4,815 research outputs found
The automorphism group of accessible groups
In this article, we study the outer automorphism group of a group G
decomposed as a finite graph of group with finite edge groups and finitely
generated vertex groups with at most one end. We show that Out(G) is
essentially obtained by taking extensions of relative automorphism groups of
vertex groups, groups of Dehn twists and groups of automorphisms of free
products. We apply this description and obtain a criterion for Out(G) to be
finitely presented, as well as a necessary and sufficient condition for Out(G)
to be finite. Consequences for hyperbolic groups are discussed.Comment: 18 pages, 3 figures. Section 4 rewritten and corrected, added
reference
On the Uniform Random Generation of Non Deterministic Automata Up to Isomorphism
In this paper we address the problem of the uniform random generation of non
deterministic automata (NFA) up to isomorphism. First, we show how to use a
Monte-Carlo approach to uniformly sample a NFA. Secondly, we show how to use
the Metropolis-Hastings Algorithm to uniformly generate NFAs up to isomorphism.
Using labeling techniques, we show that in practice it is possible to move into
the modified Markov Chain efficiently, allowing the random generation of NFAs
up to isomorphism with dozens of states. This general approach is also applied
to several interesting subclasses of NFAs (up to isomorphism), such as NFAs
having a unique initial states and a bounded output degree. Finally, we prove
that for these interesting subclasses of NFAs, moving into the Metropolis
Markov chain can be done in polynomial time. Promising experimental results
constitute a practical contribution.Comment: Frank Drewes. CIAA 2015, Aug 2015, Umea, Sweden. Springer, 9223,
pp.12, 2015, Implementation and Application of Automata - 20th International
Conferenc
Graphs, permutations and topological groups
Various connections between the theory of permutation groups and the theory
of topological groups are described. These connections are applied in
permutation group theory and in the structure theory of topological groups.
The first draft of these notes was written for lectures at the conference
Totally disconnected groups, graphs and geometry in Blaubeuren, Germany, 2007.Comment: 39 pages (The statement of Krophollers conjecture (item 4.30) has
been corrected
Vertex finiteness for splittings of relatively hyperbolic groups
Consider a group G and a family of subgroups of G. We say that
vertex finiteness holds for splittings of G over if, up to
isomorphism, there are only finitely many possibilities for vertex stabilizers
of minimal G-trees with edge stabilizers in .
We show vertex finiteness when G is a toral relatively hyperbolic group and
is the family of abelian subgroups.
We also show vertex finiteness when G is hyperbolic relative to virtually
polycyclic subgroups and is the family of virtually cyclic
subgroups; if moreover G is one-ended, there are only finitely many minimal
G-trees with virtually cyclic edge stabilizers, up to automorphisms of G.Comment: Minor modifications following referee's comments. To appear in Israel
Journal of Mathematic
Black Box White Arrow
The present paper proposes a new and systematic approach to the so-called
black box group methods in computational group theory. Instead of a single
black box, we consider categories of black boxes and their morphisms. This
makes new classes of black box problems accessible. For example, we can enrich
black box groups by actions of outer automorphisms.
As an example of application of this technique, we construct Frobenius maps
on black box groups of untwisted Lie type in odd characteristic (Section 6) and
inverse-transpose automorphisms on black box groups encrypting .
One of the advantages of our approach is that it allows us to work in black
box groups over finite fields of big characteristic. Another advantage is
explanatory power of our methods; as an example, we explain Kantor's and
Kassabov's construction of an involution in black box groups encrypting .
Due to the nature of our work we also have to discuss a few methodological
issues of the black box group theory.
The paper is further development of our text "Fifty shades of black"
[arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black
box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248
Lepton Mixing Predictions including Majorana Phases from Flavour Symmetry and Generalised CP
Generalised CP transformations are the only known framework which allows to
predict Majorana phases in a flavour model purely from symmetry. For the first
time generalised CP transformations are investigated for an infinite series of
finite groups, . In direct models the
mixing angles and Dirac CP phase are solely predicted from symmetry.
flavour symmetry provides many examples of viable predictions
for mixing angles. For all groups the mixing matrix has a trimaximal middle
column and the Dirac CP phase is 0 or . The Majorana phases are predicted
from residual flavour and CP symmetries where can take several
discrete values for each and the Majorana phase is a multiple
of . We discuss constraints on the groups and CP transformations from
measurements of the neutrino mixing angles and from neutrinoless double-beta
decay and find that predictions for mixing angles and all phases are accessible
to experiments in the near future.Comment: 16 pages, 8 figures; references added; clarification in section 2.3
added; results are unchange
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