5,141 research outputs found
Homological Domination in Large Random Simplicial Complexes
In this paper we state the homological domination principle for random
multi-parameter simplicial complexes, claiming that the Betti number in one
specific dimension (which is explicitly determined by the probability
multi-parameter) significantly dominates the Betti numbers in all other
dimensions. We also state and discuss evidence for two interesting conjectures
which would imply a stronger version of the homological domination principle,
namely that generically homology of a random simplicial complex coincides with
that of a wedges of k-dimensional spheres. These two conjectures imply that
under an additional assumption (specified in the paper) a random simplicial
complex collapses to a k-dimensional complex homotopy equivalent to a wedge of
spheres of dimension k.Comment: 8 pages, 1 figur
L^2 torsion without the determinant class condition and extended L^2 cohomology
We associate determinant lines to objects of the extended abelian category
built out of a von Neumann category with a trace. Using this we suggest
constructions of the combinatorial and the analytic L^2 torsions which, unlike
the work of the previous authors, requires no additional assumptions; in
particular we do not impose the determinant class condition. The resulting
torsions are elements of the determinant line of the extended L^2 cohomology.
Under the determinant class assumption the L^2 torsions of this paper
specialize to the invariants studied in our previous work. Applying a recent
theorem of D. Burghelea, L. Friedlander and T. Kappeler we obtain a Cheeger -
Muller type theorem stating the equality between the combinatorial and the
analytic L^2 torsions.Comment: 39 page
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
Large random simplicial complexes, II; the fundamental group
In our recent work we described conditions under which a multi-parameter
random simplicial complex is connected and simply connected. We showed that the
Betti numbers of multi-parameter random simplicial complexes in one specific
dimension dominate significantly the Betti numbers in all other dimensions. In
this paper we focus mainly on the properties of fundamental groups of
multi-parameter random simplicial complexes, which can be viewed as a new class
of random groups. We describe thresholds for nontrivially and hyperbolicity (in
the sense of Gromov) for these groups. Besides, we find domains in the
multi-parameter space where these groups have 2-torsion. We also prove that
these groups have never odd-prime torsion and their geometric and cohomological
dimensions are either 0,1, 2 or infinity. Another result presented in this
paper states that aspherical 2-dimensional subcomplexes of random complexes
satisfy the Whitehead Conjecture, i.e. all their subcomplexes are also
aspherical (with probability tending to one).Comment: arXiv admin note: text overlap with arXiv:1312.1208, arXiv:1307.361
Large random simplicial complexes, III the critical dimension
In this paper we study the notion of critical dimension of random simplicial
complexes in the general multi-parameter model described in our previous papers
of this series. This model includes as special cases the
Linial-Meshulam-Wallach model as well as the clique complexes of random graphs.
We characterise the concept of critical dimension in terms of various geometric
and topological properties of random simplicial complexes such as their Betti
numbers, the fundamental group, the size of minimal cycles and the degrees of
simplexes. We mention in the text a few interesting open questions.Comment: 27 pages, 2 figure
Evidence for a Bulk Complex Order-Parameter in Y0.9Ca0.1Ba2Cu3O7-delta Thin Films
We have measured the penetration depth of overdoped Y0.9Ca0.1Ba2Cu3O7-delta
(Ca-YBCO) thin films using two different methods. The change of the penetration
depth as a function of temperature has been measured using the parallel plate
resonator (PPR), while its absolute value was obtained from a quasi-optical
transmission measurements. Both sets of measurements are compatible with an
order parameter of the form: Delta*dx2-y2+i*delta*dxy, with Delta=14.5 +- 1.5
meV and delta=1.8 meV, indicating a finite gap at low temperature. Below 15 K
the drop of the scattering rate of uncondensed carriers becomes steeper in
contrast to a flattening observed for optimally doped YBCO films. This decrease
supports our results on the penetration depth temperature dependence. The
findings are in agreement with tunneling measurements on similar Ca-YBCO thin
films.Comment: 11 pages, 4 figure
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