Outer Automorphism Group Research Articles

Dynamics of fully irreducible automorphisms of free products

Let G be a free product and Out(G) the outer automorphism group of G. In this article using the theory of laminations we give a criterion for a subgroup H of Out(G) to contain a nonabelian free subgroup. We also study the centraliser of a fully irreducible element of Out(G) and the stabiliser of its associated lamination.

Flux Landscape with enhanced symmetry not on SL(2, ℤ) elliptic points

We study structures of solutions for SUSY Minkowski F-term equations on two toroidal orientifolds with h2,1 = 1. Following our previous study [1], with fixed upper bounds of a flux D3-brane charge Nflux, we obtain a whole Landscape and a distribution of degeneracies of physically-distinct solutions for each case. In contrast to our previous study, we consider a non-factorizable toroidal orientifold and its Landscape on which SL(2, ℤ) is violated into a certain congruence subgroup, as it had been known in past studies. We find that it is not the entire duality group that a complex-structure modulus U enjoys but its outer semi-direct product with a “scaling” outer automorphism group. The fundamental region is enlarged to include the |U| < 1 region. In addition, we find that high degeneracy is observed at an elliptic point, not of SL(2, Z) but of the outer automorphism group. Furthermore, ℤ2-enhanced symmetry is realized on the elliptic point. The outer automorphism group is exceptional in the sense that it is consistent with a symplectic basis transformation of background three-cycles, as opposed to the outer automorphism group of SL(2, ℤ). We also compare this result with Landscape of another factorizable toroidal orientifold.

On virtual indicability and property (T) for outer automorphism groups of RAAGs

We give a condition on the defining graph of a right-angled Artin group, which implies that its automorphism group is virtually indicable, that is, it has a finite index subgroup that admits a homomorphism onto \mathbb{Z} . We use this as a part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence, we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group.

The group of splendid Morita equivalences of principal 2-blocks with dihedral and generalised quaternion defect groups

Let $k$ be an algebraically closed field of characteristic $2$, let $G$ be a finite group and let $B$ be the principal $2$-block of $kG$ with a dihedral or a generalised quaternion defect group $P$. Let also $\calT(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that \begin{align*} \calT(B)\cong \Out_P(A)\rtimes \Out(P,\calF), \end{align*} where $\Out(P,\calF)$ is the group of outer automorphisms of $P$ which stabilize the fusion system $\calF$ of $G$ on $P$ and $\Out_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by $(A^P)^\times$.

Characterizations of normal cancellative monoids

&lt;abstract&gt;&lt;p&gt;Normal cancellative monoids were introduced to explore the general structure of cancellative monoids, which are innovative and open up new possibilities. Specifically, we pointed out that the Green's relations in a cancellative monoid $ S $ are determined by its unitary subgroup $ U $ to a great extent. The specific composition of egg boxes in $ S $, derived from the general semigroup theory, can be settled by the subgroups of $ U $. We call a cancellative monoid normal when these subgroups are normal and characterize it as an NCM-system. This NCM-system was created in this article and can be obtained by combining a group and a condensed cancellative monoid. Furthermore, we introduced the concept of torsion extension and proved that a special kind of normal cancellative monoids can be constructed delicately by the outer automorphism groups of given groups and some simplified cancellative monoids.&lt;/p&gt;&lt;/abstract&gt;

Right‐angled Artin groups as finite‐index subgroups of their outer automorphism groups

AbstractWe prove that every right‐angled Artin group occurs as a finite‐index subgroup of the outer automorphism group of another right‐angled Artin group. We furthermore show that the latter group can be chosen in such a way that the quotient is isomorphic to for some . For these, we give explicit constructions using the group of pure symmetric outer automorphisms. Moreover, we need two conditions by Day–Wade and Wade–Brück about when this group is a right‐angled Artin group and when it has finite index.

The Euler characteristic of the moduli space of graphs

The moduli space of rank n graphs, the outer automorphism group of the free group of rank n and Kontsevich's Lie graph complex have the same rational cohomology. We show that the associated Euler characteristic grows like −e−1/4(n/e)n/(nlog⁡n)2 as n goes to infinity, and thereby prove that the total dimension of this cohomology grows rapidly with n.

Finite Groups Acting on Compact Complex Parallelizable Manifolds

Abstract We prove that the automorphism group of a compact complex parallelizable manifold is Jordan. In the course of the proof, we show that outer automorphism groups of cocompact lattices in complex Lie groups have bounded finite subgroups.

Commensurations of the outer automorphism group of a universal Coxeter group

This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank n , which is the free product W_n of n copies of \mathbb{Z}/2\mathbb{Z} . We prove that for n\geq 5 the natural map \mathrm{Out}(W_n) \to \mathrm{Comm}(\mathrm{Out}(W_n)) is an isomorphism and that every isomorphism between finite index subgroups of \mathrm{Out}(W_n) is given by a conjugation by an element of \mathrm{Out}(W_n) .

North-South type dynamics of relative atoroidal automorphisms of free groups on a relative space of currents

This paper, which is the second of a series of three papers, studies dynamical properties of elements of Out ⁡ ( F n ) \operatorname {Out}(F_{\mathtt n}) , the outer automorphism group of a nonabelian free group F n F_{\mathtt n} . We prove that, for every exponentially growing outer automorphism of F n F_{\mathtt n} , there exists a preferred compact topological space, the space of currents relative to a malnormal subgroup system, on which ϕ \phi acts by homeomorphism with a North-South dynamics behavior.

Quotients by countable normal subgroups are hyperfinite

We show that for any Polish group G and any countable normal subgroup \Gamma \triangleleft G , the coset equivalence relation G/\Gamma is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.

A note on virtual duality and automorphism groups of right-angled Artin groups

AbstractA theorem of Brady and Meier states that a right-angled Artin group is a duality group if and only if the flag complex of the defining graph is Cohen–Macaulay. We use this to give an example of a RAAG with the property that its outer automorphism group is not a virtual duality group. This gives a partial answer to a question of Vogtmann. In an appendix, Brück describes how he used a computer-assisted search to find further examples.

Spectral gap and strict outerness for actions of locally compact groups on full factors

We prove that an outer action of a locally compact group G on a full factor M is automatically strictly outer, meaning that the relative commutant of M in the crossed product is trivial. If moreover the image of G in the outer automorphism group Out M is closed, we prove that the crossed product remains full. We obtain this result by proving that the inclusion of M in the crossed product automatically has a spectral gap property. Such results had only been proven for actions of discrete groups and for actions of compact groups, by using quite different methods in both cases. Even for the canonical free Bogoljubov actions on free group factors or free Araki–Woods factors, these results are new.

Tate–Shafarevich groups and algebras

The Tate–Shafarevich set of a group [Formula: see text] defined by Takashi Ono coincides, in the case where [Formula: see text] is finite, with the group of outer class-preserving automorphisms of [Formula: see text] introduced by Burnside. We consider analogs of this important group-theoretic object for Lie algebras and associative algebras and establish some new structure properties thereof. We also discuss open problems and eventual generalizations to other algebraic structures.

Formal matrix rings: Automorphisms

Automorphism groups of formal matrix algebras with zero trace ideals are studied. Such an algebra is represented as a splitting extension of some ring by some nilpotent ideal. Using this extension, the study of the structure of the automorphism group of an algebra in a certain sense is reduced to the study of the structure of some its subgroups and quotient groups. Then the structure of these subgroups and factor groups is found. The case of formal triangular matrix rings is specially considered. At the end of the paper we study automorphisms of ordinary matrix rings. In this case, a well-known homomorphism from the group of outer automorphisms to the Picard group of a certain ring is used.

On the injectivity of the homomorphisms from the automorphism groups of fields to the outer automorphism groups of the absolute Galois groups

On the injectivity of the homomorphisms from the automorphism groups of fields to the outer automorphism groups of the absolute Galois groups

Homology Representations of Compactified Configurations on Graphs Applied to 𝓜2,n

We obtain new calculations of the top weight rational cohomology of the moduli spaces M2,n , equivalently the rational homology of the tropical moduli spaces , as a representation of Sn . These calculations are achieved fully for all , and partially—for specific irreducible representations of Sn —for . We also present conjectures, verified up to n = 22, for the multiplicities of the irreducible representations and . We achieve our calculations via a comparison with the homology of compactified configuration spaces of graphs. These homology groups are equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. In this paper, we construct an efficient free resolution for these homology representations, from which we extract calculations on irreducible representations one at a time, simplifying the calculation of these homology representations.

Belyi injectivity for outer representations on certain quotients of étale fundamental groups of hyperbolic curves of genus zero

In the present paper, we study certain quotients of the étale fundamental group of a hyperbolic curve over a field. We prove that the action of the outer automorphism group of a certain quotient of the étale fundamental group of a hyperbolic curve over an algebraically closed field on its conjugacy classes of open subgroups is faithful. Also, we prove that, if k is either a number field or a p-adic local field, then the outer Galois representation associated to a certain quotient of the geometric fundamental group of ℙk1\{0,1,∞} is injective.

Embedding universality for II1 factors with property (T)

We prove that every separable tracial von Neumann algebra embeds into a II1 factor with property (T) which can be taken to have trivial outer automorphism and fundamental groups. We also establish an analogous result for the trivial extension over a non-atomic probability space of every countable p.m.p. equivalence relation. In addition, we obtain two new results concerning the structure of infinitely generic II1 factors. These results are obtained by using the class of wreath-like product groups introduced recently in [8].

A generalization of cellular automata over groups

Let G be a group and let A be a finite set with at least two elements. A cellular automaton (CA) over AG is a function defined via a finite memory set and a local function . The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) , where H is another arbitrary group, via a group homomorphism . Our definition preserves the essence of CA, as we prove analogous versions of three key results in the theory of CA: a generalized Curtis-Hedlund Theorem for GCA, a Theorem of Composition for GCA, and a Theorem of Invertibility for GCA. When G = H, we prove that the group of invertible GCA over AG is isomorphic to a semidirect product of and the group of invertible CA. Finally, we apply our results to study automorphisms of the monoid consisting of all CA over AG . In particular, we show that every defines an automorphism of via conjugation by the invertible GCA defined by , and that, when G is abelian, is embedded in the outer automorphism group of . Communicated by Pedro Garcia-Sanchez