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John Guaschi

Publications and source records attributed to John Guaschi.

35 records · Page 2Linked to original sources
arXiv2015

The inclusion of configuration spaces of surfaces in Cartesian products, its induced homomorphism, and the virtual cohomological dimension of the braid groups of S^2 and RP^2

Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration space F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\_\#: P\_n(M) --\textgreater{} (π\_1(M))^n, where P\_n(M) is the n-string pure braid group of M. Both i and i\_\# were studied initially by J.Birman who conjectured that Ker(i\_\#) is equal to the normal closure of the Artin pure braid group P\_n in P\_n(M). The conjecture was later proved by C.Goldberg for compact surfaces without boundary different from the 2-sphere S^2 and the projective plane RP^2. In this paper, we prove the conjecture for S^2 and RP^2. In the case of RP^2, we prove that Ker(i\_\#) is equal to the commutator subgroup of P\_n(RP^2), we show that it may be decomposed in a manner similar to that of P\_n(S^2) as a direct sum of a torsion-free subgroup L\_n and the finite cyclic group generated by the full twist braid, and we prove that L\_n may be written as an iterated semi-direct product of free groups. Finally, we show that the groups B\_n(S^2) and P\_n(S^2) (resp. B\_n(RP^2) and P\_n(RP^2)) have finite virtual cohomological dimension equal to n-3 (resp. n-2), where B\_n(M) denotes the full n-string braid group of M. This allows us to determine the virtual cohomological dimension of the mapping class groups of the mapping class groups of S^2 and RP^2 with marked points, which in the case of S^2, reproves a result due to J.Harer.

math.GT↗
arXiv2015

Quotients of the Artin braid groups and crystallographic groups

Let n be greater than or equal to 3. We study the quotient group B\_n/[P n,P\_n] of the Artin braid group B\_n by the commutator subgroup of its pure Artin braid group P\_n. We show that B\_n/[P n,P\_n] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that B\_n/[P n,P\_n] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B\_n/[P n,P\_n] with the conjugacy classes of the elements of odd order of the symmetric group S\_n, and that the isomorphism class of any Abelian subgroup of odd order of S\_n is realised by a subgroup of B\_n/[P n,P\_n]. Finally, we discuss the realisation of non-Abelian subgroups of S\_n of odd order as subgroups of B\_n/[P n,P\_n], and we show that the Frobenius group of order 21, which is the smallest non-Abelian group of odd order, embeds in B\_n/[P n,P\_n] for all n greater than or equal to 7.

math.GR↗
arXiv2014

Abelian and metabelian quotients of surface braid groups

In this paper we study abelian and metabelian quotients of braid groups on oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell-Neuwirth fibrations.

math.GR↗
arXiv2013

A survey of surface braid groups and the lower algebraic K-theory of their group rings

We give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surface braid groups, such as the existence of torsion, orderability, linearity, and their relation both with mapping class groups and with the homotopy groups of the 2-sphere. The braid groups of the 2-sphere and the real projective plane are of particular interest because they possess elements of finite order, and we discuss in detail their torsion and the classification of their finite and virtually cyclic subgroups. Finally, we outline the methods used to study the lower algebraic K-theory of the group rings of surface braid groups, highlighting recent results concerning the braid groups of the 2-sphere and the real projective plane.

math.GT↗
arXiv2012

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.

math.GT↗
arXiv2011

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.

math.GT↗
arXiv2011

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.

math.GT↗
arXiv2010

The lower central and derived series of the braid groups of the projective plane

We determine the lower central and derived series of the n-string braid groups B_n(RP^2) of the real projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. For n=1,2, B_n(RP^2) is finite and its lower central and derived series are known. If n>2 (resp. n>4), we show that the lower central (resp. derived) series of B_n(RP^2) is constant from the commutator subgroup onwards, and we exhibit a presentation of the commutator subgroup. In the exceptional cases n=3,4, we determine explicitly the complete derived series of B_3(RP^2), we calculate the derived series of B_4(RP^2) up to and including its fifth term, and we obtain many of the derived series quotients in these two cases.

math.GT↗
arXiv2010

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 π_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 π_1(X/t) is isomorphic to π_1(RP^2), or π_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 π_1(X/t)-->Z_2 induced by the double covering X-->X/t. The cases S=S^2,RP^2 are treated separately.

math.GT↗
arXiv2009

Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give several applications of this result. First, we classify the finite subgroups of the n-th braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of the mapping class group of the n-punctured real projective plane due to Bujalance, Cirre and Gamboa. Secondly, using the linearity of B_{n} due to Bigelow and Krammer, we show that the braid groups of compact, connected surfaces of low genus are linear.

math.GT↗
arXiv2009

Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence

Let M be a compact, connected non-orientable surface without boundary and of genus g greater than or equal to 3. We investigate the pure braid groups P_n(M) of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 --> P_m(M {x_1,...,x_n}) --> P_{n+m}(M) --> P_n(M) --> 1, where m,n are positive integers, and the homomorphism p*:P_{n+m}(M) --> P_n(M) corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration p:F_{n+m}(M)} --> F_n(M) of configuration spaces, defined by p((x_1,...,x_n,..., x_{n+m}))= (x_1, ..., x_n). We show that p and p* admit a section if and only if n=1. Together with previous results, this completes the resolution of the splitting problem for surfaces pure braid groups.

math.GT↗
arXiv2007

Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups $P_{n}(RP^2)$ of the projective plane. The maximal finite subgroups of $P_{n}(RP^2)$ are isomorphic to the quaternion group of order 8 if $n=3$, and to $\Z_{4}$ if $n\geq 4$. Further, for all $n\geq 3$, up to isomorphism, the following groups are the infinite virtually cyclic subgroups of $P_{n}(RP^2)$: $\Z$, $\Z_{2} \times \Z$ and the amalgamated product $\Z_{4} \ast_{\Z_{2}} \Z_{4}$.

math.GR↗
arXiv2007

The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence

We study the pure braid groups $P_n(RP^2)$ of the real projective plane $RP^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1 \to P_m(RP^2 {x_1,...,x_n} \to P_{n+m}(RP^2) \stackrel{p_{\ast}}{\to} P_n(RP^2) \to 1$, where $n\geq 2$ and $m\geq 1$, and $p_{\ast}$ is the homomorphism which corresponds geometrically to forgetting the last $m$ strings. This problem is equivalent to that of the existence of a section for the associated fibration $p: F_{n+m}(RP^2) \to F_n(RP^2)$ of configuration spaces. Van Buskirk proved in 1966 that $p$ and $p_{\ast}$ admit a section if $n=2$ and $m=1$. Our main result in this paper is to prove that there is no section if $n\geq 3$. As a corollary, it follows that $n=2$ and $m=1$ are the only values for which a section exists. As part of the proof, we derive a presentation of $P_n(RP^2)$: this appears to be the first time that such a presentation has been given in the literature.

math.GT↗
arXiv2006

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 Γ\_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 Γ\_i/Γ\_{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 Γ\_i/Γ\_{i+1} are 2-elementary finitely-generated groups.

math.GT↗
arXiv2006

The quaternion group as a subgroup of the sphere braid groups

Let n be greater than or equal to 3. We prove that the quaternion group of order 8 is realised as a subgroup of the sphere braid group B\_n(S^2) if and only if n is even. If n is divisible by 4 then the commutator subgroup of B\_n(S^2) contains such a subgroup. Further, for all n greater than or equal to 3, B\_n(S^2) contains a subgroup isomorphic to the dicyclic group of order 4n.

math.GT↗
arXiv2004

The braid groups of the projective plane

Let B_n(RP^2)$ (respectively P_n(RP^2)) denote the braid group (respectively pure braid group) on n strings of the real projective plane RP^2. In this paper we study these braid groups, in particular the associated pure braid group short exact sequence of Fadell and Neuwirth, their torsion elements and the roots of the `full twist' braid. Our main results may be summarised as follows: first, the pure braid group short exact sequence 1 --> P_{m-n}(RP^2 - {x_1,...,x_n}) --> P_m(RP^2) --> P_n(RP^2) --> 1 does not split if m > 3 and n=2,3. Now let n > 1. Then in B_n(RP^2), there is a k-torsion element if and only if k divides either 4n or 4(n-1). Finally, the full twist braid has a k-th root if and only if k divides either 2n or 2(n-1).

math.AT↗