31,363 research outputs found
Cusps of lattices in rank 1 Lie groups over local fields
Let G be the group of rational points of a semisimple algebraic group of rank
1 over a nonarchimedean local field. We improve upon Lubotzky's analysis of
graphs of groups describing the action of lattices in G on its Bruhat-Tits tree
assuming a condition on unipotents in G. The condition holds for all but a few
types of rank 1 groups. A fairly straightforward simplification of Lubotzky's
definition of a cusp of a lattice is the key step to our results. We take the
opportunity to reprove Lubotzky's part in the analysis from this foundation.Comment: to appear in Geometriae Dedicat
Third rank Killing tensors in general relativity. The (1+1)-dimensional case
Third rank Killing tensors in (1+1)-dimensional geometries are investigated
and classified. It is found that a necessary and sufficient condition for such
a geometry to admit a third rank Killing tensor can always be formulated as a
quadratic PDE, of order three or lower, in a Kahler type potential for the
metric. This is in contrast to the case of first and second rank Killing
tensors for which the integrability condition is a linear PDE. The motivation
for studying higher rank Killing tensors in (1+1)-geometries, is the fact that
exact solutions of the Einstein equations are often associated with a first or
second rank Killing tensor symmetry in the geodesic flow formulation of the
dynamics. This is in particular true for the many models of interest for which
this formulation is (1+1)-dimensional, where just one additional constant of
motion suffices for complete integrability. We show that new exact solutions
can be found by classifying geometries admitting higher rank Killing tensors.Comment: 16 pages, LaTe
Opposition diagrams for automorphisms of large spherical buildings
Let be an automorphism of a thick irreducible spherical building
of rank at least with no Fano plane residues. We prove that if
there exist both type and simplices of mapped onto
opposite simplices by , then there exists a type simplex
of mapped onto an opposite simplex by . This property is
called "cappedness". We give applications of cappedness to opposition diagrams,
domesticity, and the calculation of displacement in spherical buildings. In a
companion piece to this paper we study the thick irreducible spherical
buildings containing Fano plane residues. In these buildings automorphisms are
not necessarily capped
Finite-Temperature Screening of U(1) Fractons
We investigate the finite-temperature screening behavior of three-dimensional
U(1) spin liquid phases with fracton excitations. Several features are shared
with the conventional U(1) spin liquid. The system can exhibit spin liquid
physics over macroscopic length scales at low temperatures, but screening
effects eventually lead to a smooth finite-temperature crossover to a trivial
phase at sufficiently large distances. However, unlike more conventional U(1)
spin liquids, we find that complete low-temperature screening of fractons
requires not only very large distances, but also very long timescales. At the
longest timescales, a charged disturbance (fracton) will acquire a screening
cloud of other fractons, resulting in only short-range correlations in the
system. At intermediate timescales, on the other hand, a fracton can only be
partially screened by a cloud of mobile excitations, leaving weak power-law
correlations in the system. Such residual power-law correlations may be a
useful diagnostic in an experimental search for U(1) fracton phases.Comment: 8+2 page
Permutation of elements in double semigroups
Double semigroups have two associative operations related by
the interchange relation: . Kock \cite{Kock2007} (2007) discovered a
commutativity property in degree 16 for double semigroups: associativity and
the interchange relation combine to produce permutations of elements. We show
that such properties can be expressed in terms of cycles in directed graphs
with edges labelled by permutations. We use computer algebra to show that 9 is
the lowest degree for which commutativity occurs, and we give self-contained
proofs of the commutativity properties in degree 9.Comment: 24 pages, 11 figures, 4 tables. Final version accepted by Semigroup
Forum on 12 March 201
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