# Publications

### Work in Progress

Hyperbolic Dynamics and Centralizers in the Brin-Thompson Group

*2V,*with J. Belk, C. Martinez-Perez, B. Nucinkis,*in preparation*Intersection growth in nilpotent groups, with I.Biringer, K.Bou-Rabee, M.Kassabov,

*in preparation*Free subgroups of Thompson's group

*V*and ping-pong, with C. Bleak, E. Bieniecka,*in progress*Groups with polynomial Intersection growth, M. Kassabov, I. Snopce,

*in progress*

### Research publications

Conjugator length in Thompson's groups, with J.Belk,

*submitted*(arXiv)We prove Thompson's group

*F*has quadratic conjugator length function. That is, for any two conjugate elements of*F*of length*n*or less, there exists an element of*F*of length*O(n^2)*that conjugates one to the other. Moreover, there exist conjugate pairs of elements of*F*of length at most*n*such that the shortest conjugator between them has length*Ω(nˆ2)*. This latter statement holds for*T*and*V*as well.

Integrals of groups II, with J.Araújo, C.Casolo, P. Cameron,

*submitted*(arXiv)An

*integral*of a group G is a group H whose commutator subgroup is isomorphic to G. This paper continues the investigation on integrals of groups started in the work arXiv:1803.10179. We study: (1) A sufficient condition for a bound on the order of an integral for a finite integrable group and a necessary condition for a group to be integrable. (2) The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups. (3) Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups. (4) The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class. (5) Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups. (6) Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral. We end the paper with a number of open problems.

Embeddings of ℚ into Some Finitely Presented Groups, with J.Belk, J.Hyde,

*submitted*(arXiv)We observe that the lift of Thompson's group

*T*to the real line is finitely presented and contains the additive group ℚ of the rational numbers. This gives an explicit realization of the Higman embedding theorem for ℚ, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe. We also prove that ℚ can be embedded into a finitely presented simple group, explicitly realizing the Boone-Higman theorem for ℚ. Specifically, we describe a simple group*TA*of homeomorphisms of the circle that contains ℚ, and we prove that*TA*is two-generated and has type F∞. Our method for proving finiteness properties extends existing techniques to allow for groups of homeomorphisms that are locally more complicated than the standard Thompson groups.*A discussion of this problem**can be found on Mathoverflow**.*

Conjugacy and Centralizers in Groups of Piecewise Projective Homeomorphisms, with A.Santos de Oliveira-Tosti,

*submitted*(arXiv)Monod introduced in (

*Proc. Natl. Acad. Sci. USA***110**, 12 (2013), 4524--4527) a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and an invariant and use them to compute centralizers in one group of this family.

Rational embeddings of hyperbolic groups, with J.Belk, C.Bleak, accepted by the

*Journal of Combinatorial Algebra*(arXiv and journal list of forthcoming papers)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*.

Extensions of automorphisms of self-similar groups, with P. V. Silva, accepted by the

*Journal of Group Theory*(arXiv and online publication)In this work we study automorphisms of synchronous self-similar groups and the existence of extensions to continuous automorphisms over the closure of these groups with respect to the depth metric. We obtain conditions for the continuity of such extensions, but we also construct examples of groups where such extensions do not exist. We study in detail the case of the lamplighter group L_k = Zk \wr Z.

Embedding Right-Angled Artin Groups into Brin-Thompson Groups, with J.Belk, C.Bleak,

*Mathematical Proceedings of the Cambridge Philosophical Society,***169**(2), 2020, 225-229 (arXiv and journal article)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.

Integrals of groups, with J.Araújo, C.Casolo, P. Cameron,

*Israel Journal of Mathematics,***234(**1), 2019, 149-178An 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.

On the asynchronous rational group, with J.Belk, J.Hyde,

*Groups, Geometry and Dynamics.***13(**4), 2019, 1271-1284We 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.

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.

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 most*n*, 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.

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.

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*.

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*.

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∞. We show that, provided that*V_r(Σ)*is of type F∞, 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.

Rover's Simple Group is of Type F∞ , with J.Belk,

*Publicacions Matematiques,***60**(2016), no.2, 501-524 (arXiv and journal article)

We prove that Claas Rover's Thompson-Grigorchuk simple group

*VG*has type F∞. The proof involves constructing a filtered simplicial complex on which*VG*acts, and then analyzing the descending links.

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.

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*.

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))).

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.

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*.

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.

Mather invariant in groups of piecewise-linear homeomorphisms,

*Trends in Mathematics*, Combinatorial and Geometric Group Theory (2010) 251--260 (arXiv and book chapter)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.

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.

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.

### Conference proceedings

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*books@ocg.at***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.

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.

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.

### 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)