The Zieschang-McCool method for generating algebraic mapping-class groups

Let g and p be non-negative integers. Let A(g,p) denote the group consisting of all those automorphisms of the free group on {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} which fix the element t_1t_2...t_p[x_1,y_1]...[x_g,y_g] and permute the set of conjugacy classes {[t_1],....,[t_p]}. Labru\`ere and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A(g,p) is generated by a set that is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. Labru\`ere and Paris also gave defining relations for the ADLH set in A(g,p); we do not know an algebraic proof of this for g > 1. Consider an orientable surface S(g,p) of genus g with p punctures, such that (g,p) is not (0,0) or (0,1). The algebraic mapping-class group of S(g,p), denoted M(g,p), is defined as the group of all those outer automorphisms of the one-relator group with generating set {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} and relator t_1t_2...t_p[x_1,y_1]...[x_g,y_g] which permute the set of conjugacy classes {[t_1],....,[t_p]}. It now follows from a result of Nielsen that M(g,p) is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that M(g,p) equals the (topological) mapping-class group of S(g,p), along lines suggested by Magnus, Karrass, and Solitar in 1966.Comment: 21 pages, 0 figure

Automorphisms of one-relator groups

It is a well-known fact that every group $G$ has a presentation of the form $G = F/R$, where $F$ is a free group and $R$ the kernel of the natural epimorphism from $F$ onto $G$. Driven by the desire to obtain a similar presentation of the group of automorphisms $Aut(G)$, we can consider the subgroup $Stab(R) \subseteq Aut(F)$ of those automorphisms of $F$ that stabilize $R$, and try to figure out if the natural homomorphism $Stab(R) \to Aut(G)$ is onto, and if it is, to determine its kernel. Both parts of this task are usually quite hard. The former part received considerable attention in the past, whereas the latter, more difficult, part (determining the kernel) seemed unapproachable. Here we approach this problem for a class of one-relator groups with a special kind of small cancellation condition. Then, we address a somewhat easier case of 2-generator (not necessarily one-relator) groups, and determine the kernel of the above mentioned homomorphism for a rather general class of those groups.Comment: LaTex file, 8 page

Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we call mapping class groups over graphs, are indexed over labeled simplicial graphs with 2g vertices. The mapping class group over the graph Gamma is defined to be a subgroup of the automorphism group of the right-angled Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a theorem of Magnus.Comment: 45 page

Full-featured peak reduction in right-angled Artin groups

We prove a new version of the classical peak-reduction theorem for automorphisms of free groups in the setting of right-angled Artin groups. We use this peak-reduction theorem to prove two important corollaries about the action of the automorphism group of a right-angled Artin group $A_\Gamma$ on the set of $k$-tuples of conjugacy classes from $A_\Gamma$: orbit membership is decidable, and stabilizers are finitely presentable. Further, we explain procedures for checking orbit membership and building presentations of stabilizers. This improves on a previous result of the author's. We overcome a technical difficulty from the previous work by considering infinite generating sets for the automorphism groups. The method also involves a variation on the Hermite normal form for matrices.Comment: 72 pages, 1 figure. Updated to incorporate referee comment

Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfaces

Accepted and to appearInternational audienceWe present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney-Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semi-pure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the three-dimensional symmetries of the primitive, diamond and gyroid triply-periodic minimal surfaces, which have relevance to a variety of physical systems

The isomorphism problem for all hyperbolic groups

We give a solution to Dehn's isomorphism problem for the class of all hyperbolic groups, possibly with torsion. We also prove a relative version for groups with peripheral structures. As a corollary, we give a uniform solution to Whitehead's problem asking whether two tuples of elements of a hyperbolic group $G$ are in the same orbit under the action of \Aut(G). We also get an algorithm computing a generating set of the group of automorphisms of a hyperbolic group preserving a peripheral structure.Comment: 71 pages, 4 figure

On restricting the ambiguity in morphic images of words

For alphabets Delta_1, Delta_2, a morphism g : Delta_1* to Delta_2* is ambiguous with respect to a word u in Delta_1* if there exists a second morphism h : Delta_1* to Delta_2* such that g(u) = h(u) and g not= h. Otherwise g is unambiguous. Hence unambiguous morphisms are those whose structure is fully preserved in their morphic images. A concept so far considered in the free monoid, the first part of this thesis considers natural extensions of ambiguity of morphisms to free groups. It is shown that, while the most straightforward generalization of ambiguity to a free monoid results in a trivial situation, that all morphisms are (always) ambiguous, there exist meaningful extensions of (un)ambiguity which are non-trivial - most notably the concepts of (un)ambiguity up to inner automorphism and up to automorphism. A characterization is given of words in a free group for which there exists an injective morphism which is unambiguous up to inner automorphism in terms of fixed points of morphisms, replicating an existing result for words in the free monoid. A conjecture is presented, which if correct, is sufficient to show an equivalent characterization for unambiguity up to automorphism. A rather counterintuitive statement is also established, that for some words, the only unambiguous (up to automorphism) morphisms are non-injective (or even periodic). The second part of the thesis addresses words for which all non-periodic morphisms are unambiguous. In the free monoid, these take the form of periodicity forcing words. It is shown using morphisms that there exist ratio-primitive periodicity forcing words over arbitrary alphabets, and furthermore that it is possible to establish large and varied classes in this way. It is observed that the set of periodicity forcing words is spanned by chains of words, where each word is a morphic image of its predecessor. It is shown that the chains terminate in exactly one direction, meaning not all periodicity forcing words may be reached as the (non-trivial) morphic image of another. Such words are called prime periodicity forcing words, and some alternative methods for finding them are given. The free-group equivalent to periodicity forcing words - a special class of C-test words - is also considered, as well as the ambiguity of terminal-preserving morphisms with respect to words containing terminal symbols, or constants. Moreover, some applications to pattern languages and group pattern languages are discussed

The outer automorphism groups of three classes of groups

The theory of outer automorphism groups allows us to better understand groups through their symmetries, and in this thesis we approach outer automorphism groups from two directions. In the first direction we start with a class of groups and then classify their outer automorphism groups. In the other direction we start with a broad class of groups, for example finitely generated groups, and for each group Q in this class we construct a group G_Q such that Q is related, in a suitable sense, to the outer automorphism group of G_Q. We give a list of 14 groups which precisely classifies the outer automorphism groups of one-ended two-generator, one-relator groups with torsion. We also describe the outer automorphism groups of such groups which have more than one end. Combined with recent algorithmic results of Dahmani–Guirardel, this work yields an algorithm to compute the outer automorphism group of a two-generator, one-relator group with torsion. We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a particular kind of HNN-extension. We apply this to prove two main results. These results demonstrate a universal property of triangle groups and are as follows. Fix an arbitrary hyperbolic triangle group H. If Q is a finitely generated group then there exists an HNN-extension G_Q of H such that Q embeds with finite index into the outer automorphism group of G_Q. Moreover, if Q is residually finite then G_Q can be taken to be residually finite. Secondly, fix an equilateral triangle group H = ⟨a, b; a^i, bi, (ab)^i⟩ with i > 9 arbitrary. If Q is a countable group then there exists an HNN-extension G_Q of H such that Q is isomorphic to the outer automorphism group of G_Q. The proof of this second main result applies a theory of Wise underlying his recent work leading to the resolution of the virtually fibering and virtually Haken conjectures. We prove a technical theorem which, in a certain sense, writes down a specific subgroup of the outer automorphism group of a semi-direct product

Automorphisms of free products of groups

The symmetric automorphism group of a free product is a group rich in algebraic structure and with strong links to geometric configuration spaces. In this thesis I describe in detail and for the first time the (co)homology of the symmetric automorphism groups. To this end I construct a classifying space for the Fouxe-Rabinovitch automorphism group, a large normal subgroup of the symmetric automorphism group. This classifying space is a moduli space of 'cactus products', each of which has the homotopy type of a wedge product of spaces. To study this space we build a combinatorial theory centred around 'diagonal complexes' which may be of independent interest. The diagonal complex associated to the cactus products consists of the set of forest posets, which in turn characterise the homology of the moduli spaces of cactus products. The machinery of diagonal complexes is then turned towards the symmetric automorphism groups of a graph product of groups. I also show that symmetric automorphisms may be determined by their categorical properties and that they are in particular characteristic of the free product functor. This goes some way to explain their occurence in a range of situations. The final chapter is devoted to a class of configuration spaces of Euclidean n-spheres embedded disjointly in (n+2)-space. When n = 1 this is the configuration space of unknotted, unlinked loops in 3-space, which has been well studied. We continue this work for higher n and find that the fundamental groups remain unchanged. We then consider the homology and the higher homotopy groups of the configuration spaces. Our last contribution is an epilogue which discusses the place of these groups in the wider field of mathematics. It is the functoriality which is important here and using this new-found emphasis we argue that there should exist a generalised version of the material from the final chapter which would apply to a far wider range of configuration spaces