Commutators originated over 100 years ago as a by-product of computing group characters of nonabelian groups. They are now an established and immensely useful tool in all of group theory. Commutators became objects of interest in their own right soon after their introduction. In particular, the phenomenon that the set of commutators does not necessarily form a subgroup has been well documented ...

A doodle is a collection of immersed circles without triple intersections in the 2-sphere. It was shown by second author and Tayler that doodles induce commutator identities (identities amongst commutators) free group. In this paper, we observe idea more closely concentrating on with proper noose systems elementary identities. particular, show there bijection between cobordism classes colored w...

Let R be a prime ring of char R = 2, d a nonzero derivation of R, ρ a nonzero right ideal of R and 0 = b ∈ R such that b[[d2[x, y], [x, y]]n = 0 for all x, y ∈ ρ, n ≥ 1 fixed integer. If [ρ, ρ]ρ = 0 then either bρ = 0 or d(ρ)ρ = 0. Mathematics Subject Classification: 16W25, 16R50, 16N60

Let Mb denote the operation of multiplication by b. Let Hj denote the Hilbert transform performed in the jth coordinate, for j = 1, 2, . . .. Consider the jth order iterated commutator Cb,j = [H1, · · · , [Hj , Mb] · · · ], j = 1, 2, . . . We show that the operator norm of Cb,3 on L (R) is comparable to the norm of b in Chang–Fefferman product BMO(⊗31R 2 +). This fact has some well known equiva...

We prove that the stable commutator lengths of Dehn twists are positive, implying that the natural map from the second bounded cohomology to the ordinary cohomology is not injective, that mapping class groups are not uniformly perfect and that the growth rate of Dehn twists are linear. We then prove that the dimensions of the second bounded cohomology of hyperelliptic mapping class groups and o...