Charles Sims has asked whether or not, for a free group F , the lower central subgroup γn(F ) coincides with the normal closure in F of the set of basic commutators of weight n. In this paper, we will show that Sims’ question has a positive answer when n = 5. The longest part of the proof is showing that this is true when F has rank 3. Once this is known, the results for higher ranks follow fai...

Let Mb be the operator of pointwise multiplication by b, that is Mb f = bf . Set [A,B] = AB−BA. The Reisz potentials are the operators Rα f(x) = ∫ f(x− y) dy |y|α , 0 < α < 1. They map L 7→ L, for 1 − α + 1 q = 1 p , a fact we shall take for granted in this paper. A Theorem of Chanillo MR84j:42027 [6] states that one has the equivalence ‖[Mb,Rα]‖p→q ≃ ‖b‖BMO with the later norm being that of th...

Two new measures to assess spatial ability are presented: the mental cutting test “Schnitte” (Fay & Quaiser-Pohl, 1999; English version: Fay, Quaiser-Pohl, & Rönicke, 2003), a test for selecting people with extraordinary spatial abilities, and the Picture Rotation Test (Hinze, 2002; Hinze & Quaiser-Pohl, 2003), a mental rotation test for preschool children. Both overcome some of the limits of t...

In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for br...

In this paper we characterize when the semi-commutator TfTg − Tfg of two Toeplitz operators Tf and Tg on the Hardy space of the bidisc is zero. We also show that there is no nonzero finite rank semi-commutator on the bidisc. Furthermore explicit examples of compact semi-commutators with symbols continuous on the bitorus T 2 are given.

We discuss bipermutive cellular automata from a combinatorial and topological perspective. We prove a type of topological randomizing property for bipermutive CA, show that the commutator of a bipermutive CA is always small and that bipermutive affine CA have only affine CA in their commutator. We show the last result also in the multidimensional case, proving a conjecture of [Moore-Boykett, 97].

These brief remarks have been prepared in connection with a conference in honor of my thesis advisor, Richard Rochberg.

It is shown that there exists a finitely generated infinite simple group of infinite commutator width and infinite square width in which every conjugation-invariant norm is stably bounded, and in particular stable commutator length vanishes. Moreover, a recursive presentation of such a group with decidable word and conjugacy problems is constructed.

We consider the vector-valued extension of the Fefferman–Stein– Strömberg sharp maximal inequality under growth condition. As an application we obtain a vector-valued extension of the boundedness of the commutator. Furthermore, we prove the boundedness of the commutator. 2000 Mathematics Subject Classification: Primary 42B35; Secondary 42B25.

The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the re...