26 research outputs found
Stallings graphs for quasi-convex subgroups
We show that one can define and effectively compute Stallings graphs for
quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or
right-angled Artin groups). These Stallings graphs are finite labeled graphs,
which are canonically associated with the corresponding subgroups. We show that
this notion of Stallings graphs allows a unified approach to many algorithmic
problems: some which had already been solved like the generalized membership
problem or the computation of a quasi-convexity constant (Kapovich, 1996); and
others such as the computation of intersections, the conjugacy or the almost
malnormality problems.
Our results extend earlier algorithmic results for the more restricted class
of virtually free groups. We also extend our construction to relatively
quasi-convex subgroups of relatively hyperbolic groups, under certain
additional conditions.Comment: 40 pages. New and improved versio
Normality of one-matching semi-Cayley graphs over finite abelian groups with maximum degree three
A graph is said to be a semi-Cayley graph over a group if it
admits as a semiregular automorphism group with two orbits of equal size.
We say that is normal if is a normal subgroup of . We prove that every connected intransitive one-matching
semi-Cayley graph, with maximum degree three, over a finite abelian group is
normal and characterize all such non-normal graphs.Comment: 10 page
Algebraic degrees of -Cayley digraphs over abelian groups
A digraph is called an -Cayley digraph if its automorphism group has an
-orbit semiregular subgroup. We determine the splitting fields of -Cayley
digraphs over abelian groups and compute a bound on their algebraic degrees,
before applying our results on Cayley digraphs over non-abelian groups
On prisms, M\"obius ladders and the cycle space of dense graphs
For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum
degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense
(i.e. 1-dimensional cycle group in the sense of simplicial homology theory with
Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of
all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main
purpose of this paper is to prove the following: for every s > 0 there exists
n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >=
(1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X)
>= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits
of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all
circuits of X having length either f_0(X)-1 or f_0(X) generates all of
Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X
is Hamilton-generated. All these degree-conditions are essentially
best-possible. The implications in (1) and (2) give an asymptotic affirmative
answer to a special case of an open conjecture which according to [European J.
Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure
-trees and laminations for free groups II: The dual lamination of an -tree
This is the second part of a series of three articles which introduce
laminations for free groups (see math.GR/0609416 for the first part). Several
definition of the dual lamination of a very small action of a free group on an
-tree are given and proved to be equivalent.Comment: corrections of typos and minor updat