108 research outputs found
Simplicial decompositions of graphs: a survey of applications
AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects
Fixed-point free circle actions on 4-manifolds
This paper is concerned with fixed-point free -actions (smooth or
locally linear) on orientable 4-manifolds. We show that the fundamental group
plays a predominant role in the equivariant classification of such 4-manifolds.
In particular, it is shown that for any finitely presented group with infinite
center, there are at most finitely many distinct smooth (resp. topological)
4-manifolds which support a fixed-point free smooth (resp. locally linear)
-action and realize the given group as the fundamental group. A similar
statement holds for the number of equivalence classes of fixed-point free
-actions under some further conditions on the fundamental group. The
connection between the classification of the -manifolds and the
fundamental group is given by a certain decomposition, called fiber-sum
decomposition, of the -manifolds. More concretely, each fiber-sum
decomposition naturally gives rise to a Z-splitting of the fundamental group.
There are two technical results in this paper which play a central role in our
considerations. One states that the Z-splitting is a canonical JSJ
decomposition of the fundamental group in the sense of Rips and Sela. Another
asserts that if the fundamental group has infinite center, then the homotopy
class of principal orbits of any fixed-point free -action on the
4-manifold must be infinite, unless the 4-manifold is the mapping torus of a
periodic diffeomorphism of some elliptic 3-manifold. The paper ends with two
questions concerning the topological nature of the smooth classification and
the Seiberg-Witten invariants of 4-manifolds admitting a smooth fixed-point
free -action.Comment: 42 pages, no figures, Algebraic and Geometric Topolog
Crossed simplicial groups and structured surfaces
We propose a generalization of the concept of a Ribbon graph suitable to
provide combinatorial models for marked surfaces equipped with a G-structure.
Our main insight is that the necessary combinatorics is neatly captured in the
concept of a crossed simplicial group as introduced, independently, by
Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category
leads to Ribbon graphs while other crossed simplicial groups naturally yield
different notions of structured graphs which model unoriented, N-spin, framed,
etc, surfaces. Our main result is that structured graphs provide orbicell
decompositions of the respective G-structured moduli spaces. As an application,
we show how, building on our theory of 2-Segal spaces, the resulting theory can
be used to construct categorified state sum invariants of G-structured
surfaces.Comment: 86 pages, v2: revised versio
Quasi-isometric Rigidity of Piecewise Geometric Manifolds
Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are always quasi-isometric, and a group G is quasi-isometrically rigid if every group quasi-isometric to G is virtually isomorphic to G.
In this survey we describe quasi-isometric rigidity results for fundamental groups of manifolds which can be decomposed into geometric pieces. After stating by now classical results on lattices in semisimple Lie groups, we focus on the class of fundamental groups of 3-manifolds, and describe the behaviour of quasi-isometries with respect to the Milnor-Kneser prime decomposition (following Papasoglu and Whyte) and with respect to the JSJ decomposition (following Kapovich and Leeb). We also discuss quasi-isometric rigidity results for fundamental groups of higher dimensional graph manifolds, that were recently defined by Lafont, Sisto and the author. Our main tools are the study of geometric group actions and quasi-actions on Riemannian manifolds and on trees of spaces, via the analysis of the induced actions on asymptotic cones
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