A finitely presented infinite simple group of homeomorphisms of the circle

We construct a finitely presented, infinite, simple group that acts by homeomorphisms on the circle, but does not admit a non-trivial action by $C^1$-diffeomorphisms on the circle. The group emerges as a group of piecewise projective homeomorphisms of $S^1=\mathbf{R}\cup \{\infty\}$. However, we show that it does not admit a non-trivial action by piecewise linear homeomorphisms of the circle. Another interesting and new feature of this example is that it produces a non amenable orbit equivalence relation with respect to the Lebesgue measure.Comment: 30 pages (Some typos have been corrected and some proofs have been streamlined.

A finitely presented group of piecewise projective homeomorphisms

In this article we will describe a finitely presented subgroup of Monod's group of piecewise projective homeomorphisms of R. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian free subgroup. It is in fact the first such example which is torsion free. We will also develop a means for representing the elements of the group by labeled tree diagrams in a manner which closely parallels Richard Thompson's group F.Comment: Formerly "A geometric solution to the von Neumann-Day problem for finitely presented groups". Section added on tree diagrams. Minor revisions elsewher

Commutators in groups of piecewise projective homeomorphisms

In 2012 Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and later Lodha and Moore introduced examples of finitely presented groups with the same property. In this article we examine the normal subgroup structure of these groups. Two important cases of our results are the groups $H$ and $G_0$. We show that the group $H$ of piecewise projective homeomorphisms of $\mathbb{R}$ has the property that $H"$ is simple and that every proper quotient of $H$ is metabelian. We establish simplicity of the commutator subgroup of the group $G_0$, which admits a presentation with $3$ generators and $9$ relations. Further we show that every proper quotient of $G_0$ is abelian. It follows that the normal subgroups of these groups are in bijective correspondence with those of the abelian (or metabelian) quotient

Components of Money Multiplier and their Relative Contributions

The Money stock is the product of High Powered Money and Money Multiplier. However, money multiplier should not be regarded as a purely mechanical apparatus as is evident by the identity M = mH and argued by Majumdar (1976), Shetty et al (1976) and by the Second Working Group of the RBI (RBI Bulletin, 1977). Instead, it grows out of the interactions of banks, non-bank-public and decisions of monetary authorities. Essentially, it summarises the influences of all those factors other than changes in the high-powered money on the money stock process. Specifically, it reflects portfolio decisions of the non-bank-public when it decides on its currency and time deposit ratios; the behaviour of banks regarding the distribution of assets between excess reserves and earning assets and the behaviour of central bank when it sets reserve requirements on time and demand deposits and imposes additional reserves under the statutory provisions. Keywords : Money Multiplier, High Powered Money, Required Reserves, Excess Reserves, M2 Definition of Money and Liabilitie

A Review of Empirical Studies on Money Supply at Abroad and in India

Amongst the many issues addressed in the economic literature, the money demand has perhaps attracted the most attention. The field of money supply has remained much ignored because of the underlying assumptions that money supply is exogenously determined by the central bank. In fact “In the world where banks use all their reserves, where there is no free reserves, and where both the banks and the public do not undertake any portfolio changes, there is no need to concern ourselves with the money supply. Once we get away from the simple mechanical link between reserves, deposits and money, the supply of money has an independent existence as an economic variable determined by behavior and subject to analysis.” (Fand : 1967). For years there has been continuing debate between two prominent schools of economic thought; Monetarists and Post Keynesians. The debate, which started since the publications of Keynes’ General Theory in mid 1930s, became much heated in the late 1960s’ and in 1970s’. The Monetarist view ‘began to be recognized as a serious challenge to Post Keynesian economics. Monetarists contended that changes in money exert a strong force on aggregate demand, price level and output. The key proposition was that changes in money supply dominate short run influences on price level and on nominal aggregate demand. Bringing in continuity in the debate puts forth an argument about the role of money, which has been based upon the lack of synchronization between transactions receipts and expenditures. In such a case, it is desirable for market participants to hold an inventory of money balances. This argument can be used to develop a model, which delegates a powerful role of money in influencing the nations’ money supply and has an important influence on economic activity. Managing the nation's money supply is essential so as to assist the economy in achieving a high level of employment, output, relatively stable price level and a viable balance of payment. Now our aim is to review the literature on money supply studies at length at abroad and in India. The First part of this article deals money supply studies at aboard and second part deals with the studies conducted in India

Approximating nonabelian free groups by groups of homeomorphisms of the real line

We show that for a large class $\mathcal{C}$ of finitely generated groups of orientation preserving homeomorphisms of the real line, the following holds: Given a group $G$ of rank $k$ in $\mathcal{C}$, there is a sequence of $k$-markings $(G, S_n), n\in \mathbf{N}$ whose limit in the space of marked groups is the free group of rank $k$ with the standard marking. The class we consider consists of groups that admit actions satisfying mild dynamical conditions and a certain "self-similarity" type hypothesis. Examples include Thompson's group $F$, Higman-Thompson groups, Stein-Thompson groups, various Bieri-Strebel groups, the golden ratio Thompson group, and finitely presented non amenable groups of piecewise projective homeomorphisms. For the case of Thompson's group $F$ we provide a new and considerably simpler proof of this fact proved by Brin (Groups, Geometry, and Dynamics 2010).Comment: 8 pages. Referee comments incorporated: to appear in the Journal of Algebr