634 research outputs found
Ending Laminations and Cannon-Thurston Maps
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian
surface groups. In this paper we prove that pre-images of points are precisely
end-points of leaves of the ending lamination whenever the Cannon-Thurston map
is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one.
This completes the proof of the conjectural picture of Cannon-Thurston maps for
surface groups.Comment: v4: Final version 22pgs 2figures. Includes the main theorem of the
appendix arXiv:1002.2090 by Shubhabrata Das and Mahan Mj. To appear in
Geometric and Functional Analysi
Topological phases of fermions in one dimension
In this paper we show how the classification of topological phases in
insulators and superconductors is changed by interactions, in the case of 1D
systems. We focus on the TR-invariant Majorana chain (BDI symmetry class).
While the band classification yields an integer topological index , it is
known that phases characterized by values of in the same equivalence class
modulo 8 can be adiabatically transformed one to another by adding suitable
interaction terms. Here we show that the eight equivalence classes are distinct
and exhaustive, and provide a physical interpretation for the interacting
invariant modulo 8. The different phases realize different Altland-Zirnbauer
classes of the reduced density matrix for an entanglement bipartition into two
half-chains. We generalize these results to the classification of all one
dimensional gapped phases of fermionic systems with possible anti-unitary
symmetries, utilizing the algebraic framework of central extensions. We use
matrix product state methods to prove our results.Comment: 14 pages, 3 figures, v2: references adde
Fixed subgroups are compressed in surface groups
For a compact surface (orientable or not, and with boundary or not)
we show that the fixed subgroup, , of any family of
endomorphisms of is compressed in i.e.,
for
any subgroup . On the way, we
give a partial positive solution to the inertia conjecture, both for free and
for surface groups. We also investigate direct products, , of finitely many
free and surface groups, and give a characterization of when satisfies that
for
every
On abstract commensurators of groups
We prove that the abstract commensurator of a nonabelian free group, an
infinite surface group, or more generally of a group that splits appropriately
over a cyclic subgroup, is not finitely generated.
This applies in particular to all torsion-free word-hyperbolic groups with
infinite outer automorphism group and abelianization of rank at least 2.
We also construct a finitely generated, torsion-free group which can be
mapped onto Z and which has a finitely generated commensurator.Comment: 13 pages, no figur
Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs
In this paper, we study a class of graph drawings that arise from bobbin lace
patterns. The drawings are periodic and require a combinatorial embedding with
specific properties which we outline and demonstrate can be verified in linear
time. In addition, a lace graph drawing has a topological requirement: it
contains a set of non-contractible directed cycles which must be homotopic to
, that is, when drawn on a torus, each cycle wraps once around the minor
meridian axis and zero times around the major longitude axis. We provide an
algorithm for finding the two fundamental cycles of a canonical rectangular
schema in a supergraph that enforces this topological constraint. The polygonal
schema is then used to produce a straight-line drawing of the lace graph inside
a rectangular frame. We argue that such a polygonal schema always exists for
combinatorial embeddings satisfying the conditions of bobbin lace patterns, and
that we can therefore create a pattern, given a graph with a fixed
combinatorial embedding of genus one.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Topological Designs
We give an exponential upper and a quadratic lower bound on the number of
pairwise non-isotopic simple closed curves can be placed on a closed surface of
genus g such that any two of the curves intersects at most once. Although the
gap is large, both bounds are the best known for large genus. In genus one and
two, we solve the problem exactly.
Our methods generalize to variants in which the allowed number of pairwise
intersections is odd, even, or bounded, and to surfaces with boundary
components.Comment: 14 p., 4 Figures. To appear in Geometriae Dedicat
Accidental parabolics and relatively hyperbolic groups
By constructing, in the relative case, objects analoguous to Rips and Sela's
canonical representatives, we prove that the set of images by morphisms without
accidental parabolic, of a finitely presented group in a relatively hyperbolic
group, is finite, up to conjugacy.Comment: Revision, 24 pages, 4 figure
Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps
We prove the existence of continuous boundary extensions (Cannon-Thurston
maps) for the inclusion of a vertex space into a tree of (strongly) relatively
hyperbolic spaces satisfying the qi-embedded condition. This implies the same
result for inclusion of vertex (or edge) subgroups in finite graphs of
(strongly) relatively hyperbolic groups. This generalises a result of Bowditch
for punctured surfaces in 3 manifolds and a result of Mitra for trees of
hyperbolic metric spaces.Comment: 27pgs No figs, v3: final version, incorporating referee's comments,
to appear in Geometriae Dedicat
A Combination Theorem for Metric Bundles
We define metric bundles/metric graph bundles which provide a purely
topological/coarse-geometric generalization of the notion of trees of metric
spaces a la Bestvina-Feighn in the special case that the inclusions of the edge
spaces into the vertex spaces are uniform coarsely surjective quasi-isometries.
We prove the existence of quasi-isometric sections in this generality. Then we
prove a combination theorem for metric (graph) bundles (including exact
sequences of groups) that establishes sufficient conditions, particularly
flaring, under which the metric bundles are hyperbolic. We use this to give
examples of surface bundles over hyperbolic disks, whose universal cover is
Gromov-hyperbolic. We also show that in typical situations, flaring is also a
necessary condition.Comment: v3: Major revision: 56 pages 5 figures. Many details added.
Characterization of convex cocompact subgroups of mapping class groups of
surfaces with punctures in terms of relative hyperbolicity given v4: Final
version incorporating referee comments: 63 pages 5 figures. To appear in
Geom. Funct. Ana
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