26 research outputs found
On the geometry of a proposed curve complex analogue for
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
On rigidity and the isomorphism problem for tree braid groups
We solve the isomorphism problem for braid groups on trees with or 5
strands. We do so in three main steps, each of which is interesting in its own
right. First, we establish some tools and terminology for dealing with
computations using the cohomology of tree braid groups, couching our discussion
in the language of differential forms. Second, we show that, given a tree braid
group on or 5 strands, is an exterior face algebra.
Finally, we prove that one may reconstruct the tree from a tree braid group
for or 5. Among other corollaries, this third step shows that,
when or 5, tree braid groups and trees (up to homeomorphism)
are in bijective correspondence. That such a bijection exists is not true for
higher dimensional spaces, and is an artifact of the 1-dimensionality of trees.
We end by stating the results for right-angled Artin groups corresponding to
the main theorems, some of which do not yet appear in the literature.Comment: 48 pages, 11 figures
Discrete Morse theory and graph braid groups
If Gamma is any finite graph, then the unlabelled configuration space of n
points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of
Gamma. The braid group of Gamma on n strands is the fundamental group of
UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma),
for any n and any Gamma, and provide a clear description of the critical cells
in every case. As a result, we can calculate a presentation for the braid group
of any tree, for any number of strands. We also give a simple proof of a
theorem due to Ghrist: the space UC^n(Gamma) strong deformation retracts onto a
CW complex of dimension at most k, where k is the number of vertices in Gamma
of degree at least 3 (and k is thus independent of n).Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-44.abs.htm
Face vectors of subdivided simplicial complexes
Brenti and Welker have shown that for any simplicial complex X, the face
vectors of successive barycentric subdivisions of X have roots which converge
to fixed values depending only on the dimension of X. We improve and generalize
this result here. We begin with an alternative proof based on geometric
intuition. We then prove an interesting symmetry of these roots about the real
number -2. This symmetry can be seen via a nice algebraic realization of
barycentric subdivision as a simple map on formal power series in two
variables. Finally, we use this algebraic machinery with some geometric
motivation to generalize the combinatorial statements to arbitrary subdivision
methods: any subdivision method will exhibit similar limit behavior and
symmetry. Our techniques allow us to compute explicit formulas for the values
of the limit roots in the case of barycentric subdivision.Comment: 13 pages, final version, appears in Discrete Mathematics 201
On the cohomology rings of tree braid groups
Let be a finite connected graph. The (unlabelled) configuration
space of points on is the space of -element
subsets of . The -strand braid group of , denoted
, is the fundamental group of .
We use the methods and results of our paper "Discrete Morse theory and graph
braid groups" to get a partial description of the cohomology rings , where is a tree. Our results are then used to prove that is a
right-angled Artin group if and only if is linear or . This gives a
large number of counterexamples to Ghrist's conjecture that braid groups of
planar graphs are right-angled Artin groups.Comment: 25 pages, 7 figures. Revised version, accepted by the Journal of Pure
and Applied Algebr
Embedding right-angled Artin groups into graph braid groups
We construct an embedding of any right-angled Artin group defined
by a graph into a graph braid group. The number of strands required
for the braid group is equal to the chromatic number of . This
construction yields an example of a hyperbolic surface subgroup embedded in a
two strand planar graph braid group.Comment: 8 pages. Final version, appears in Geometriae Dedicata