Kleinian groups and the complex of curves

We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have `bounded geometry' (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed using the ending invariants of the group and the complex of curves. These results are directly analogous to those obtained in the case of punctured-torus surface groups. In that setting the ending invariants are points in the closed unit disk and the coefficients are closely related to classical continued-fraction coefficients. The estimates obtained play an essential role in the solution of Thurston's ending lamination conjecture in that case.Comment: 32 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper3.abs.htm

The classification of punctured-torus groups

Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free two-generator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus. As a consequence we verify the conjectural topological description of the deformation space of punctured-torus groups (including Bers' conjecture that the quasi-Fuchsian groups are dense in this space) and prove a rigidity theorem: two punctured-torus groups are quasi-conformally conjugate if and only if they are topologically conjugate.Comment: 67 pages, published versio

"Finance and Stability: The Limits of Capitalism"

Once again the United States economy is facing a crisis, resolution of which first requires the realization that there are many types of capitalism: Solutions implemented in the past, therefore, may or may not be an appropriate solution today, as they could have been implemented as an answer to a problem posed within the context of a different model. Alternatively, the solution may lie in the implementation of a totally new economic regime in answer to reoccurring problems inherent in capitalism in general. The implementation of a new model is not a unique happening in United States economic history. The interventionist model-set in motion by President Roosevelt in answer to the failure of the laissez-faire model in the 1930s-dealt with the obvious flaw inherent in capitalism in general namely, its inability to maintain a level of aggregate demand consistent with full employment. Implementation of the interventionist model prevented a massive depression of the type experienced in the 1930s from being repeated due to the larger role played by the government sector in maintaining demand via active fiscal policy, while moderating inflation through the use of monetary policy. The interventionist model also recognized the less obvious, deeper flaw of capitalism-namely, the manner in which the financial system can adversely affect the price of assets relative to that of current output. Absent any interventionist policy, the resulting decline in private investment and profits leads to a downward spiral and collapse of the financial sector. The institutional roadblocks included in the interventionist model were sufficient to avert large disequilibriums in asset and output prices, thereby sustaining profits and precluding a deep recession. (Indeed, the Federal Reserve was not forced to act to avert a financial crisis until 1968, when problems arose in the commercial paper market.) The interventionist model, however, was abrogated during the 1980s with the reinstitution of a new laissez-faire model. The new model eliminated many of the restrictions imposed on financial sector, massive increases in national deficits through unproductive public sector spending (made even more inefficient by the resulting interest on the debt), and the growth of speculative financing schemes that left us with too many highly indebted firms. A large, financially induced depression was contained only through the reintroduction of massive governing monetary and fiscal intervention in the form of the S&L bailout and the maintenance of profits with massive deficits. Although the subsequent drop in interest rates has resulted in a rise in asset values and somewhat abated the turmoil in the financial markets, the economy continues to stagnate.

"The Capitalist Development of the Economy and the Structure of Financial Institutions"

This paper evolves from the sharp contrast in Smithian and Keynesian views about the relationship between the financial structure and the economy. The Smithian perspective implies that the financial structure is irrelevant, whereas the Keynesian position concludes that effective financing is necessary for the "capital development of the economy"- there is also a need to constrain any tendency of what Keynes referred to as speculation to dominate. Thus, the essential elements of equilibrium in Keynesian theory, the financial theory of investment and the investment theory of business cycles, are most apt when examined as outcomes of processes that operate over time. During the 1980s, there was a sharp increase in speculative financing resulting from the trend toward leveraged buyouts and the rising demand for short-term marketable corporate liabilities. A main characteristic of a capitalist economy that is stagnant or immersed in a depression is that the capital development of the economy is not progressing. The 1980s were filled with examples of financing inept investments, while the current climate is one of grossly inadequate investment levels to create a progressive full-employment economy. The financial instability interpretation of Keynes rests upon the profitability of debt financing, and incorporates the potential collapse of asset values in an environment of speculative and Ponzi financing. Consequently, the financial structure is significantly more fragile today than earlier in the post World War II era.