1,205 research outputs found
Long path and cycle decompositions of even hypercubes
We consider edge decompositions of the -dimensional hypercube into
isomorphic copies of a given graph . While a number of results are known
about decomposing into graphs from various classes, the simplest cases of
paths and cycles of a given length are far from being understood. A conjecture
of Erde asserts that if is even, and divides the number
of edges of , then the path of length decomposes . Tapadia et
al.\ proved that any path of length , where , satisfying these
conditions decomposes . Here, we make progress toward resolving Erde's
conjecture by showing that cycles of certain lengths up to
decompose . As a consequence, we show that can be decomposed into
copies of any path of length at most dividing the number of edges of
, thereby settling Erde's conjecture up to a linear factor
On the automorphisms group of the asymptotic pants complex of an infinite surface of genus zero
The braided Thompson group is an asymptotic mapping class group
of a sphere punctured along the standard Cantor set, endowed with a rigid
structure. Inspired from the case of finite type surfaces we consider a
Hatcher-Thurston cell complex whose vertices are asymptotically trivial pants
decompositions. We prove that the automorphism group of this complex is also an asymptotic mapping class group in
a weaker sense. Moreover is obtained by
by first adding new elements called half-twists and further
completing it.Comment: revised version,17p., 13 figure
Deciding Isomorphy using Dehn fillings, the splitting case
We solve Dehn's isomorphism problem for virtually torsion-free relatively
hyperbolic groups with nilpotent parabolic subgroups.
We do so by reducing the isomorphism problem to three algorithmic problems in
the parabolic subgroups, namely the isomorphism problem, separation of torsion
(in their outer automorphism groups) by congruences, and the mixed Whitehead
problem, an automorphism group orbit problem. The first step of the reduction
is to compute canonical JSJ decompositions. Dehn fillings and the given
solutions of the algorithmic problems in the parabolic groups are then used to
decide if the graphs of groups have isomorphic vertex groups and, if so,
whether a global isomorphism can be assembled.
For the class of finitely generated nilpotent groups, we give solutions to
these algorithmic problems by using the arithmetic nature of these groups and
of their automorphism groups.Comment: 76 pages. This version incorporates referee comments and corrections.
The main changes to the previous version are a better treatment of the
algorithmic recognition and presentation of virtually cyclic subgroups and a
new proof of a rigidity criterion obtained by passing to a torsion-free
finite index subgroup. The previous proof relied on an incorrect result. To
appear in Inventiones Mathematica
Complexity volumes of splittable groups
Using graph of groups decompositions of finitely generated groups, we define
Euler characteristic type invariants which are non-zero in many interesting
classes of finitely presented, hyperbolic, limit and CSA groups, including
elementarily free groups and one-ended torsion-free hyperbolic groups whose JSJ
decomposition contains a maximal hanging Fuchsian vertex group.Comment: Major revisions following the suggestions of the referee. To appear
in Journal of Algebr
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