1,079 research outputs found
JSJ decompositions of Quadratic Baumslag-Solitar groups
Generalized Baumslag-Solitar groups are defined as fundamental groups of
graphs of groups with infinite cyclic vertex and edge groups. Forester proved
(in "On uniqueness of JSJ decompositions of finitely generated groups",
Comment. Math. Helv. 78 (2003) pp 740-751) that in most cases the defining
graphs are cyclic JSJ decompositions, in the sense of Rips and Sela. Here we
extend Forester's results to graphs of groups with vertex groups that can be
either infinite cyclic or quadratically hanging surface groups.Comment: 20 pages, 2 figures. Several corrections and improvements from
referee's report. Imprtant changes in Definition 5.1, and the proof of
Theorem 5.5 (previously 5.4). Lemma 5.4 was adde
Whitehead moves for G-trees
We generalize the familiar notion of a Whitehead move from Culler and
Vogtmann's Outer space to the setting of deformation spaces of G-trees.
Specifically, we show that there are two moves, each of which transforms a
reduced G-tree into another reduced G-tree, that suffice to relate any two
reduced trees in the same deformation space. These two moves further factor
into three moves between reduced trees that have simple descriptions in terms
of graphs of groups. This result has several applications.Comment: v1: 9 pages; v2: 10 pages, minor revisions and one added referenc
Deformation and rigidity of simplicial group actions on trees
We study a notion of deformation for simplicial trees with group actions
(G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a
deformation if there is a finite sequence of collapse and expansion moves
joining them. We show that this relation on the set of G-trees has several
characterizations, in terms of dynamics, coarse geometry, and length functions.
Next we study the deformation space of a fixed G-tree X. We show that if X is
`strongly slide-free' then it is the unique reduced tree in its deformation
space.
These methods allow us to extend the rigidity theorem of Bass and Lubotzky to
trees that are not locally finite. This yields a unique factorization theorem
for certain graphs of groups. We apply the theory to generalized
Baumslag-Solitar groups and show that many have canonical decompositions. We
also prove a quasi-isometric rigidity theorem for strongly slide-free G-trees.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper8.abs.htm
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
Simple crystallizations of 4-manifolds
Minimal crystallizations of simply connected PL 4-manifolds are very natural
objects. Many of their topological features are reflected in their
combinatorial structure which, in addition, is preserved under the connected
sum operation. We present a minimal crystallization of the standard PL K3
surface. In combination with known results this yields minimal crystallizations
of all simply connected PL 4-manifolds of "standard" type, that is, all
connected sums of , , and the K3 surface. In
particular, we obtain minimal crystallizations of a pair of homeomorphic but
non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that
the minimal 8-vertex crystallization of is unique and its
associated pseudotriangulation is related to the 9-vertex combinatorial
triangulation of by the minimum of four edge contractions.Comment: 23 pages, 7 figures. Minor update, replacement of Figure 7. To appear
in Advances in Geometr
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