Embedding right-angled Artin groups into Brin–Thompson groups

We prove that every finitely-generated right-angled Artin group embeds into some Brin–Thompson group nV. It follows that any virtually special group can be embedded into some nV, a class that includes surface groups, all finitely-generated Coxeter groups, and many one-ended hyperbolic groups.

Numerical studies of Thompson’s group F and related groups

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups [Formula: see text], the lamplighter group, [Formula: see text] and the Brin–Navas group [Formula: see text]. We have also given an improved algorithm for the coefficients of Thompson’s group [Formula: see text], giving 32 terms of the cogrowth series. We develop numerical techniques to extract the asymptotics of these various cogrowth series. We present improved rigorous lower bounds on the growth-rate of the cogrowth series for Thompson’s group F using the method from [S. Haagerup, U. Haagerup and M. Ramirez-Solano, A computational approach to the Thompson group F, Int. J. Alg. Comp. 25 (2015) 381–432] applied to our extended series. We also generalise their method by showing that it applies to loops on any locally finite graph. Unfortunately, lower bounds less than 16 do not help in determining amenability. Again for Thompson’s group F we prove that, if the group is amenable, there cannot be a sub-dominant stretched exponential term in the asymptotics. Yet the numerical data provides compelling evidence for the presence of such a term. This observation suggests a potential path to a proof of non-amenability: If the universality class of the cogrowth sequence can be determined rigorously, it will likely prove non-amenability. We estimate the asymptotics of the cogrowth coefficients of F to be [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The growth constant [Formula: see text] must be 16 for amenability. These two approaches, plus a third based on extrapolating lower bounds, support the conjecture [M. Elder, A. Rechnitzer and E. J. Janse van Rensburg, Random sampling of trivial words in finitely presented groups, Expr. Math. 24 (2015) 391–409, S. Haagerup, U. Haagerup and M. Ramirez-Solano, A computational approach to the Thompson group F, Int. J. Alg. Comp. 25 (2015) 381–432] that the group is not amenable.

On Jones’ subgroup of R. Thompson’s group T

Jones introduced unitary representations for the Thompson groups [Formula: see text] and [Formula: see text] from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups [Formula: see text] and [Formula: see text]. Golan and Sapir studied [Formula: see text] and identified it as a copy of the Thompson group [Formula: see text]. In this paper, we completely describe [Formula: see text] and show that [Formula: see text] coincides with its commensurator in [Formula: see text], implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to [Formula: see text], showing that [Formula: see text] and [Formula: see text] are not isomorphic, but as annular diagram groups they have very similar presentations.

Polynomial-time right-ideal morphisms and congruences

We continue with the functional approach to the P -versus- NP problem, begun in [J. C. Birget, Semigroups and one-way functions, Int. J. Algebra Comput. 25(1–2) (2015) 3–36; J. C. Birget, Infinitely generated semigroups and polynomial complexity, Int. J. Algebra Comput. 26(04) (2016) 727–750.] We previously constructed a monoid [Formula: see text] that is non-regular iff NP [Formula: see text] P . We now construct homomorphic images of [Formula: see text] with interesting properties. In particular, the homomorphic image [Formula: see text] of [Formula: see text] is finitely generated, and is non-regular iff P [Formula: see text] NP . The group of units of [Formula: see text] is the famous Richard Thompson group [Formula: see text].