We investigate subgroups of the group PLo(I) of piecewise-linear, orientation-preserving homeomorphisms of the unit interval with finitely many breaks in slope, under the operation of composition, and also subgroups of generalized Thompson groups Fn. We find geometric criteria determining the derived length of any such group, and use this criteria to produce a geometric classification of the so...

An extraordinary theorem of Gromov, [4], characterizes the finitely generated groups of polynomial growth; a group has polynomial growth iff it is nilpotent by finite. This theorem went a long way from its roots in the class of discrete subgroups of solvable Lie groups. Wolf, [11], proved that a polycyclic group of polynomial growth is nilpotent by finite. This theorem is primarily about linear...

A conjecture raised by Cossey in 2007 asserts that if G is a finite p-solvable group and φ an irreducible p-Brauer character of with vertex Q, then the number lifts at most |Q:Q′|. This now known to be true several situations for p odd, but there has been little progress even. The main obstacle appeared characteristic two all pairs lift might neither linear nor conjugate. In this paper we show ...

Let k be an algebraic number field. Let α be a root of a polynomial f ∈ k[x] which is solvable by radicals. Let L be the splitting field of α over k. Let n be a natural number divisible by the discriminant of the maximal abelian subextension of L, as well as the exponent of G(L/k), the Galois group of L over k. We show that an optimal nested radical with roots of unity for α can be effectively ...

The (asymptotic) dimension growth functions of groups were introduced by Gromov in 1999. In this paper, we show connections between dimension growth and expansion properties of graphs, Ramsey theory and the Kolmogorov-Ostrand dimension of groups and prove that all solvable subgroups of the R.Thompson group F have polynomial dimension growth. We introduce controlled dimension growth function and...

We examine the structure of the Levi component MA in a minimal parabolic subgroup P = MAN of a real reductive Lie group G and work out the cases where M is metabelian, equivalently where p is solvable. When G is a linear group we verify that p is solvable if and only if M is commutative. In the general case M is abelian modulo the center ZG , we indicate the exact structure of M and P , and we ...

Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on vector space V, is not metacyclic. Then always has regular orbit V except for few “small” cases.

For every integer g, we construct a 2-solvable and 2-bipolar knot whose topological 4-genus is greater than g. Note that knots are in particular algebraically slice have vanishing Casson-Gordon obstructions. Similarly all known smooth bounds from gauge theory Floer homology vanish for knots. Moreover, our bound smoothly embedded height four gropes $D^4$, an priori stronger condition being 2-sol...