Mathematics

We present a generalization of the Dehornoy-Brin braided Thompson group B V 2 that uses recursive braids. Our new groups are denoted by B V n,r (H) , for all n≥2,r≥1 and H≤ B n , where B n is the braid group on n strands. We give a new approach to deal with braided Thompson groups by using strand diagrams. We show that B V n,r (H) is finitely generated if H is finitely generated.

We exhibit a regular language of geodesics for a large set of elements of BS(1,n) and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of BS(1,n) , which was initially computed by Collins, Edjvet and Gill in [5]. Our methods are based on those we develop in [8] to show that BS(1,n) has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan, Duchin and Kropholler in [1].

Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions in the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. The proof follows an induction on dimension strategy; new ingredients include an understanding of the role played by maximal compact subgroups of Lie groups, a necessary modified form of the inequality which is also applicable to nonunimodular locally compact groups, and a proportionated averaging trick.

We reprove a theorem of Bunn, Grow, Insall, and Thiem, which asserts that a minimal congruence lattice representation for M p+1 has size 2p , and is an expansion of a regular D 2p -set.

This note aims to introduce a left adjoint functor to the functor which assigns a heap to a group. The adjunction is monadic. It is explained how one can decompose a free group functor through the previously introduced adjoint and employ it to describe a slightly different construction of free groups.

We study the action of the groups H(λ) generated by the linear fractional transformations x:z↦− 1 z and w:z↦z+λ , where λ is a positive integer, on the subsets Q ∗ ( n − − √ )={ a+ n √ c |a,b= a 2 −n c ,c∈Z} , where n is a square-free integer. We prove that this action has a finite number of orbits if and only if λ=1 or λ=2 , and we give an upper bound for the number of orbits for λ=2 .

A theorem of Brady and Meier states that a right-angled Artin group is a duality group if and only if the flag complex of the defining graph is Cohen--Macaulay. We use this to give an example of a RAAG with the property that its outer automorphism group is not a virtual duality group. This gives a partial answer to a question of Vogtmann. In an appendix, Brück describes how he used a computer-assisted search to find further examples.

We describe a minimal presentation for the group of planar pure braids on n strands.

We construct a 2 -generated pro- 2 group with full normal Hausdorff spectrum [0,1] , with respect to each of the four standard filtration series: the 2 -power series, the lower 2 -series, the Frattini series, and the dimension subgroup series. This answers a question of Klopsch and the second author, for the even prime case; the odd prime case was settled by the first author and Klopsch. Also, our construction gives the first example of a finitely generated pro- 2 group with full Hausdorff spectrum with respect to the lower 2 -series.

We say that a subgroup H is isolated in a group G if for each x∈G we have either x∈H or ⟨x⟩∩H=1 . Z. Janko, in his paper [J. Algebra, 465(2016), 41--61], determined certain classes of finite nonabelian p -groups which possess some isolated subgroups. In this note, a theorem of his paper is refined.