On the geometry of a proposed curve complex analogue for Out(Fn)Out(F_n)

The group \Out of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which \Out acts, in analogy with the curve complex for the mapping class group. Here, we focus on one of these proposed analogues: the edge splitting complex \ESC, equivalently known as the separating sphere complex. We characterize geodesic paths in its 1-skeleton algebraically, and use our characterization to find lower bounds on distances between points in this graph. Our distance calculations allow us to find quasiflats of arbitrary dimension in \ESC. This shows that \ESC: is not hyperbolic, has infinite asymptotic dimension, and is such that every asymptotic cone is infinite dimensional. These quasiflats contain an unbounded orbit of a reducible element of \Out. As a consequence, there is no coarsely \Out-equivariant quasiisometry between \ESC and other proposed curve complex analogues, including the regular free splitting complex \FSC, the (nontrivial intersection) free factorization complex \FFZC, and the free factor complex \FFC, leaving hope that some of these complexes are hyperbolic.Comment: 23 pages, 6 figure

Mapping Class Groups and Moduli Spaces of Curves

This is a survey paper that also contains some new results. It will appear in the proceedings of the AMS summer research institute on Algebraic Geometry at Santa Cruz.Comment: We expanded section 7 and rewrote parts of section 10. We also did some editing and made some minor corrections. latex2e, 46 page

On a theorem of Kontsevich

In two seminal papers M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads `commutative,' `associative' and `Lie.' We generalize his theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich's original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(F_r) outlined by Kontsevich. In [`Infinitesimal operations on chain complexes of graphs', Mathematische Annalen, 327 (2003) 545-573] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [James Conant, `Fusion and fission in graph complexes', Pac. J. 209 (2003), 219-230] to the case of all cyclic operads. These operations form a Lie bi-algebra on a natural subcomplex. We show that in the associative and Lie cases the subcomplex on which the bi-algebra structure exists carries all of the homology, and we explain why the subcomplex in the commutative case does not.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-42.abs.htm

From rubber bands to rational maps: A research report

This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. One one hand, this lets us tell when one rubber band network is looser than another, and on the other hand tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular "self-embedding" property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example

Landscape History and Theory: from Subject Matter to Analytic Tool

This essay explores how landscape history can engage methodologically with the adjacent disciplines of art history and visual/cultural studies. Central to the methodological problem is the mapping of the beholder � spatially, temporally and phenomenologically. In this mapping process, landscape history is transformed from subject matter to analytical tool. As a result, landscape history no longer simply imports and applies ideas from other disciplines but develops its own methodologies to engage and influence them. Landscape history, like art history, thereby takes on a creative cultural presence. Through that process, landscape architecture and garden design regain the cultural power now carried by the arts and museum studies, and has an effect on the innovative capabilities of contemporary landscape design