388 research outputs found
Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures
This arXived paper has two independant parts, that are improved and corrected
versions of different parts of a single paper once named "On equations in
relatively hyperbolic groups".
The first part is entitled "Existential questions in (relatively) hyperbolic
groups". We study there the existential theory of torsion free hyperbolic and
relatively hyperbolic groups, in particular those with virtually abelian
parabolic subgroups. We show that the satisfiability of systems of equations
and inequations is decidable in these groups.
In the second part, called "Finding relative hyperbolic structures", we
provide a general algorithm that recognizes the class of groups that are
hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in
Israel J. Math, and Bull. London Math. Soc. respectivel
F-sets and finite automata
The classical notion of a k-automatic subset of the natural numbers is here
extended to that of an F-automatic subset of an arbitrary finitely generated
abelian group equipped with an arbitrary endomorphism F. This is
applied to the isotrivial positive characteristic Mordell-Lang context where F
is the Frobenius action on a commutative algebraic group G over a finite field,
and is a finitely generated F-invariant subgroup of G. It is shown
that the F-subsets of introduced by the second author and Scanlon are
F-automatic. It follows that when G is semiabelian and X is a closed subvariety
then X intersect is F-automatic. Derksen's notion of a k-normal subset
of the natural numbers is also here extended to the above abstract setting, and
it is shown that F-subsets are F-normal. In particular, the X intersect
appearing in the Mordell-Lang problem are F-normal. This generalises
Derksen's Skolem-Mahler-Lech theorem to the Mordell-Lang context.Comment: The final section is revised following an error discovered by
Christopher Hawthorne; it is no longer claimed that an F-normal subset has a
finite symmetric difference with an F-subset. The main theorems of the paper
remain unchange
On Sloane's persistence problem
We investigate the so-called persistence problem of Sloane, exploiting
connections with the dynamics of circle maps and the ergodic theory of
actions. We also formulate a conjecture concerning the
asymptotic distribution of digits in long products of finitely many primes
whose truth would, in particular, solve the persistence problem. The heuristics
that we propose to complement our numerical studies can be thought in terms of
a simple model in statistical mechanics.Comment: 5 figure
On Sushchansky p-groups
We study Sushchansky p-groups. We recall the original definition and
translate it into the language of automata groups. The original actions of
Sushchansky groups on p-ary tree are not level-transitive and we describe their
orbit trees. This allows us to simplify the definition and prove that these
groups admit faithful level-transitive actions on the same tree. Certain branch
structures in their self-similar closures are established. We provide the
connection with, so-called, G groups that shows that all Sushchansky groups
have intermediate growth and allows to obtain an upper bound on their period
growth functions.Comment: 14 pages, 3 figure
Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for A^{(1)}_n
We study an integrable vertex model with a periodic boundary condition
associated with U_q(A_n^{(1)}) at the crystallizing point q=0. It is an
(n+1)-state cellular automaton describing the factorized scattering of
solitons. The dynamics originates in the commuting family of fusion transfer
matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the
periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum
group, we develop an inverse scattering/spectral formalism and solve the
initial value problem based on several conjectures. The action-angle variables
are constructed representing the amplitudes and phases of solitons. By the
direct and inverse scattering maps, separation of variables into solitons is
achieved and nonlinear dynamics is transformed into a straight motion on a
tropical analogue of the Jacobi variety. We decompose the level set into
connected components under the commuting family of time evolutions and identify
each of them with the set of integer points on a torus. The weight multiplicity
formula derived from the q=0 Bethe equation acquires an elegant interpretation
as the volume of the phase space expressed by the size and multiplicity of
these tori. The dynamical period is determined as an explicit arithmetical
function of the n-tuple of Young diagrams specifying the level set. The inverse
map, i.e., tropical Jacobi inversion is expressed in terms of a tropical
Riemann theta function associated with the Bethe ansatz data. As an
application, time average of some local variable is calculated
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