Uniform growth rate

In an evolutionary system in which the rules of mutation are local in nature, the number of possible outcomes after $m$ mutations is an exponential function of $m$ but with a rate that depends only on the set of rules and not the size of the original object. We apply this principle to find a uniform upper bound for the growth rate of certain groups including the mapping class group. We also find a uniform upper bound for the growth rate of the number of homotopy classes of triangulations of an oriented surface that can be obtained from a given triangulation using $m$ diagonal flips.Comment: 13 pages, 5 figures, minor revisions, final version appears in Proc. Amer. Math. So

Growth Tight Actions of Product Groups

A group action on a metric space is called growth tight if the exponential growth rate of the group with respect to the induced pseudo-metric is strictly greater than that of its quotients. A prototypical example is the action of a free group on its Cayley graph with respect to a free generating set. More generally, with Arzhantseva we have shown that group actions with strongly contracting elements are growth tight. Examples of non-growth tight actions are product groups acting on the $L^1$ products of Cayley graphs of the factors. In this paper we consider actions of product groups on product spaces, where each factor group acts with a strongly contracting element on its respective factor space. We show that this action is growth tight with respect to the $L^p$ metric on the product space, for all $1<p\leq \infty$. In particular, the $L^\infty$ metric on a product of Cayley graphs corresponds to a word metric on the product group. This gives the first examples of groups that are growth tight with respect to an action on one of their Cayley graphs and non-growth tight with respect to an action on another, answering a question of Grigorchuk and de la Harpe.Comment: 13 pages v2 15 pages, minor changes, to appear in Groups, Geometry, and Dynamic

Veech surfaces and simple closed curves

We study the SL(2,R)-infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the "no small virtual triangles" theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spectrum of a half-translation surface either has a gap above zero or is dense in a neighborhood of zero. These results make use of the auxiliary polygon associated to a curve on a half-translation surface, as introduced by Tang and Webb.Comment: 12 pages. v2: added proof of continuity of infimal length functions on quadratic differential space; 16 pages, one figure; to appear in Israel J. Mat

Maximum Smoothed Likelihood Component Density Estimation in Mixture Models with Known Mixing Proportions

In this paper, we propose a maximum smoothed likelihood method to estimate the component density functions of mixture models, in which the mixing proportions are known and may differ among observations. The proposed estimates maximize a smoothed log likelihood function and inherit all the important properties of probability density functions. A majorization-minimization algorithm is suggested to compute the proposed estimates numerically. In theory, we show that starting from any initial value, this algorithm increases the smoothed likelihood function and further leads to estimates that maximize the smoothed likelihood function. This indicates the convergence of the algorithm. Furthermore, we theoretically establish the asymptotic convergence rate of our proposed estimators. An adaptive procedure is suggested to choose the bandwidths in our estimation procedure. Simulation studies show that the proposed method is more efficient than the existing method in terms of integrated squared errors. A real data example is further analyzed