Volume growth in the component of the Dehn–Seidel twist

We consider an entropy-type invariant which measures the polynomial volume growth of submanifolds under the iterates of a map, and we show that this invariant is at least 1 for every diffeomorphism in the symplectic isotopy class of the Dehn–Seidel twist. Received: April 2004 Revision: June 2004 Accepted: October 2004     

The conjugacy problem for Dehn twist automorphisms of free groups

A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. For , a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn twist automorphisms of . Received: February 12, 1996.     

Computing commutator length in free groups

We study commutator length in free groups. (By a commutator lengthcl(g) of an element g in a derived subgroup G′ of a group G we mean the least natural number k such that g is a product of k commutators.) A purely algebraic algorithm is constructed for computing commutator length in a free group F₂ (Thm. 1). Moreover, for every element z ε F′₂ and for any natural m, the following estimate derives:cl(z^m) ≥ (ms(z) + 6)/12, where s(z) is a nonnegative number defined by an element z (Thm. 2). This estimate is used to compute commutator length of some particular elements. By analogy with the concept of width of a derived subgroup known in group theory, we define the concept of width of a derived subalgebra. The width of a derived subalgebra is computed for an algebra P of pairs, and also for its corresponding Lie algebra P_L. The algebra of pairs arises naturally in proving Theorem 2 and enjoys a number of interesting properties. We state that in a free group F_2k with free generators a₁, b₁, ..., a_k, b_k, k εN, every natural m satisfiescl(([a₁, b₁] ... [a_k, b_k])^m)=[(2 − m)/2] + mk. For k=1, this entails a known result of Culler. The notion of a growth function as applied to a finitely generated group G is well known. Associated with a derived subgroup of F₂ is some series depending on two variables which bears information not only on the number of elements of prescribed length but also on the number of elements of prescribed commutator length. A number of open questions are formulated. Supported by RFFR grant No. 98-01-00699. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 395–440, July–August, 2000.     

可约的Dehn手术

本文将利用C.Gordon和J.Luecke发展起来的组合拓扑的方法来证明缆式猜想对一类特殊纽结而言是正确的。     

On commutator subgroups

Acta Mathematica Hungarica -     

On quadratic Dehn functions

We confirm with new examples that Solvable groups of high -rank are expected to satisfy a polynomial isoperimetric inequality ([Gro93] 5A9). To that end we study invariant quasi-geodesic foliations in simply connected solvable Lie groups, endowed with left-invariant Riemannian metrics, whose leaves are isometric to closed subgroups. We establish a decomposition theorem which implies upper bounds on the Dehn (or filling) function (of loops by disks) of the solvable group in terms of the Dehn functions of the leaves. We obtain examples of metabelian polycyclic groups with exponential growth and quadratic Dehn functions. We also deduce that the horospheres in SL(4,)/SO(4,) which bound an invariant core for SL(4, ) and that the horospheres which bound an invariant core for Hilbert modular groups in have quadratic filling functions. The main theorem also applies to some solvable Lie groups which are not quasi-isometric to horospheres in symmetric spaces.Mathematics Subject Classification (2000): 20F65, 20F69, 22E15, 22E40, 53C35Délégation CNRS, UMR 5580     

On the commutator subgroup of a nonconnected central group

In this paperG denotes a central topologicalT ₂-group—G/Z(G) compact, whereZ(G) is the center. There are some results concerning compactness of the commutator subgroupG; in general (G)^– is compact ([3]), but not necessarilyG ([7]). If in additionG is a Lie group or ifG is connected,G is compact ([6], [5]). The purpose of this paper is to show, that if the componentG ₀ of the identity is open,G must be compact, and to give an example of a compact group with (G/G ₀) compact, whileG is not compact.Dedicated to Prof. R. Inzinger on his 70th birthday     

On the commutator of the Marcinkiewicz integral

boundedness is considered for the commutator of higher-dimensional Marcinkiewicz integral. Some conditions implying the and the boundedness for the commutator of the Marcinkiewicz integral are obtained.     

On the positive commutator in the radical

In this paper we prove that a positive commutator between a positive compact operator A and a positive operator B is in the radical of the Banach algebra generated by A and B. Furthermore, on every at least three-dimensional Banach lattice we construct finite rank operators A and B satisfying \(AB\ge BA\ge 0\) such that the commutator \(AB-BA\) is not contained in the radical of the Banach algebra generated by A and B. These two results now completely answer to two open questions published in (Bra?i? et al., Positivity 14:431–439, 2010). We also obtain relevant results in the case of the Volterra and the Donoghue operator.     

On a variant of commutator estimates in spectral theory

Completeness is proved for systems of two or three quantum particles. The proof is based on the following statement on operators in a Hilbert space. If an operator A is bounded with respect to a self-adjoint operator H and Re((H – )f, Af) Bf², then B is smooth with respect to H.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 29–36, 1987.     

On a commutator theorem of robert c. thompson

A very short proof of the following special case of R. C. Thompson's Theorem is presented. If A is an n × n mairix over a field with more than n elements and if det A = 1. then Ais a multiplicative commutator.     

On a commutator theorem of robert c. thompson

A very short proof of the following special case of R. C. Thompson's Theorem is presented. If A is an n × n mairix over a field with more than n elements and if det A = 1. then Ais a multiplicative commutator.     

On conciseness of some commutator words

Archiv der Mathematik - We prove that a commutator of two non-commutator words is a concise word.