96 research outputs found
Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane
We consider the (pure) braid groups B_{n}(M) and P_{n}(M), where M is the
2-sphere S^2 or the real projective plane RP^2. We determine the minimal
cardinality of (normal) generating sets X of these groups, first when there is
no restriction on X, and secondly when X consists of elements of finite order.
This improves on results of Berrick and Matthey in the case of S^2, and extends
them in the case of RP^2. We begin by recalling the situation for the Artin
braid groups. As applications of our results, we answer the corresponding
questions for the associated mapping class groups, and we show that for M=S^2
or RP^2, the induced action of B_n(M) on H_3 of the universal covering of the n
th configuration space of M is trivial.Comment: 18 page
The lower central and derived series of the braid groups of the sphere and the punctured sphere
Our aim is to determine the lower central series (LCS) and derived series
(DS) for the braid groups of the sphere and of the finitely-punctured sphere.
We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string
braid group B\_n(S^2) is constant from the commutator subgroup onwards, and
that \Gamma\_2(B\_4(S^2)) is a semi-direct product of the quaternion group by a
free group of rank 2. For n=4, we determine the DS of B\_4(S^2), as well as its
quotients. For n \geq 1, the class of m-string braid groups B\_m(S^2) \
{x\_1,...,x\_n} of the n-punctured sphere includes the Artin braid groups B\_m,
those of the annulus, and certain Artin and affine Artin groups. We extend
results of Gorin and Lin, and show that the LCS (resp. DS) of B\_m is
determined for all m (resp. for all m\neq 4). For m=4, we obtain some elements
of the DS. When n\geq 2, we prove that the LCS (resp. DS) of B\_m(S^2) \
{x\_1,...,x\_n} is constant from the commutator subgroup onwards for all m\geq
3 (resp. m\geq 5). We then show that B\_2(S^2\{x\_1,x\_2}) is residually
nilpotent, that its LCS coincides with that of Z\_2*Z, and that the
\Gamma\_i/\Gamma\_{i+1} are 2-elementary finitely-generated groups. For m\geq 3
and n=2, we obtain a presentation of the derived subgroup and its
Abelianisation. For n=3, we see that the quotients \Gamma\_i/\Gamma\_{i+1} are
2-elementary finitely-generated groups.Comment: 103 page
The Borsuk-Ulam theorem for maps into a surface
Let (X, t, S) be a triple, where S is a compact, connected surface without
boundary, and t is a free cellular involution on a CW-complex X. The triple (X,
t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map
f:X-->S, there exists a point x belonging to X satisfying f(t(x))=f(x). In this
paper, we formulate this property in terms of a relation in the 2-string braid
group B_2(S) of S. If X is a compact, connected surface without boundary, we
use this criterion to classify all triples (X, t, S) for which the Borsuk-Ulam
property holds. We also consider various cases where X is not necessarily a
surface without boundary, but has the property that \pi_1(X/t) is isomorphic to
the fundamental group of such a surface. If S is different from the 2-sphere
S^2 and the real projective plane RP^2, then we show that the Borsuk-Ulam
property does not hold for (X, t, S) unless either \pi_1(X/t) is isomorphic to
\pi_1(RP^2), or \pi_1(X/t) is isomorphic to the fundamental group of a compact,
connected non-orientable surface of genus 2 or 3 and S is orientable. In the
latter case, the veracity of the Borsuk-Ulam property depends further on the
choice of involution t; we give a necessary and sufficient condition for it to
hold in terms of the surjective homomorphism \pi_1(X/t)-->Z_2 induced by the
double covering X-->X/t. The cases S=S^2,RP^2 are treated separately.Comment: 31 page
Nielsen theory, braids and fixed points of surface homeomorphisms
AbstractWe study two problems in Nielsen fixed point theory using Artin's braid groups and the Nielsen–Thurston classification of surface homeomorphismsup to isotopy. The first is that of distinguishing Reidemeister classes of free group automorphisms realized by a braid (and thus induced by homeomorphismsof the 2-disc relative to a finite invariant set), for which we give a necessary and sufficient condition in terms of a conjugacy problem in the braid group. Consequently, one may use any braid conjugacy invariant (those of Garside's algorithm, linking numbers, topological entropy, etc.) and any link invariant (Alexander polynomial, splittability, etc.) to distinguish Reidemeisterclasses, giving much stronger criteria than those already known.The second problem is that of deciding when two fixed points of a surface homeomorphismbelong to the same Nielsen fixed point class. We give two criteria, the first in terms of certain reducing curves which can be checked using the Bestvina–Handel algorithm, the second using the multi-variable Alexander polynomial of a link associated with the suspension of the homeomorphism.Finally we consider generalizations of Sharkovskii's theorem on the coexistence of periodic orbits of interval maps to homeomorphismsof the 2-disc. We show that for each n⩾5 there exists a pseudo-Anosovorientation-preservinghomeomorphismof the 2-disc relative to a periodic orbit of period n that does not have periodic orbits of all periods, with an analogous result for the 2-sphere
Embeddings of finite groups in for
Let . In this paper, we study the problem of whether a given finite
group embeds in a quotient of the form , where is
the -string Artin braid group, , and
is the lower central series of the
-string pure braid group . Previous results show that a necessary
condition for such an embedding to exist is that is odd (resp. is
relatively prime with ) if (resp. ), where denotes the
order of . We show that any finite group of odd order (resp. of order
relatively prime with ) embeds in (resp. in
). The result in the case of
has been proved independently by Beck and Marin.
One may then ask whether embeds in a quotient of the form
, where and . If is of the
form , where the action
is injective, is an odd prime (resp. is prime) is
odd (resp. is relatively prime with ) and divides , we show that
embeds in (resp. in
). In the case , this extends a result of Marin
concerning the embedding of the Frobenius groups in , and is
a special case of another result of Beck and Marin. Finally, we construct an
explicit embedding in of the two non-Abelian groups of
order , namely the semi-direct product ,
where the action is given by multiplication by , and the Heisenberg group
mod
The classification of the virtually cyclic subgroups of the sphere braid groups
We study the problem of determining the isomorphism classes of the virtually
cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If
n is odd, or if n is even and sufficiently large, we obtain the complete
classification. For small even values of n, the classification is complete up
to an explicit finite number of open cases. In order to prove our main theorem,
we obtain a number of other results of independent interest, notably the
characterisation of the centralisers and normalisers of the finite cyclic and
dicyclic subgroups of B_n(S^2), a result concerning conjugate powers of finite
order elements, an analysis of the isomorphism classes of the amalgamated
products that occur as subgroups of B_n(S^2), as well as an alternative proof
of the fact that the universal covering space of the n-th configuration space
of S^2 has the homotopy type of S^3 if n is greater than or equal to three.Comment: 96 pages, 11 figure
Exact sequences, lower central series and representations of surface braid groups
We consider exact sequences and lower central series of surface braid groups
and we explain how they can prove to be useful for obtaining representations
for surface braid groups. In particular, using a completely algebraic
framework, we describe the notion of extension of a representation introduced
and studied recently by An and Ko and independently by Blanchet.Comment: 23 pages, 4 figure
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