53,321 research outputs found
Admitting a coarse embedding is not preserved under group extensions
We construct a finitely generated group which is an extension of two finitely
generated groups coarsely embeddable into Hilbert space but which itself does
not coarsely embed into Hilbert space. Our construction also provides a new
infinite monster group: the first example of a finitely generated group that
does not coarsely embed into Hilbert space and yet does not contain a weakly
embedded expander.Comment: 15 pages; Proposition 3.3(v1) was modified following a comment of D.
Sawicki; Theorem 2(v3) is new and gives an extension of finitely generated
group
Cohomology of Local Cochains
We prove that for generalised partitions of unity and
coverings of a
topological space the cohomology of abstract -local cochains
coincides with the cohomology of continuous -local cochains,
provided the coefficients are loop contractible. Furthermore we show that for
each locally contractible group and loop contractible coefficient group
the complex of germs of continuous functions on left-invariant diagonal
neighbourhoods computes the Alexander-Spanier and singular cohomology; similar
results are obtained for -groups and for germs of smooth functions on Lie
groups .Comment: 38 pages, LaTeX, elucidated argument for Corollaries 2.8 and 4.5,
added Corollary 4.6 and examples in section 4, results unchange
On the o-minimal Hilbert's fifth problem
Let be an arbitrary o-minimal structure. Let be a definably
compact definably connected abelian definable group of dimension . Here we
compute the new the intrinsic o-minimal fundamental group of for each
, the -torsion subgroups of the o-minimal cohomology algebra over
of As a corollary we obtain a new uniform proof of Pillay's
conjecture, an o-minimal analogue of Hilbert's fifth problem, relating
definably compact groups to compact real Lie groups, extending the proof
already known in o-minimal expansions of ordered fields
Controlled Connectivity for Semi-Direct Products Acting on Locally Finite Trees
In 2003 Bieri and Geoghegan generalized the Bieri-Neuman-Strebel invariant
by defining , an isometric action by a
finitely generated group on a proper CAT(0) space . In this paper, we
show how the natural and well-known connection between Bass-Serre theory and
covering space theory provides a framework for the calculation of
when is a cocompact action by , a
finitely generated group, on a locally finite Bass-Serre tree for . This
framework leads to a theorem providing conditions for including an endpoint in,
or excluding an endpoint from, . When is a finitely
generated free group acting on its Cayley graph, we can restate this theorem
from a more algebraic perspective, which leads to some general results on
for such actions
Geometric structures on orbifolds and holonomy representations
An orbifold is a topological space modeled on quotient spaces of a finite
group actions. We can define the universal cover of an orbifold and the
fundamental group as the deck transformation group. Let be a Lie group
acting on a space . We show that the space of isotopy-equivalence classes of
-structures on a compact orbifold is locally homeomorphic to
the space of representations of the orbifold fundamental group of to
following the work of Thurston, Morgan, and Lok. This implies that the
deformation space of -structures on is locally homeomorphic to
the space of representations of the orbifold fundamental group to when
restricted to the region of proper conjugation action by .Comment: 35 page
The nuclear dimension of C*-algebras associated to homeomorphisms
We show that if X is a finite dimensional locally compact Hausdorff space,
then the crossed product of C_0(X) by any automorphism has finite nuclear
dimension. This generalizes previous results, in which the automorphism was
required to be free. As an application, we show that group C*-algebras of
certain non-nilpotent groups have finite nuclear dimension.Comment: With an appendix by Gabor Szabo. 28 pages. Minor typos corrected. To
appear, Adv. Math. arXiv admin note: text overlap with arXiv:1308.5418 by
other author
Finite Spaces and Schemes
A ringed finite space is a ringed space whose underlying topological space is
finite. The category of ringed finite spaces contains, fully faithfully, the
category of finite topological spaces and the category of affine schemes. Any
ringed space, endowed with a finite open covering, produces a ringed finite
space. We introduce the notions of schematic finite space and schematic
morphism, showing that they behave, with respect to quasi-coherence, like
schemes and morphisms of schemes do. Finally, we construct a fully faithful and
essentially surjective functor from a localization of a full subcategory of the
category of schematic finite spaces and schematic morphisms to the category of
quasi-compact and quasi-separated schemes.Comment: This is a simplified version of the paper Ringed Finite Spaces,
arXiv:1409.4574v
Simple groups of dynamical origin
We associate with every etale groupoid G two normal subgroups S(G) and A(G)
of the topological full group of G, which are analogs of the symmetric and
alternating groups. We prove that if G is a minimal groupoid of germs (e.g., of
a group action), then A(G) is simple and is contained in every non-trivial
normal subgroup of the full group. We show that if G is expansive (e.g., is the
groupoid of germs of an expansive action of a group), then A(G) is finitely
generated. We also show that S(G)/A(G) is a quotient of H_0(G, Z/2Z).Comment: 25 pages, 3 figure
Final group topologies, Kac-Moody groups and Pontryagin duality
We study final group topologies and their relations to compactness
properties. In particular, we are interested in situations where a colimit or
direct limit is locally compact, a k_\omega-space, or locally k_\omega. As a
first application, we show that unitary forms of complex Kac-Moody groups can
be described as the colimit of an amalgam of subgroups (in the category of
Hausdorff topological groups, and the category of k_\omega-groups). Our second
application concerns Pontryagin duality theory for the classes of almost
metrizable topological abelian groups, resp., locally k_\omega topological
abelian groups, which are dual to each other. In particular, we explore the
relations between countable projective limits of almost metrizable abelian
groups and countable direct limits of locally k_\omega abelian groups.Comment: v3: exposition improved; former title "Final group topologies, Phan
systems and Pontryagin duality'' replaced by new titl
Examples of buildings constructed via covering spaces
Covering space theory is used to construct new examples of buildings
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