9,214 research outputs found
On tree-decompositions of one-ended graphs
A graph is one-ended if it contains a ray (a one way infinite path) and
whenever we remove a finite number of vertices from the graph then what remains
has only one component which contains rays. A vertex {\em dominates} a ray
in the end if there are infinitely many paths connecting to the ray such
that any two of these paths have only the vertex in common. We prove that
if a one-ended graph contains no ray which is dominated by a vertex and no
infinite family of pairwise disjoint rays, then it has a tree-decomposition
such that the decomposition tree is one-ended and the tree-decomposition is
invariant under the group of automorphisms.
This can be applied to prove a conjecture of Halin from 2000 that the
automorphism group of such a graph cannot be countably infinite and solves a
recent problem of Boutin and Imrich. Furthermore, it implies that every
transitive one-ended graph contains an infinite family of pairwise disjoint
rays
Triangulated surfaces in triangulated categories
For a triangulated category A with a 2-periodic dg-enhancement and a
triangulated oriented marked surface S we introduce a dg-category F(S,A)
parametrizing systems of exact triangles in A labelled by triangles of S. Our
main result is that F(S,A) is independent on the choice of a triangulation of S
up to essentially unique Morita equivalence. In particular, it admits a
canonical action of the mapping class group. The proof is based on general
properties of cyclic 2-Segal spaces.
In the simplest case, where A is the category of 2-periodic complexes of
vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya
category of the surface S. Therefore, our construction can be seen as
implementing a 2-dimensional instance of Kontsevich's program on localizing the
Fukaya category along a singular Lagrangian spine.Comment: 55 pages, v2: references added and typos corrected, v3: expanded
version, comments welcom
Self-avoiding walks and amenability
The connective constant of an infinite transitive graph is the
exponential growth rate of the number of self-avoiding walks from a given
origin. The relationship between connective constants and amenability is
explored in the current work.
Various properties of connective constants depend on the existence of
so-called 'graph height functions', namely: (i) whether is a local
function on certain graphs derived from , (ii) the equality of and
the asymptotic growth rate of bridges, and (iii) whether there exists a
terminating algorithm for approximating to a given degree of accuracy.
In the context of amenable groups, it is proved that the Cayley graphs of
infinite, finitely generated, elementary amenable groups support graph height
functions, which are in addition harmonic. In contrast, the Cayley graph of the
Grigorchuk group, which is amenable but not elementary amenable, does not have
a graph height function.
In the context of non-amenable, transitive graphs, a lower bound is presented
for the connective constant in terms of the spectral bottom of the graph. This
is a strengthening of an earlier result of the same authors. Secondly, using a
percolation inequality of Benjamini, Nachmias, and Peres, it is explained that
the connective constant of a non-amenable, transitive graph with large girth is
close to that of a regular tree. Examples are given of non-amenable groups
without graph height functions, of which one is the Higman group.Comment: v2 differs from v1 in the inclusion of further material concerning
non-amenable graphs, notably an improved lower bound for the connective
constan
- …