Outer Automorphism Group Research Articles

OUTER AUTOMORPHISMS OF MAPPING CLASS GROUPS OF NONORIENTABLE SURFACES

Let Ng be the connected closed nonorientable surface of genus g ≥ 5 and Mod (Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod (Ng) is cyclic.

Outer automorphism groups of certain 1-relator groups

Grossman first showed that outer automorphism groups of 1-relator groups given by orientable surface groups are residually finite, whence mapping class groups of orientable surfaces are residually finite. Allenby, Kim and Tang showed that outer automorphism groups of cyclically pinched 1-relator groups are residually finite, whence mapping class groups of orientable and non-orientable surfaces are residually finite. In this paper we show that outer automorphism groups of certain conjugacy separable 1-relator groups are residually finite.

Outer automorphism groups of certain 1-relator groups

Outer automorphism groups of certain 1-relator groups

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.

Hypermap operations of finite order

Duality and chirality are examples of operations of order 2 on hypermaps. James showed that the groups of all operations on hypermaps and on oriented hypermaps can be identified with the outer automorphism groups Out Δ ≅ P G L 2 ( Z ) and Out Δ + ≅ G L 2 ( Z ) of the groups Δ = C 2 ∗ C 2 ∗ C 2 and Δ + = F 2 . We will consider the elements of finite order in these two groups, and the operations they induce.

The cohomology groups of the outer Whitehead automorphism group of a free product

ji 2 Aut() induced by gj i (gk )= g k if k 6= i g g j k if k = i (where gk 2 Gk and g g j k is shorthand for conjugating gk by gj )i s anelementary Whitehead automorphism. The Whitehead automorphism group, Wh(), is the subgroup of Aut() generated by the g j i . If none of the Gi are infinite cyclic, then Wh() is the kernel of the map Aut() Out(G1 ◊···◊ Gn). In particular, if the Gi are finite, then Wh() is a finite-index subgroup of Aut(). In this paper we compute the cohomology groups of OWh(), the quotient of Wh() in the outer-automorphism group, with field coecients. Our approach is to analyze the equivariant spectral sequence associated to the action of OWh() on a contractible, simplicial complex introduced by McCullough and Miller [10]. Throughout the paper, cohomology groups will be assumed to have field coecients unless otherwise indicated, and () n denotes the n-fold product ◊ ···◊ | {z } n copies

On abstract commensurators of groups

We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated. This applies in particular to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2. We also construct a finitely generated, torsion-free group which can be mapped onto Z and which has a finitely generated commensurator.

Automorphisms of two-dimensional RAAGS and partially symmetric automorphisms of free groups

We compute the virtual cohomological dimension of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a 2 -dimensional right-angled Artin group. In the case of a right-angled Artin group with defining graph a tree, the bounds agree.

Cohomology of Invariant Drinfeld Twists on Group Algebras

We show how to compute a certain group of equivalence classes of invariant Drinfeld twists on the algebra of a finite group G over a field k of characteristic zero. This group is naturally isomorphic to the second lazy cohomology group of the Hopf algebra of k-valued functions on G. When k is algebraically closed, the answer involves the group of outer automorphisms of G induced by conjugation in the group algebra as well as the set of all pairs (A, b), where A is an abelian normal subgroup of G and is a k×-valued G-invariant nondegenerate alternating bilinear form on the dual ⁠. When the ground field k is not algebraically closed, we use algebraic group techniques to reduce the computation of Hℓ2(G) to a computation over the algebraic closure. As an application of our results, we compute Hℓ2(G) for a number of groups.

Graded Brauer tree algebras

In this paper we construct non-negative gradings on a basic Brauer tree algebra A Γ corresponding to an arbitrary Brauer tree Γ of type ( m , e ) . We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra A S , whose tree is a star with the exceptional vertex in the middle, to A Γ . The grading on A S comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green’s walk around Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras. We also compute Out K ( A Γ ) , the group of outer automorphisms that fix the isomorphism classes of simple A Γ -modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on A Γ up to graded Morita equivalence and rescaling.

Symmetry of Reidemeister torsion on [formula omitted]-representation spaces of knots

We study two sorts of actions on the space of conjugacy classes of irreducible SU 2 -representations of a knot group. One of them is an involution which comes from the algebraic structure of SU 2 and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on a 1-dimensional smooth part of the space, which is canonically oriented and metrized via a Reidemeister torsion volume form. As an application we show that the Reidemeister torsion function on the 1-dimensional subspace has symmetry about the metrization.

Homotopical theory of periodic points of periodic homeomorphisms on closed surfaces

In this work we show that the Wecken theorem for periodic points holds for periodic homeomorphisms on closed surfaces, which therefore completes the periodic point theory in such a special case. Using it we derive the set of homotopy minimal periods for such homeomorphisms. Moreover we show that the results hold for homotopically periodic self-maps of closed surfaces. This let us to re-formulate our results as a statement on properties of elements of finite order in the group of outer automorphisms of the fundamental group of a surface with non-positive Euler characteristic.

Groups with finitely many conjugacy classes and their automorphisms

We combine classical methods of combinatorial group theory with the theory of small cancellation over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreover, we present several results concerning embeddings into such groups. As another application of these techniques, we prove that every countable group C can be realized as a group of outer automorphisms of a group N, where N is a finitely generated group having Kazhdan’s property (T) and containing exactly two conjugacy classes.

Outer automorphism groups of Seifert 3-manifold groups over non-orientable surfaces

In this paper we show that fundamental groups of Seifert fibered spaces over non-orientable surfaces have Property E. From this we deduce that the outer automorphism groups of almost all Seifert groups are residually finite. We conjecture that all outer automorphism groups of Seifert groups are residually finite. It follows that almost all mapping class groups of Seifert fibered spaces are residually finite.

Residual finiteness of outer automorphism groups of certain 1-relator groups

We prove that certain 1-relator groups have Property E. Using this fact, we characterize all conjugacy separable 1-relator groups of the form 〈a, b; (a -α b β a α b γ) t 〉, t ⩾ 1, having residually finite outer automorphism groups.

Finiteness properties of automorphism groups of right-angled Artin groups

We study the algebraic structure of the outer automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved earlier by the authors and Crisp for 2-dimensional right-angled Artin groups.

Lattices of some solvable Lie groups and actions of products of affine groups

We consider solvable Lie groups which are isomorphic to unimodularizations of products of affine groups. It is shown that a lattice of such a Lie group is determined, up to commensurability, by a totally real algebraic number field. We also show that the outer automorphism group of the lattice is represented faithfully in the automorphism group of the number field. As an application, we obtain a classification of codimension one, volume preserving, locally free actions of products of affine groups.

Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be non-hyperbolic relative to any non-trivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of non-relatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil–Peterson metric, in contrast with Brock–Farb’s hyperbolicity result in low complexity.

OUTER AUTOMORPHISM GROUPS OF POLYGONAL PRODUCTS OF CERTAIN CONJUGACY SEPARABLE GROUPS

Grossman [7] showed that certain cyclically pinched 1-relator groups have residually finite outer automorphism groups. In this paper we prove that tree products of finitely generated free groups amalgamating maximal cyclic subgroups have residually finite outer automorphism groups. We also prove that polygonal products of finitely generated central subgroup separable groups amalgamating trivial intersecting central subgroups have residually finite outer automorphism groups.

Outer automorphism groups of equivalence relations for mapping class group actions

We study outer automorphism groups Out(ℛ) of equivalence relations ℛ arising from ergodic, essentially free and measure-preserving actions on standard probability spaces of mapping class groups, and compute them explicitly for some examples. We show that the subgroups of Out(ℛ) associated with action-automorphisms are always finite index ones. Moreover, a sharp upper bound of the indices among all such actions of mapping class groups is obtained.