Lie theory of finite simple groups and the Roth property

In noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset $\mathcal{C}$ stable under inversion. We study the associated Killing form K. For the universal calculus associated to $\mathcal{C}$ = G \ {e} we show that the magnitude $\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$ of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in any Sn has invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groups D2n with n odd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

Manifold Conjugation and Discrete Gear Design

This paper presents a new conjugation theory and a new design methodology for the advanced conjugation design based on CAD (computer aided design) and CAE (computer aided engineering). A general conjugation theory called manifold conjugation is established based on the classic conjugation theory and manifold theory. It leads to both smooth conjugation modeling and discrete conjugation modeling. The former one has been studied in the previous work for design optimization of analytical conjugation properties. The latter one is a desirable methodology for the conjugation design based on CAE simulation. The paper brings up a new perspective towards conjugation representation theory, leading to a new design process. Contrary to the traditional idea that CAD has to be known before and separated from CAE, the new theory suggests using discrete manifold to bridge CAD and CAE. Furthermore, discrete modeling techniques from CG (computer graphics) such as subdivision can be employed for conjugation modeling and design. Two planar examples that minimize gear PPTE (peak to peak transmission error) show that the methodology is capable of and effective in conjugation optimization.

Locally compact groups whose conjugation representations satisfy a Kazhdan type property or have countable support

For a locally compact group G with left Haar measure and modular function δ the conjugation representation γG of G on L2(G) is defined byf ∈ L2(G), x, y ∈ G. γG has been investigated recently (see [19, 20, 21, 24, 32, 35]). For semi-simple Lie groups, a related representation has been studied in [25]. γG is of interest not least because of its connection to questions on inner invariant means on L∞(G). In what follows suppγG denotes the support of γG in the dual space Ĝ, that is the closed subset of all equivalence classes of irreducible representations which are weakly contained in γG. The purpose of this paper is to establish relations between properties such as a variant of Kazhdan's property and discreteness or countability of supp γG and the structure of G.