that F has property R_\infty and which can be extended to show that Thompson's group T also has property R_\infty.
  11. Cohomological finiteness conditions and centralisers in generalisations of Thompson's group, with C.Martinez-Perez, B.Nucinkis, Forum Mathematicum 28 (2016), no. 5, 909-921 (arXiv and journal article)
      • We consider generalisations of Thompson's group V, denoted V_r(Σ), which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, V_r(Σ) is the full automorphism group of a Cantor-algebra. Under some further minor restrictions, we prove that these groups are of type F_\infty. We show that, provided that V_r(Σ) is of type F_\infty, centralisers of finite subgroups are also of type F_\infty, and give an explicit finite presentation of these centralisers in this case. Upon weakening the hypotheses on the Cantor-algebra, it can be shown that the groups V_r(Σ) are still finitely generated.
  12. Rover's Simple Group is of Type F_\infty , with J.Belk, Publicacions Matematiques, 60 (2016), no.2, 501-524 (arXiv and journal article)
      1. We prove that Claas Rover's Thompson-Grigorchuk simple group VG has type F_\infty. The proof involves constructing a filtered simplicial complex on which VG acts, and then analyzing the descending links.
  13. The tree completions of an automaton group, with P.V. Silva, Proceedings of the Seventh Workshop on Non-Classical Models of Automata and Applications (NCMA 2015), in the series 318 (2015), Austrian Computer Society, ISBN 978-3-903035-07-2 (conference website, publisher website and proceedings volume).
      • If d denotes the depth metric on the regular A-ary rooted tree T_A , then (Aut(T_A), d_A) is a compact complete metric space. Each automaton group G on A determines a compact complete subgroup G of (Aut(T_A), d_A), named the tree completion of G. Several natural problems arise in connection with Aut(G) and G. We report on results obtained for the subclass of automata groups defined by the Cayley machine of a finite abelian group H, including the famous lamplighter group.
  14. Conjugacy and Dynamics in Thompson's groups, with J.Belk, Geometriae Dedicata 169, No. 1 (2014) 239-261 (arXiv and journal article)
      • We give a unified solution to the conjugacy problem for Thompson's groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.
  15. Implementation of a Solution to the Conjugacy Problem in Thompson’s Group F, with J. Belk, N. Hossain, R. McGrail, ACM Communications in Computer Algebra 47, 3/4 (2014), 120-121 (paper and proceedings website)
      • We present an efficient implementation of the solution to the conjugacy problem in Thompson's group F. This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. We provide a description of our solution algorithm, including the data structure that represents strand diagrams and supports simplifications.
  16. Deciding Conjugacy in Thompson's Group F in Linear Time, with J. Belk, N. Hossain, R. McGrail, 2013 15th International Symposium on International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 89-96 (paper, slides and proceedings website)
      • We present an efficient implementation of the solution to the conjugacy problem in Thompson's group F, a certain infinite group whose elements are piecewise-linear homeomorphisms of the unit interval [0,1]. This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. We provide a comprehensive description of our solution algorithm, including the data structure that stores strand diagrams and methods to simplify them. We prove that our algorithm theoretically achieves an O(n) bound in the size of the input, and we present a O(n^2) working solution.
  17. Centralizers in R.Thompson’s group V_n, with C.Bleak, H.Bowman, A.Gordon, G.Graham, J.Hughes, J.Sapir, Groups, Geometry and Dynamics 7, No. 4 (2013), 821-865 (arXiv and journal article)
      • Let n be bigger than 1 and let a be an element in the Higman-Thompson group V_n. We study the structure of the centralizer of a in V_n through a careful analysis of the action of the group generated by a on the Cantor set C. We make use of revealing tree pairs as developed by Brin and Salazar from which we derive discrete train tracks to assist us in our analysis. A consequence of our structure theorem is that centralizers are finitely generated. Along the way we give a short argument using revealing tree pairs which shows that cyclic groups are undistorted in V_n.
  18. Presentations for the Higher Dimensional Thompson's groups nV , with J.Hennig, Pacific Journal of Mathematics 257, No. 1 (2012) 53--74 (arXiv and journal article)
      • In his papers (Geom. Dedicata 108 (2004) 163-192. and J. Algebra 284 (2005) no. 2, 520--558) Brin introduced the higher dimensional Thompson groups nV which are generalizations to the Thompson's group V of self-homeomorphisms of the Cantor set and found a finite set of generators and relations in the case n = 2. We show how to generalize his construction to obtain a finite presentation for every positive integer n. As a corollary, we obtain another proof that the groups nV are simple (first proved by Brin in (Publ. Mat. 54 (2010) 433-439))).
  19. The Simultaneous Conjugacy Problem in Groups of Piecewise Linear Functions, with M.Kassabov, Groups, Geometry and Dynamics 6, No. 2 (2012) 279--315 (arXiv and journal article)
      • Guba and Sapir asked if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give a solution to the latter question using elementary techniques which rely purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit interval. The techniques we develop extend the ones used by Brin and Squier allowing us to compute roots and centralizers as well. Moreover, these techniques can be generalized to solve the same question in larger groups of piecewise-linear homeomorphisms.
  20. Structure Theorems for Subgroups of Homeomorphisms Groups, with C.Bleak, M.Kassabov, International Journal of Algebra and Computation 21, no. 6 (2011), 1007--1036, (arXiv and journal article)
      • In this partly expository paper, we study the set A of groups of orientation-preserving homeomorphisms of the circle S^1 which do not admit non-abelian free subgroups. We use classical results about homeomorphisms of the circle and elementary dynamical methods to derive various new and old results about the groups in A. Of the known results, we include some results from a family of results of Beklaryan and Malyutin, and we also give a new proof of a theorem of Margulis. Our primary new results include a detailed classification of the solvable subgroups of R. Thompson's group T.
  21. Bounding the residual finiteness of free groups, with M.Kassabov, Proceedings of the American Mathematical Society 139 (2011), no. 7, 2281--2286 (arXiv and journal article)
      • We find a lower bound to the size of finite groups detecting a given word in the free group, more precisely we construct a word w_n of length n in non-abelian free groups with the property that w_n is the identity on all finite quotients of size ~ n^{2/3} or less. This improves on a previous result of Bou-Rabee and McReynolds quantifying the lower bound of the residual finiteness of free groups.
  22. Mather invariant in groups of piecewise-linear homeomorphisms, Trends in Mathematics, Combinatorial and Geometric Group Theory (2010) 251--260 (arXiv and journal article)
      • We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in (Comm. Algebra, 29, no. 10, 4557--4596, 2001) and Kassabov and Matucci in (Groups, Geometry and Dynamics 6, No. 2 (2012) 279--315). Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows us to easily recover centralizers and lends itself to generalization.
  23. Finite solvable groups whose Quillen complex is Cohen-Macaulay, Journal of Algebra 322 (2009) 969--982 (arXiv and journal article)
      • We prove that the p-Quillen complex of a finite solvable group with cyclic derived group is Cohen-Macaulay, if p is an odd prime. If p = 2 we prove a similar conclusion, but there is a discussion to be made.
  24. Cryptanalysis of the Shpilrain-Ushakov Protocol in Thompson's Group, Journal of Cryptology, 21(3) (2008) 458--468 (arXiv and journal article)
      • This paper shows that an eavesdropper can always recover efficiently the private key of one of the two parts of the public key cryptography protocol introduced by Shpilrain and Ushakov in (ACNS 2005, Lecture Notes in Comput. Sci. 3531, 151--163, 2005). Thus an eavesdropper can always recover the shared secret key, making the protocol insecure.

PhD thesis

  • Algorithms and Classification in Groups of Piecewise-Linear Homeomorphisms, PhD Thesis, Cornell University, 2008, Advisors: Ken Brown and Martin Kassabov (arXiv and ProQuest database)

Selected Talks

  • Bounding the Residual Finiteness of Free Groups, Ithaca 2010 (slides)
  • Structure Theorems for Groups of Piecewise-Linear Homeomorphisms, Charlottesville 2009 (slides)
  • All flavors of conjugacy in Thompson's groups, Barcelona 2007 (slides)
  • Cryptanalysis of the Shpilrain-Ushakov protocol in Thompson's group F, Barcelona 2007 (slides)