223 research outputs found
A Generalized Axis Theorem for Cube Complexes
We consider a finitely generated virtually abelian group acting properly
and without inversions on a CAT(0) cube complex . We prove that
stabilizes a finite dimensional CAT(0) subcomplex that is
isometrically embedded in the combinatorial metric. Moreover, we show that
is a product of finitely many quasilines. The result represents a higher
dimensional generalization of Haglund's axis theorem.Comment: 14 pages Corrected proof of Corollary 1.4. Various other corrections
made following referee report and comments made by thesis examiner. Appendix
added giving a proof of a theorem by Gerasimo
Classifying Finite Dimensional Cubulations of Tubular Groups
A tubular group is a group that acts on a tree with vertex
stabilizers and edge stabilizers. This paper develops further a
criterion of Wise and determines when a tubular group acts freely on a finite
dimensional CAT(0) cube complex. As a consequence we offer a unified
explanation of the failure of separability by revisiting the non-separable
3-manifold group of Burns, Karrass and Solitar and relating it to the work of
Rubinstein and Wang. We also prove that if an immersed wall yields an infinite
dimensional cubulation then the corresponding subgroup is quadratically
distorted.Comment: 24 pages, 11 figures. Minor corrections and clarifications. Some
figures are redrawn. The proof of Theorem 6.1 is rewritten for clarity and to
correct error
Quasi-isometric groups with no common model geometry
A simple surface amalgam is the union of a finite collection of surfaces with
precisely one boundary component each and which have their boundary curves
identified. We prove if two fundamental groups of simple surface amalgams act
properly and cocompactly by isometries on the same proper geodesic metric
space, then the groups are commensurable. Consequently, there are infinitely
many fundamental groups of simple surface amalgams that are quasi-isometric,
but which do not act properly and cocompactly on the same proper geodesic
metric space.Comment: v2: 19 pages, 6 figures; minor changes. To appear in Journal of the
London Mathematical Societ
A Cubical Flat Torus Theorem and the Bounded Packing Property
We prove the bounded packing property for any abelian subgroup of a group
acting properly and cocompactly on a CAT(0) cube complex. A main ingredient of
the proof is a cubical flat torus theorem. This ingredient is also used to show
that central HNN extensions of maximal free-abelian subgroups of compact
special groups are virtually special, and to produce various examples of groups
that are not cocompactly cubulated.Comment: 14 pages, 2 figures, submitted May 2015 Minor corrections and swapped
sections 2 and 3 Corrected an unfortunate typo in Theorem 2.1 - the
hypothesis that the cube complex be finite dimensional has now been adde
Hyperbolic groups that are not commensurably coHopfian
Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We
prove that there exist torsion-free one-ended hyperbolic groups that are not
commensurably coHopfian. In particular, we show that the fundamental group of
every simple surface amalgam is not commensurably coHopfian.Comment: v3: 14 pages, 4 figures; minor changes. To appear in International
Mathematics Research Notice
Topology-dependent density optima for efficient simultaneous network exploration
A random search process in a networked environment is governed by the time it takes to visit every node, termed the cover time. Often, a networked process does not proceed in isolation but competes with many instances of itself within the same environment. A key unanswered question is how to optimise this process: how many concurrent searchers can a topology support before the benefits of parallelism are outweighed by competition for space? Here, we introduce the searcher-averaged parallel cover time (APCT) to quantify these economies of scale. We show that the APCT of the networked symmetric exclusion process is optimised at a searcher density that is well predicted by the spectral gap. Furthermore, we find that non-equilibrium processes, realised through the addition of bias, can support significantly increased density optima. Our results suggest novel hybrid strategies of serial and parallel search for efficient information gathering in social interaction and biological transport networks.This work was supported by the EPSRC Systems Biology DTC Grant No. EP/G03706X/1 (D.B.W.), a Royal Society Wolfson Research Merit Award (R.E.B.), a Leverhulme Research Fellowship (R.E.B.), the BBSRC UK Multi-Scale Biology Network Grant No. BB/M025888/1 (R.E.B. and F.G.W.), and Trinity College, Cambridge (F.G.W.)
Displacement of transport processes on networked topologies
Consider a particle whose position evolves along the edges of a network. One
definition for the displacement of a particle is the length of the shortest
path on the network between the current and initial positions of the particle.
Such a definition fails to incorporate information of the actual path the
particle traversed. In this work we consider another definition for the
displacement of a particle on networked topologies. Using this definition,
which we term the winding distance, we demonstrate that for Brownian particles,
confinement to a network can induce a transition in the mean squared
displacement from diffusive to ballistic behaviour, for long times. A multiple scales approach is used to derive a
macroscopic evolution equation for the displacement of a particle and uncover a
topological condition for whether this transition in the mean squared
displacement will occur. Furthermore, for networks satisfying this topological
condition, we identify a prediction of the timescale upon which the
displacement transitions to long-time behaviour. Finally, we extend the
investigation of displacement on networks to a class of anomalously diffusive
transport processes, where we find that the mean squared displacement at long
times is affected by both network topology and the character of the transport
process.Comment: 22 pages, 8 figure
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