The Kervaire-Laudenbach conjecture and presentations of simple groups

The statement ``no nonabelian simple group can be obtained from a nonsimple group by adding one generator and one relator" 1) is equivalent to the Kervaire--Laudenbach conjecture; 2) becomes true under the additional assumption that the initial nonsimple group is either finite or torsion-free. Key words: Kervaire--Laudenbach conjecture, relative presentations, simple groups, car motion, cocar comotion. AMS MSC: 20E32, 20F05, 20F06.Comment: 20 pages, 13 figure

Easy Monitored Entangled States

We discuss the generation and monitoring of durable atomic entangled state via Raman-type process, which can be used in the quantum information processing.Comment: 9 pages, 2 figures, the previous title "Durable Entanglement in Atomic Systems" is replaced by new title, accepted to Appl. Phys. Let

Economical adjunction of square roots to groups

How large must an overgroup of a given group be in order to contain a square root of any element of the initial group? We give an almost exact answer to this question (the obtained estimate is at most twice worse than the best possible) and state several related open questions.Comment: 5 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V2: minor correction

Free subgroups of one-relator relative presentations

Suppose that G is a nontrivial torsion-free group and w is a word over the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the group \~G= always contains a nonabelian free subgroup. For n=1 the question about the existence of nonabelian free subgroups in \~G is answered completely in the unimodular case (i.e., when the exponent sum of x_1 in w is one). Some generalisations of these results are discussed.Comment: V3: A small correction in the last phrase of the proof of Theorem 1. 4 page

Connecting N-representability to Weyl's problem: The one particle density matrix for N = 3 and R = 6

An analytic proof is given of the necessity of the Borland-Dennis conditions for 3-representability of a one particle density matrix with rank 6. This may shed some light on Klyachko's recent use of Schubert calculus to find general conditions for N-representability