The Zieschang-McCool method for generating algebraic mapping-class groups

Let g and p be non-negative integers. Let A(g,p) denote the group consisting of all those automorphisms of the free group on {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} which fix the element t_1t_2...t_p[x_1,y_1]...[x_g,y_g] and permute the set of conjugacy classes {[t_1],....,[t_p]}. Labru\`ere and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A(g,p) is generated by a set that is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. Labru\`ere and Paris also gave defining relations for the ADLH set in A(g,p); we do not know an algebraic proof of this for g > 1. Consider an orientable surface S(g,p) of genus g with p punctures, such that (g,p) is not (0,0) or (0,1). The algebraic mapping-class group of S(g,p), denoted M(g,p), is defined as the group of all those outer automorphisms of the one-relator group with generating set {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} and relator t_1t_2...t_p[x_1,y_1]...[x_g,y_g] which permute the set of conjugacy classes {[t_1],....,[t_p]}. It now follows from a result of Nielsen that M(g,p) is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that M(g,p) equals the (topological) mapping-class group of S(g,p), along lines suggested by Magnus, Karrass, and Solitar in 1966.Comment: 21 pages, 0 figure

Export and Import Price Indices

Export and import price indices are essential for assessing the impact of international trade on the domestic economy. Among their most important uses are analyzing developments in the trade balance, measuring foreign prices' contribution to domestic inflation, and deflating nominal values of exports and imports for estimating the volume of gross domestic product. This paper discusses economic concepts for trade price indices at some length. We note the need for reasonably frequent chaining in view of the fluctuation in the conditioning variables of trade price indices. We characterize the effect of the residency orientation of the index on the substitution biases of the commonly used Laspeyres and Paasche formulas, and superlative formulas, which greatly attenuate these biases. Finally, we consider the data sources and methods used to compile them. Copyright 2004, International Monetary Fund

JSJ decompositions of Quadratic Baumslag-Solitar groups

Generalized Baumslag-Solitar groups are defined as fundamental groups of graphs of groups with infinite cyclic vertex and edge groups. Forester proved (in "On uniqueness of JSJ decompositions of finitely generated groups", Comment. Math. Helv. 78 (2003) pp 740-751) that in most cases the defining graphs are cyclic JSJ decompositions, in the sense of Rips and Sela. Here we extend Forester's results to graphs of groups with vertex groups that can be either infinite cyclic or quadratically hanging surface groups.Comment: 20 pages, 2 figures. Several corrections and improvements from referee's report. Imprtant changes in Definition 5.1, and the proof of Theorem 5.5 (previously 5.4). Lemma 5.4 was adde

Some quadratic equations in the free group of rank 2

For a given quadratic equation with any number of unknowns in any free group F, with right-hand side an arbitrary element of F, an algorithm for solving the problem of the existence of a solution was given by Culler. The problem has been studied by the authors for parametric families of quadratic equations arising from continuous maps between closed surfaces, with certain conjugation factors as the parameters running through the group F. In particular, for a one-parameter family of quadratic equations in the free group F_2 of rank 2, corresponding to maps of absolute degree 2 between closed surfaces of Euler characteristic 0, the problem of the existence of faithful solutions has been solved in terms of the value of the self-intersection index mu: F_2 --> Z[F_2] on the conjugation parameter. The present paper investigates the existence of faithful, or non-faithful, solutions of similar families of quadratic equations corresponding to maps of absolute degree 0.Comment: This is the version published by Geometry & Topology Monographs on 29 April 200

The Measurement of Banking Services in the System of National Accounts

The paper considers some of the problems associated with the indirectly measured components of financial service outputs in the System of National Accounts (SNA), termed FISIM (Financial Intermediation Services Indirectly Measured). The paper characterizes FISIM by a user cost and supplier benefit approach determining the price and quantity of various financial services in the banking sector. We examine the need for FISIM in the context of plausible alternative accounting schemes that could be used to account for financial services. The alternative accounting frameworks have implications for the labour and multifactor productivity of both the financial and nonfinancial sectors.User costs, banking services, deposit services, loan services, Total Factor Productivity growth, production accounts, System of National Accounts, FIS

On Dihedral Configurations and their Coxeter Geometries

AbstractWithin the theory of homogeneous coherent configurations, the dihedral configurations play the role which is played by the finite dihedral groups in the theory of finite groups. Imitating Tits’ construction of a geometry from a set of subgroups of a given group, we assign a geometry of rank 2 to each dihedral configuration, its ‘Coxeter geometry’. (Each finite generalized polygon is a Coxeter geometry in this sense.)Apart from general results on the relationship between dihedral configurations and their Coxeter geometries, we settle completely the (ordinary) representation theory of the dihedral configurations of rank 7. We obtain three major classes. The Coxeter geometries of the first class are exactly the non-symmetric 2-designs withI=1. The other two classes lead to questions which require a further combinatorial treatment

Geometric Intersection Number and analogues of the Curve Complex for free groups

For the free group $F_{N}$ of finite rank $N \geq 2$ we construct a canonical Bonahon-type continuous and $Out(F_N)$-invariant \emph{geometric intersection form} $$: \bar{cv}(F_N)\times Curr(F_N)\to \mathbb R_{\ge 0}.$$ Here $\bar{cv}(F_N)$ is the closure of unprojectivized Culler-Vogtmann's Outer space $cv(F_N)$ in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that $\bar{cv}(F_N)$ consists of all \emph{very small} minimal isometric actions of $F_N$ on $\mathbb R$-trees. The projectivization of $\bar{cv}(F_N)$ provides a free group analogue of Thurston's compactification of the Teichm\"uller space. As an application, using the \emph{intersection graph} determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.Comment: Revised version, to appear in Geometry & Topolog

Manifolds with small Heegaard Floer ranks

We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floer homology of the branched double cover, our results show that Khovanov homology detects the unknot if and only if it detects the two component unlink.Comment: 19 pages, 1 figur