17,491 research outputs found
The asymptotic dimension of a curve graph is finite
We find an upper bound for the asymptotic dimension of a hyperbolic metric
space with a set of geodesics satisfying a certain boundedness condition
studied by Bowditch. The primary example is a collection of tight geodesics on
the curve graph of a compact orientable surface. We use this to conclude that a
curve graph has finite asymptotic dimension. It follows then that a curve graph
has property . We also compute the asymptotic dimension of mapping class
groups of orientable surfaces with genus .Comment: 19 pages. Made some minor revisions. The section on mapping class
groups has been rewritten; in particular we compute the asdim of Mod(S) where
S has genus at most 2. The last section on open questions has been modified
to reflect recent developments. References have been update
Trees and the dynamics of polynomials
The basin of infinity of a polynomial map f : {\bf C} \arrow {\bf C}
carries a natural foliation and a flat metric with singularities, making it
into a metrized Riemann surface . As diverges in the moduli space of
polynomials, the surface collapses along its foliation to yield a
metrized simplicial tree , with limiting dynamics F : T \arrow T.
In this paper we characterize the trees that arise as limits, and show they
provide a natural boundary \PT_d compactifying the moduli space of
polynomials of degree . We show that records the limiting
behavior of multipliers at periodic points, and that any divergent meromorphic
family of polynomials \{f_t(z) : t \mem \Delta^* \} can be completed by a
unique tree at its central fiber. Finally we show that in the cubic case, the
boundary of moduli space \PT_3 is itself a tree.
The metrized trees provide a counterpart, in the setting of
iterated rational maps, to the -trees that arise as limits of
hyperbolic manifolds.Comment: 60 page
Helicoidal minimal surfaces of prescribed genus, I
For every genus g, we prove that S^2 x R contains complete, properly
embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids
of any prescribed pitch. We also show that as the radius of the S^2 tends to
infinity, these examples converge smoothly to complete, properly embedded
minimal surfaces in Euclidean 3-space R^3 that are helicoidal at infinity. In a
companion paper, we prove that helicoidal surfaces in R^3 of every prescribed
genus occur as such limits of examples in S^2 x R.Comment: 53 pages, 5 figure
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