arXiv research

Abstract commensurators of ${ m{Aut}}(F_N)$ and ${ m{IA}}_N$ are the same.

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…

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If $m$ is less than $n$ then a homomorphism $Aut(F_n)\to Aut(F_m)$ can have cardinality at most 2. More generally, this is true of homomorphisms from $\Aut(F_n)$ to any group that does not contain an isomorph…

We study the complex of partial bases of a free group, which is an analogue for $\Aut(F_n)$ of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of $\Aut(F_n)$ is highly connected. Using these results, we give a…

The free factor graph for Aut(F_N) is not hyperbolic.

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT$(0)$ spaces. If $n\ge 4$ then each of the Nielsen generators of Aut$(F_n)$ has a fixed point. If $n=3$ then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated $…

Study automorphism groups of geodesic currents and measured laminations on surfaces.

We develop an analogue of the Birman exact sequence for the Torelli subgroup of Aut(F_n). This builds on earlier work of the authors who studied an analogue of the Birman exact sequence for the entire group Aut(F_n). These results play an important role in the authors' recent work on the second homology group of the To…

For a real, non-singular, 2-step nilpotent Lie algebra $\mathfrak{n}$, the group \Aut(\mathfrak{n})/\Aut_0(\mathfrak{n})$, where$\Aut_0(\mathfrak{n})$ is the group of automorphisms which act trivially on the center, is the direct product of a compact group with the 1-dimensional group of dilations. Maximality of some …

Let $F_n$ be the free group on $n$ generators and $Γ_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Γ_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-in…

Automorphisms of Hessenberg varieties are algebraic tori of dimension n-1.

In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q={\rm Conj}(G)$ of a group $G$ we determine several subgroups of ${\rm Aut}(Q)$ and find necessary and sufficient conditions when these subgroups coincide with the whole group ${\rm Aut}(Q)$. In particular, we p…

The paper studies the non-discrete automorphisms of projective manifolds.

The paper proves properties of complex Finsler metrics on specific domains.

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex X of the curve complex C(S) such that every locally injective simplicial map from X into C(S) is the restriction of an element of Aut(C(S)), unique up to the (finite) …

We give another proof of a theorem of Hatcher and Vogtmann stating that the sequence $Aut(F_n)$ satisfies integral homological stability. The paper is for the most part expository, and we also explain Quillen's method for proving homological stability.

Study extends JB-algebra structure group results to infinite dimensions.

In this paper we conjecture the stability and vanishing of a large piece of the unstable rational cohomology of SL_n Z, of mapping class groups, and of Aut(F_n).

Certain subgroups of the groups $Aut(F_n)$ of automorphisms of a free group $F_n$ are considered. Comparing Alexander polynomials of two poly-free groups $Cb_4^+$ and $P_4$ we prove that these groups are not isomorphic, despite the fact that they have a lot of common properties. This answers the question of Cohen-Pakia…

Let $X$ be any $\mathbb{Q}$-Fano variety and $\mathrm{Aut}(X)_0$ be the identity component of the automorphism group of $X$. Let $\mathbb{G}$ be a connected reductive subgroup of $\mathrm{Aut}(X)_0$ that contains a maximal torus of $\mathrm{Aut}(X)_0$. We prove that $X$ admits a Kähler-Einstein metric if and only if $X…

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),\c…

We show that there is no uniform upper bound on |Out(Aut(A))| when A ranges over all right-angled Artin groups. This is in contrast with the cases where A is free or free abelian: for all n, Dyer-Formanek and Bridson-Vogtmann showed that Out(Aut(F_n)) = 1, while Hua-Reiner showed |Out(Aut(Z^n)| = |Out(GL(n,Z))| < 5. We…

Let IA_n be the Torelli subgroup of Aut(F_n). We give an explicit finite set of generators for H_2(IA_n) as a GL_n(Z)-module. Corollaries include a version of surjective representation stability for H_2(IA_n), the vanishing of the GL_n(Z)-coinvariants of H_2(IA_n), and the vanishing of the second rational homology grou…

Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice Λ_{24} we define Λ_{24}(v…

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 $w_n$. W…

The paper studies the automorphism groups of specific 3-manifolds and finds upper bounds for their sizes.

Study on a metric for disk automorphisms with maximal modulus.

Let $A_1,...,A_k$ be a system of free factors of $F_n$. The group of relative automorphisms $Aut(F_n;A_1,...,A_k)$ is the group given by the automorphisms of $F_n$ that restricted to each $A_i$ are conjugations by elements in $F_n$. The group of relative outer automorphisms is defined as $Out(F_n;A_1,...,A_k) = Aut(F_n…

Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature Kähler metrics in $c_1(L)$ and its automorphism group Aut$(M,L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case when Aut$(V,L)$ i…

A closed Riemann surface $S$ (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map $β:S \to E$ with at most one branch value, where $E$ is a genus one Riemann surface. In this case, $(S,β)$ is called an origami pair and ${\rm Aut}(S,β)$ is the group of conformal automorphisms $φ…

Let $\mathcal{A} = {A_1, ..., A_k}$ be a system of free factors of $F_n$. The group of relative automorphisms $\mathrm{Aut}(F_n; \mathcal{A})$ is the group given by the automorphisms of $F_n$ that restricted to each $A_i$ are conjugations by elements in $F_n$. The group of relative outer automorphisms is defined as $\m…

In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a new topology on …

Given a free factor A of the rank n free group F_n, we characterize when the subgroup of Out(F_n) that stabilizes the conjugacy class of A is distorted in Out(F_n). We also prove that the image of the natural embedding of Aut(F_{n-1}) in Aut(F_n) is nondistorted, that the stabilizer in Out(F_n) of the conjugacy class o…

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Aut_0(M,J)\neq0$. Moreover, we also show that when $Aut_0(M,J)\neq0$, the properness of …

Study automorphism group actions on Jacobi diagrams spaces.

The twin group $T_n$ is a right angled Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection from $T_n$ onto the symmetric group on $n$ symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conj…

Given a CAT(0) cube complex X, we show that if Aut(X) $\neq$ Isom(X) then there exists a full subcomplex of X which decomposes as a product with $\mathbb{R}^n$. As applications, we prove that if X is $δ$-hyperbolic, cocompact and 1-ended, then Aut(X) $=$ Isom(X) unless X is quasi-isometric to $\mathbb{H}^2$, and extend…

For n > 2, the Dehn functions of Aut(F_n) and Out(F_n) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case n=3 was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for n>4 to the case n=3. In …

The paper classifies groups containing incommensurable lattices in Baumslag-Solitar complexes.

Study embeddings of free group products into automorphism groups.

We construct locally homogeneous 6-dimensional nearly Kähler manifolds as quotients of homogeneous nearly Kähler manifolds $M$ by freely acting finite subgroups of $Aut_0(M)$. We show that non-trivial such groups do only exists if $M=S^3\times S^3$. In that case we classify all freely acting subgroups of $Aut_0(M)=SU (…

Graph theory connects automorphisms to cohomology.

We refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an ex…

Normal closures of certain finite subgroups in automorphism and outer automorphism groups of free groups are determined.

We describe natural abelian extensions of the Lie algebra $\aut(P)$ of infinitesimal automorphisms of a principal bundle over a compact manifold $M$ and discuss their integrability to corresponding Lie group extensions. Already the case of a trivial bundle $P = M \times K$ is quite interesting. In this case, we show th…

It is shown that a simple Lie group $G$ ($\neq {\rm SL}_2$) can be locally characterised by an integrability condition on an $\operatorname{Aut}(\mathfrak{g})$ structure on the tangent bundle, where $\operatorname{Aut}(\mathfrak{g})$ is the automorphism group of the Lie algebra of $G$. The integrability condition is t…

Let $M$ be a connected smooth manifold, let $\operatorname{Aut}(p)$ be the group automorphisms of the bundle $p\colon \mathbb{R}\times M\to \mathbb{R}$, and let $q\colon J^1(\mathbb{R},M)\times \mathbb{R\to }J^1(\mathbb{R},M)$ be the canonical projection. Invariant functions on $J^r(q)$ under the natural action of $\op…

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 t…