In this paper we compute the automorphism groups $\operatorname{Aut}(\mathbf{P}_n(Σ))$ and $\operatorname{Aut}(\mathbf{B}_n(Σ))$ of braid groups $\mathbf{P}_n(Σ)$ and $\mathbf{B}_n(Σ)$ on every orientable surface $Σ$, which are isomorphic to group extensions of the extended mapping class group $\mathcal{M}^*_n(Σ)$ by the transvection subgroup except for a few cases.
We also prove that $\mathbf{P}_n(Σ)$ is always a characteristic subgroup of $\mathbf{B}_n(Σ)$ unless $Σ$ is a twice-punctured sphere and $n=2$.
In this paper we show that the singular braid monoid of an orientable surface can be embedded in a group. The proof is purely topological, making no use of the monoid presentation.
In this paper we study the minimum dilatation pseudo-Anosov mapping classes coming from fibrations over the circle of a single 3-manifold, the mapping torus for the "simplest pseudo-Anosov braid". The dilatations that arise include the minimum dilatations for orientable mapping classes for genus g=2,3,4,5,8 as well as Lanneau and Thiffeault's conjectural minima for orientable mapping classes, when g = 2,4 (mod 6). Our examples also show that the minimum dilatation for orientable mapping classes is strictly greater than the minimum dilatation for non-orientable ones when g = 4,6,8.
We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that the standard generators of the braid group are not mapped to Dehn twists.
Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity element of a cyclic group of automorphisms acting on a closed non-orientable surface with one exception; namely, when this element has order 2. In this case the fixed-point set may have simple closed curves (called ovals) as well as fixed points. In this note we extend Macbeath'…
We prove that the pure braid groups on closed, orientable surfaces are bi-orderable, and that the pure braid groups on closed, non-orientable surfaces have generalized torsion, thus they are not bi-orderable.
It is proved by Sakuma and Brooks that any closed orientable $3$-manifold with a Heegaard splitting of genus $g$ admits a $2$-fold branched cover that is a hyperbolic $3$-manifold and a genus $g$ surface bundle over the circle. This paper concerns entropy of pseudo-Anosov monodromies for hyperbolic fibered $3$-manifolds. We prove that there exist infinitely many closed orientable $3$-manifolds $M$ such that the minimal entropy over all hyperbolic, genus $g$ surface bundles over the circle as $2$-fold branched covers of the $3$-manifold $M$ is comparable to $1/g$.
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 b…
We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group.
We prove that if g and n are integers at least two, then the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components.
In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of Birman and Chillingworth and of Gonçalves and Guaschi to show that the mapping class group MCG(N ; k) of N relative to a k-point subset embeds in the mapping class group MCG(S; 2k) of S relative to a 2k-point subset. We then compute the cohomological dimension of the braid groups of all…
In this paper we determine the automorphism groups of the profinite braid groups with four or more strings in terms of the profinite Grothendieck-Teichmüller group.
Based on a normal form for braid group elements suggested by Dehornoy, we prove several representations of braid groups by automorphisms of a free group to be faithful. This includes a simple proof of the standard Artin's representation being faithful.
Let $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$ be the group of automorphisms on the first homology group with $\mathbb Z_2$ coefficient of a closed non-orientable surface $N_g$ preserving the mod $2$ intersection form. In this paper, we obtain a finite presentation for $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$. As applications we calculate the first homology group and the second homology group of $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$.
In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to…
In this paper, we prove that each automorphism of a surface braid group is induced by a homeomorphism of the underlying surface, provided that this surface is a closed, connected, orientable surface of genus at least 2, and the number of strings is at least three. This result generalizes previous results for classical braid groups, mapping class groups, and Torelli groups.
We present and discuss some open problems formulated by participants of the International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in Novosibirsk, 2014. Problems are related to palindromic and commutator widths of groups; properties of Brunnian braids and two-colored braids, corresponding to an amalgamation of groups; extreme properties of hyperbolic 3-orbifold groups, relations between inner and quasi-inner automorphisms of groups; and Staic's construction of symmetric cohomology of groups.
Let M be a compact surface, either orientable or non-orientable. We study the lower central and derived series of the braid and pure braid groups of M in order to determine the values of n for which B\_n(M) and P\_n(M) are residually nilpotent or residually soluble. First, we solve this problem for the case where M is the 2-torus. We then give a general description of these series for an arbitrary semi-direct product that allows us to calculate explicitly the lower central series of P\_2(K), where K is the Klein bottle, and to give an estimate for the derived series of P\_n(K). Finally, if M i…
Let $PB_n(S_{g,p})$ be the pure braid group of a genus $g>1$ surface with $p$ punctures. In this paper we prove that any surjective homomorphism $PB_n(S_{g,p})\to PB_m(S_{g,p})$ factors through one of the forgetful homomorphisms. We then compute the automorphism group of $PB_n(S_{g,p})$, extending Irmak, Ivanov and McCarthy's result to the punctured case. Surprisingly, in contrast to the $n=1$ case, any automorphism of $PB_n(S_{g,p})$, $n>1$ is geometric.
In this article, we give a necessary and sufficient condition for embedding a finite index subgroup of Artin's braid group into the mapping class group of a connected orientable surface.
Let $UVB_n$ and $UVP_n$ be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands respectively. We study the groups $UVB_n$ and $UVP_n$, and our main results are as follows: for $n\geq 5$, we give a complete description, up to conjugation, to all possible homomorphisms from $UVB_n$ to the symmetric group $S_n$. For $n\geq 3$, we characterise all possible images of $UVB_n$, under a group homomorphism, to any finite group $G$. For $n\geq 5$, we prove that $UVP_n$ is a characteristic subgroup of $UVB_n$. In addition, we determine the automorphism group of …
In this paper, we generalize the tools that were introduced in [Dar19b] in order to study the Andreadakis problem for subgroups of IAn. In particular, we study the behaviour of the Andreadakis problem when we add inner automorphisms to a subgroup of IAn. We notably use this to show that the Andreadakis equality holds for the pure braid group on n strands modulo its center acting on the free group on n-1 generators , that is, for the (pure, based) mapping class group of the n-punctured sphere acting on its fundamental group.
Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S,f) to be conformally equivalent to (S',f') where S' is a bordered orientable Klein surface embedded in the Euclidean space and f' is the restriction to S' of a prime order rotation. We represent two famous automorphisms using rotations of R^4 and S^4 : the order seven automorphisms of the Klein quartic and the Wiman surface.
We show that many normal subgroups of the braid group modulo its centre, and of the mapping class group of a sphere with marked points, have the property that their automorphism and abstract commensurator groups are mapping class groups of such spheres. As one application, we establish the automorphism groups of each term in the lower central series and derived series of the pure braid group. We also obtain new proofs of results of Dyer-Grossman and Orevkov, showing that the automorphism groups of the braid group and of its commutator subgroup are isomorphic. We then calculate the automorphism…
Let $G$ be a finite group acting on a connected compact surface $Σ$, and $M$ be an integer homology 3-sphere. We show that if each element of $G$ is extendable over $M$ with respect to a fixed embedding $Σ\rightarrow M$, then $G$ is extendable over some $M'$ which is 1-dominated by $M$. From this result, in the orientable category we classify all periodic automorphisms of closed surfaces that are extendable over the 3-sphere. The corresponding embedded surface of such an automorphism can always be a Heegaard surface.
Let g and n be integers at least two, and let G be the pure braid group with n strands on a closed orientable surface of genus g. We describe any injective homomorphism from a finite index subgroup of G into G. As a consequence, we show that any finite index subgroup of G is co-Hopfian.
We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.
For every group genetic code with finite number of generating and at most with one defining relation we introduce the braid group of this genetic code. This construction includes the braid group of Euclidean plane, the braid groups of closed orientable surfaces, B type groups of Artin-Brieskorn, and allow us to study all these groups with common point of view. We present some results on the structure of the braid groups of genetic codes, describe normal form of words, torsion, and some results on widths of the verbal subgroups. Also we prove that Scott's system of defining relations for the br…
A dessin is a 2-cell embedding of a connected bipartite graph into an orientable closed surface. An automorphism of a dessin is a permutation of the edges of the underlying graph which preserves the colouring of the vertices and extends to an orientation-preserving self-homeomorphism of the supporting surface. A dessin is regular if its automorphism group is transitive on the edges, and a regular dessin is totally symmetric if it is invariant under all dessin operations. Thus totally symmetric dessins possesses the highest level of external symmetry. In this paper we present a classification o…
The $d$-fold ($d \geq 3$) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator $\widetildeβ_i$ of braid group inside mapping class group induced by $d$-fold covering is t…
Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）；Departamento de Matemática y Estadística, Universidad de La Frontera（拉弗龙特拉大学数学与统计系）
From a group $H$ and a non-trivial element $h$ of $H$, we define a representation $ρ: B_n \to \Aut(G)$, where $B_n$ denotes the braid group on $n$ strands, and $G$ denotes the free product of $n$ copies of $H$. Such a representation shall be called the Artin type representation associated to the pair $(H,h)$. The goal of the present paper is to study different aspects of these representations. Firstly, we associate to each braid $β$ a group $Γ_{(H,h)} (β)$ and prove that the operator $Γ_{(H,h)}$ determines a group invariant of oriented links. We then give a topological construction of the Arti…
An automorphism $f$ of a closed orientable surface $Σ$ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, Σ)$ with respect to some embedding $Σ\hookrightarrow S^3$. We prove that if an automorphism of a genus-2 surface $Σ$ is extendable over $S^3$, then $f$ extends to an automorphism of the pair $(S^3, Σ)$ with respect to an embedding $Σ\hookrightarrow S^3$ such that $Σ$ bounds genus-2 handlebodies on both sides. The classification of essential annuli in the exterior of genus-2 handlebodies embedded in $S^3$ due to Ozawa and the second author …
We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism $φ$ and use it to characterize when the homological stretch factor of $φ$ is equal to its geometric stretch factor.
In this paper we introduce, for each closed orientable surface, an analogue of Tits buildings adjusted to investigation of the Torelli group of this surface. It is a simplicial complex with some additional structure. We call this complex with its additional structure the Torelli building of the surface in question. The main result of this paper shows that Torelli buildings of surfaces of genus at least 5 have only obvious automorphisms, and identifies its group of automorphisms. Namely, we prove that for such a surface every automorphism of its Torelli building is induced by a diffeomorphism o…
Let $M$ be a compact surface without boundary, and $n\geq 2$. We analyse the quotient group $B_n(M)/Γ_2(P_n(M))$ of the surface braid group $B_{n}(M)$ by the commutator subgroup $Γ_2(P_n(M))$ of the pure braid group $P_{n}(M)$. If $M$ is different from the $2$-sphere $\mathbb{S}^2$, we prove that $B_n(M)/Γ_2(P_n(M))$ is isomorphic rho $P_n(M)/Γ_2(P_n(M)) \rtimes_φ S_n$, and that $B_n(M)/Γ_2(P_n(M))$ is a crystallographic group if and only if $M$ is orientable. If $M$ is orientable, we prove a number of results regarding the structure of $B_n(M)/Γ_2(P_n(M))$. We characterise the finite-order el…
It is well-known that there is a faithful representation of braid groups on automorphism groups of free groups, and it is also well-known that free groups are bi-orderable. We investigate which n-strand braids give rise to automorphisms which preserve some bi-ordering of the free group rank n. As a consequence of our work we find that of the two minimal volume hyperbolic 2-cusped orientable 3-manifolds, one has bi-orderable fundamental group whereas the other does not. We prove a similar result for the 1-cusped case, and have further results for more cusps. In addition, we study pseudo-Anosov …
Let S be any orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S) and the Schmutz graph G(S) of S. When all the topological ends of S carry genus, we show that all elements in the automorphism groups Aut(C(S)), Aut(N(S)) and Aut(G(S)) are geometric, i.e. these groups are naturally isomorphic to the extended mapping class group MCG*(S) of the infinite surface S. Finally, we study rigidity phenomena within Aut(C(S)) and Aut(N(S))
In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathematics and mathematical physics, and have been called, in addition to loop braid groups, with several names such as of motion groups, groups of permutation-conjugacy automorphisms, braid-permutation groups, welded braid groups and untwisted ring groups. In parallel to this, we int…
In this paper, we prove that each automorphism of the Torelli group of a surface is induced by a diffeomorphism of the surface, provided that the surface is a closed, connected, orientable surface of genus at least 3. This result was previously announced by Benson Farb for genus at least 4 and has recently been announced by Benson Farb and Nikolai Ivanov for subgroups of finite index in the Torelli group for surfaces of genus at least 5. This result is also directly analogous to previous results established for classical braid groups, mapping class groups, and surface braid groups.
Let S be a compact, connected, orientable surface of positive genus. Let HT(S) be the Hatcher-Thurston complex of S. We prove that Aut(HT(S)) is isomorphic to the extended mapping class group of S modulo its center.
We show that the smallest non-cyclic quotients of braid groups are symmetric groups, proving a conjecture of Margalit. Moreover we recover results of Artin and Lin about the classification of homomorphisms from braid groups on n strands to symmetric groups on k letters, where k is at most n. Unlike the original proofs, our method does not use the Bertrand-Chebyshev theorem, answering a question of Artin. Similarly for mapping class group of closed orientable surfaces, the smallest non-cyclic quotient is given by the mod two reduction of the symplectic representation. We provide an elementary p…
