Outer Automorphism Group Research Articles

Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap

We study equivalence relations \mathcal R(\Gamma\curvearrowright G) that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action \Gamma\curvearrowright G is free and has spectral gap, we describe precisely when \mathcal R(\Gamma\curvearrowright G) is orbit equivalent or Borel reducible to another such equivalence relation \mathcal R(\Lambda\curvearrowright H) . As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of SL_2(\mathbb Z) and its non-amenable subgroups (e.g. \mathbb F_n , with 2\leqslant n\leqslant\infty ) whose orbit equivalence relations are mutually not orbit equivalent and not Borel reducible. In particular, we show that if S and T are distinct sets of primes, then the orbit equivalence relations associated to the actions SL_2(\mathbb Z)\curvearrowright\prod_{p\in S}SL_2(\mathbb Z_p) and SL_2(\mathbb Z)\curvearrowright\prod_{p\in T}SL_2(\mathbb Z_p) are neither orbit equivalent nor Borel reducible. This settles a conjecture of S. Thomas [Th01,Th06]. Other applications include the first calculations of outer automorphism groups for concrete treeable p.m.p. equivalence relations, and the first concrete examples of free ergodic p.m.p. actions of \mathbb F_{\infty} whose orbit equivalence relations have trivial fundamental group.

Principal spin-bundles and triality

In this paper we construct a family of spin Lie groups G with an outer automorphism of order three (triality automorphism) and we describe the subgroups of fixed points for this kind of automorphisms. We will take advantage of this work to study the action of the group of outer automorphisms of G on the moduli space of principal G-bundles and describe the subvariety of fixed points in M(G) for the action of the outer automorphism of order three of G. Finally, we further study the case of Spin(8, C).

Automorphisms of p-local compact groups

Self equivalences of classifying spaces of p-local compact groups are well understood by means of the algebraic structure that gives rise to them, but explicit descriptions are lacking. In this paper we use Robinson's construction of an amalgam G, realising a given fusion system, to produce a split epimorphism from the outer automorphism group of G to the group of homotopy classes of self homotopy equivalences of the classifying space of the corresponding p-local compact group.

Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams

Abstract Let 𝔤 ${{\mathfrak{g}}}$ be a simple Lie algebra. Let Aut ⁢ ( 𝔤 ) ${{\rm Aut}(\mathfrak{g})}$ be the group of all automorphisms on 𝔤 ${\mathfrak{g}}$ , and let Int ⁢ ( 𝔤 ) ${{\rm Int}(\mathfrak{g})}$ be its identity component. The outer automorphism group of 𝔤 ${\mathfrak{g}}$ is defined as Aut ⁢ ( 𝔤 ) / Int ⁢ ( 𝔤 ) ${{\rm Aut}(\mathfrak{g})/{\rm Int}(\mathfrak{g})}$ . If 𝔤 ${\mathfrak{g}}$ is complex and has Dynkin diagram D ${{\rm D}}$ , then Aut ⁢ ( 𝔤 ) / Int ⁢ ( 𝔤 ) ${{\rm Aut}(\mathfrak{g})/{\rm Int}(\mathfrak{g})}$ is isomorphic to Aut ⁢ ( D ) ${{\rm Aut}({\rm D})}$ . We provide an analogous result for the real case. For 𝔤 ${\mathfrak{g}}$ real, we let 𝔤 ${\mathfrak{g}}$ be represented by a painted diagram P ${{\rm P}}$ . Depending on whether the Cartan involution of 𝔤 ${\mathfrak{g}}$ belongs to Int ⁢ ( 𝔤 ) ${{\rm Int}(\mathfrak{g})}$ , we show that Aut ⁢ ( 𝔤 ) / Int ⁢ ( 𝔤 ) ${{\rm Aut}(\mathfrak{g})/{\rm Int}(\mathfrak{g})}$ is isomorphic to Aut ⁢ ( P ) ${{\rm Aut}({\rm P})}$ or Aut ⁢ ( P ) × ℤ 2 ${{\rm Aut}({\rm P})\times\mathbb{Z}_{2}}$ . This result extends to the outer automorphism groups of all real semisimple Lie algebras.

Existence of Gradings on Associative Algebras

In this article, we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra A does not have a nontrivial grading if and only if A is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist nontrivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.

Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

We show that if $G$ is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous ergodic invariant random subgroups. If $G$ is a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous $G$-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary $p$-groups of countably infinite rank.

The outer automorphism groups of two-generator, one-relator groups with torsion

The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani–Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.

McCool groups of toral relatively hyperbolic groups

The outer automorphism group Out(G) of a group G acts on the set of conjugacy classes of elements of G. McCool proved that the stabilizer $Mc(c_1,...,c_n)$ of a finite set of conjugacy classes is finitely presented when G is free. More generally, we consider the group $Mc(H_1,...,H_n)$ of outer automorphisms $\Phi$ of G acting trivially on a family of subgroups $H_i$, in the sense that $\Phi$ has representatives $\alpha_i$ with $\alpha_i$ equal to the identity on $H_i$. When G is a toral relatively hyperbolic group, we show that these two definitions lead to the same subgroups of Out(G), which we call "McCool groups" of G. We prove that such McCool groups are of type VF (some finite index subgroup has a finite classifying space). Being of type VF also holds for the group of automorphisms of G preserving a splitting of G over abelian groups. We show that McCool groups satisfy a uniform chain condition: there is a bound, depending only on G, for the length of a strictly decreasing sequence of McCool groups of G. Similarly, fixed subgroups of automorphisms of G satisfy a uniform chain condition.

Combinatorial categories and permutation groups

The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group Γ , with automorphism group isomorphic to Γ / N . It is shown how to enumerate such objects with a given finite automorphism group G , how to represent them all as quotients of a single regular object U ( G ) , and how the outer automorphism group of Γ acts on them. Examples constructed include kaleidoscopic maps with trinity symmetry.

On nilpotency of the group of outer class-preserving automorphisms of a group

Let G be a group and Zj(G), for j ≥ 0, be the jth term in the upper central series of G. We prove that if Outc(G/Zj(G)), the group of outer class-preserving automorphisms of G/Zj(G), is nilpotent of class k, then Outc(G) is nilpotent of class at most j + k. Moreover, if Outc(G/Zj(G)) is a trivial group, then Outc(G) is nilpotent of class at most j. As an application we prove that if γi(G)/γi(G) ∩ Zj(G) is cyclic then Outc(G) is nilpotent of class at most i + j, where γi(G), for i ≥ 1, denotes the ith term in the lower central series of G. This extends an earlier work of the author, where this assertion was proved for j = 0. We also improve bound on the nilpotency class of Outc(G) for some classes of nilpotent groups G.

Explicit examples of equivalence relations and II₁ factors with prescribed fundamental group and outer automorphism group

In this paper we give a number of explicit constructions for II 1 _1 factors and II 1 _1 equivalence relations that have prescribed fundamental group and outer automorphism group. We construct factors and relations that have uncountable fundamental group different from R + ∗ \mathbb {R}_{+}^{\ast } . In fact, given any II 1 _1 equivalence relation, we construct a II 1 _1 factor with the same fundamental group. Given any locally compact unimodular second countable group G G , our construction gives a II 1 _1 equivalence relation R \mathcal {R} whose outer automorphism group is G G . The same construction does not give a II 1 _1 factor with G G as outer automorphism group, but when G G is a compact group or if G = S L n ± R = { g ∈ G L n R ∣ det ( g ) = ± 1 } G=\mathrm {SL}^{\pm }_n\mathbb {R}=\{g\in \mathrm {GL}_n\mathbb {R}\mid \det (g)=\pm 1\} , then we still find a type II 1 _1 factor whose outer automorphism group is G G .

Coarse median structures and homomorphisms from Kazhdan groups

We study Bowditch’s notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly left-invariant coarse median structure admits only finitely many conjugacy classes of homomorphisms from a given group with Kazhdan’s property (T). This is a common generalization of a theorem due to Paulin about the outer automorphism group of a hyperbolic group with property (T) as well as of a result of Behrstock–Druţu–Sapir on the mapping class groups of orientable surfaces. We discuss a metric approximation property of finite subsets in coarse median spaces extending the classical result on approximation of Gromov hyperbolic spaces by trees.

Group-type subfactors and Hadamard matrices

A hyperfiniteII1\mathrm {II}_1subfactor may be obtained from a symmetric commuting square via iteration of the basic construction. For certain commuting squares constructed from Hadamard matrices, we describe this subfactor as a group-type inclusionRH⊂R⋊KR^H \subset R \rtimes K, whereHHandKKare finite groups with outer actions on the hyperfiniteII1\mathrm {II}_1factorRR. We find the group of outer automorphisms generated byHHandKKand use the method of Bisch and Haagerup to determine the principal and dual principal graphs. In some cases a complete classification is obtained by examining the element ofH3(H∗K/IntR)H^3(H \ast K / \mathrm {Int} R)associated with the action.

Rholography, black holes and Scherk-Schwarz

We present a construction of a class of near-extremal asymptotically flat black hole solutions in four (or five) dimensional gauged supergravity with R-symmetry gaugings obtained from Scherk-Schwarz reductions on a circle. The entropy of these black holes is counted holographically by the well known MSW (or D1/D5) system, with certain twisted boundary conditions labeled by a twist parameter ρ. We find that the corresponding (0, 4) (or (4, 4)) superconformal algebras are exactly those studied by Schwimmer and Seiberg, using a twist on the outer automorphism group. The interplay between R-symmetries, ρ-algebras and holography leads us to name our construction “Rholography”.

An algorithm to detect full irreducibility by bounding the volume of periodic free factors

We provide an effective algorithm for determining whether an element φ of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list that can be checked for periodic proper free factors.

Right-angled Artin groups and Out($\mathbb F_n$) – I. Quasi-isometric embeddings

We construct quasi-isometric embeddings from right-angled Artin groups into the outer automorphism group of a free group. These homomorphisms are modeled on the homomorphisms into the mapping class group constructed by Clay, Leininger, and Mangahas. Toward this goal, we develop tools in the free group setting that mirror those for surface groups and discuss various analogs of subsurface projection.

Computations in formal symplectic geometry and characteristic classes of moduli spaces

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights. From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups \mathrm{Out}\, F_n of free groups for all n\leq 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.

On conjugacy of maximal abelian subalgebras and the outer automorphism group of the Cuntz algebra

We investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of maximal abelian subalgebras in . In particular, we exhibit an uncountable family of maximal abelian subalgebras, conjugate to the standard maximal abelian subalgebra via Bogolubov automorphisms, that are not inner conjugate to .

Class-preserving automorphisms of certain HNN extensions

We show that, for any integers γ,β, class-preserving automorphisms of the HNN extension 〈B,t:t−1hγt=hβ〉 of a finitely generated nilpotent or free group B are all inner. It follows from Grossman's result that outer automorphism groups of such conjugacy separable HNN extensions are residually finite.

Integral Euler Characteristic of Out F11

We show that the integral Euler characteristic of the outer automorphism group of the free group of rank 11 is −1202.