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 isomorphic copy of the symmetric group $S_{n+1}$. Strong constraints are also obtained on maps to groups that do not contain a copy of $W_n= (\Bbb Z/2)^n\rtimes S_n$, or of $\Bbb Z^{n-1}$. These results place constraints on how $\Aut(F_n)$ can act. For example, if $n\ge 3$ then any action of $\Aut(F_n)$ on the circle (by homeomorphisms) factors through $\text{\rm{det}}:Aut(F_n) \to \Bbb Z_2$ .
This is a biased survey for the Johnson homomorphisms of the automorphism groups of free groups. We just exposit some well known facts and recent developments for the Johnson homomorphisms and its related topics.
We show that the automorphism group of every zero entropy infinite shift admits a "drift" homomorphism to $(\mathbb{R},+)$ that maps the shift map to 1. This homomorphism arises as the expectation, under an invariant measure, of a cocycle defined on a space of asymptotic pairs.
We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization $V$ of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Johnson homomorphism does not extend to the automorphism group of a free group as a crossed homomorphism for the rank of the free group is greater than 4.
We give a complete classification of homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on $n$ strands when $n$ is at least 7. In particular, we show that each nontrivial homomorphism extends to an automorphism of the braid group on $n$ strands. This answers four questions of Vladimir Lin. Our main new tool is the theory of totally symmetric sets.
We study the outer automorphism group of a right-angled Artin group $A_Γ$ with finite defining graph $Γ$. We construct a subnormal series for $Out(A_Γ)$ such that each consecutive quotient is either finite, free-abelian, $GL(n,\mathbb{Z})$, or a Fouxe-Rabinovitch group. The last two types act respectively on a symmetric space or a deformation space of trees, so that there is a geometric way of studying each piece. As a consequence we prove that the group $Out(A_Γ)$ is type VF (it has a finite index subgroup with a finite classifying space). The main technical work is a study of relative outer …
On Nilpotent and Hypocentral Quandle Homomorphisms
Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University（大阪中央高等数学研究所，大阪公立大学）；Department of Mathematics, School of Science and Engineering, Waseda University（早稻田大学理工学术院数学系）
We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [Algebr. Geom. Topol. 7 (2007):1297-1326]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent …
In this paper, we consider certain descending filtrations of the $\mathrm{SL}(m,\C)$-representation algebras of free groups and free abelian groups. By using it, we introduce analogs of the Johnson homomorphisms of the automorphism groups of free groups. We show that the first homomorphisms are extended to the automorphism groups of free groups as crossed homomorphisms. Furthermore we show that the extended crossed homomorphisms induce Kawazumi's cocycles and Morita's cocycles. This works are generalization of our previous results for the $\mathrm{SL}(2,\C)$-representation algebras.
We study representations of the pure symmetric automorphism group $PAut(A_Γ)$ of a RAAG $A_Γ$ with defining graph $Γ$. We first construct a homomorphism from $PAut(A_Γ)$ to the direct product of a RAAG and a finite direct product of copies of $F_2 \times F_2$; moreover, the image of $PAut(A_Γ)$ under this homomorphism is surjective onto each factor. As a consequence, we obtain interesting actions of $PAut(A_Γ)$ on non-positively curved spaces We then exhibit, for connected $Γ$, a RAAG which property contains $Inn(A_Γ)$ and embeds as a normal subgroup of $PAut(A_Γ)$. We end with a discussion of…
We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety. We also show that every finite group and every torus appears as $Aut(X)$ for a suitable affine variety $X$, but that $Aut(X)$ cannot be isomorphic to a semisimple group. In fact, if $Aut(X)$ is finite dimensional and $X$ not isomorphic to the affine line $A^1$, then the connected component $Aut(X)^0$ is a torus. Concerning the structure of $Aut(A^n)$ we pro…
Let F_n be the free group on n generators, and PΣ_n be the group of automorphisms of F_n which send each generator to a conjugate of itself. Let K_n be the kernel of the homomorphism from PΣ_n to PΣ_{n-1} induced by mapping one of the free group generators to the identity. We show that K_n has cohomological dimension n-1, and that the ith cohomology groups are infinitely generated for all i between 2 and n-1. It follows that K_n is not finitely presentable for n>2.
Let G be a real semisimple Lie group with no compact factors and finite centre, and let $Λ$ be a lattice in G. Suppose that there exists a homomorphism from $Λ$ to the outer automorphism group of a right-angled Artin group $A_Γ$ with infinite image. We give an upper bound to the real rank of G that is determined by the structure of cliques in $Γ$. An essential tool is the Andreadakis-Johnson filtration of the Torelli subgroup $\mathcal{T}}(A_Γ)$ of $Aut(A_Γ)$. We answer a question of Day relating to the abelianisation of $\mathcal{T}}(A_Γ)$, and show that $\mathcal{T}}(A_Γ)$ and its image in $…
We classify homomorphisms of the Baumslag-Gersten group into itself. We prove it is Hopfian and co-Hopfian. We show that the group of outer automorphisms of the Baumslag-Gersten group is isomorphic to the dyadic rationals with the addition operation. These results are not new. They were obtained by Andrew M. Brunner in \cite{Brunner}. However, our exposition is self-contained and, hopefully, more accessible for some readers.
We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \to Out(F_m). We prove that if n > 8 is even and n \neq m \leq 2n, or n is odd and n \neq m \leq 2n - 2, then all such homomorphisms have finite image; in fact they factor through det: Out(F_n) \to Z/2. In contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(F_n) \to Out(F_m). In order to prove this last statement, we determine when the action of Out(F_n) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with a…
Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, double cosets of the group of label preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups are enumerated. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.
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 …
We study the existence of homomorphisms between Out(F_n) and Out(F_m) for n > 5 and m < n(n-1)/2, and conclude that if m is not equal to n then each such homomorphism factors through the finite group of order 2. In particular this provides an answer to a question of Bogopol'skii and Puga. In the course of the argument linear representations of Out(F_n) in dimension less than n(n+1)/2 over fields of characteristic zero are completely classified. It is shown that each such representation has to factor through the natural projection from Out(F_n) to GL(n,Z) coming from the action of Out(F_n) on t…
Every homomorphism from finite index subgroups of a universal lattices to mapping class groups of orientable surfaces (possibly with punctures), or to outer automorphism groups of finitely generated nonabelian free groups must have finite image. Here the universal lattice denotes the special linear group G=SL_m(Z[x1,...,xk]) with m at least 3 and k finite. Moreover, the same results hold ture if universal lattices are replaced with symplectic universal lattices Sp_{2m}(Z[x1,...,xk]) with m at least 2. These results can be regarded as a non-arithmetization of the theorems of Farb--Kaimanovich--…
We apply the method of Arzhantseva-Ol'shanskii to prove that for an exponentially generic (in the sense of Ol'shanskii) class of one-relator groups the isomorphism problem is solvable in at most exponential time. This is obtained as a corollary of our more general result that for any fixed integers $m>1, n>0$ there is an exponentially generic class of $m$-generator $n$-relator groups where every group has only one Nielsen equivalence class of $m$-tuples generating non-free subgroups. This means that a group $G$ in this class has has only one non-free $m$-generated subgroup, namely $G$ itself. …
We study fixed point properties of the automorphism group of the universal Coxeter group Aut$(W_n)$. In particular, we prove that whenever Aut$(W_n)$ acts by isometries on complete $d$-dimensional CAT$(0)$ space with $d<\lfloor\frac{n}{2}\rfloor$, then it must fix a point. We also prove that Aut$(W_n)$ does not have Kazhdan's property (T). Further, strong restrictions are obtained on homomorphisms of Aut$(W_n)$ to groups that do not contain a copy of Sym(n).
Let F_n be the free group on n generators. Define IA_n to be group of automorphisms of F_n that act trivially on first homology. The Johnson homomorphism in this setting is a map from IA_n to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology of IA_n. A descending central series of IA_n is given by the subgroups K_n^(i) which act trivially on F_n/F_n^(i+1), the free rank n, degree i nilpotent group. It is a conjecture of Andreadakis that K_n^(i) is equal to the lower central series of IA_n; indeed K_n^(2) is known to…
We give a condition on the defining graph of a right-angled Artin group which implies its automorphism group is virtually indicable, that is, it has a finite-index subgroup that admits a homomorphism onto $\Z$. We use this as 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 paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the c…
Let $f:G\rightarrow H$ be a homomorphism of groups, we construct a topological space $X_f$ such that its group of homeomorphisms is isomorphic to $G$, its group of homotopy classes of self-homotopy equivalences is isomorphic to $H$ and the natural map between the group of homeomorphisms of $X_f$ and the group of homotopy classes of self-homotopy equivalences of $X_f$ is precisely $f$. In addition, realization problems involving homology, homotopy groups and groups of automorphisms are considered.
If $G$ is a semisimple Lie group of real rank at least 2 and $Γ$ is an irreducible lattice in $G$, then every homomorphism from $Γ$ to the outer automorphism group of a finitely generated free group has finite image.
Let $φ\colonΓ\to G$ be a homomorphism of groups. We consider factorizations $Γ\xrightarrow{f} M\xrightarrow{g} G$ of $φ$ such that either $g$ or $f$ are universal normal maps (namely, crossed modules). These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup. Iterating these universal factorizations yield towers related, on the one hand, to hypercentral group extensions, Bousfield's localizations, and relative Schur multipliers. Dually, they extend to a relative situation, the automorphisms tower of a centerless group. Our constructions have…
We discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their abelianizations.
The Johnson filtration of the automorphism group of a free group is composed of those automorphisms which act trivially on nilpotent quotients of the free group. We compute cohomology classes as follows: (i) we analyze analogous classes for a subgroup of the pure symmetric automorphism group of a free group, and (ii) we analyze features of these classes which are preserved by the Johnson homomorphism. One consequence is that the ranks of the cohomology groups in any fixed dimension between 1 and n-1 increase without bound for terms deep in the Johnson filtraton.
Let G be any finitely generated infinite group. Denote by K(G) the FC-centre of G, i.e., the subgroup of all elements of G whose centralizers are of finite index in G. Let QI(G) denote the group of quasi-isometries of G with respect to word metric. We observe that the natural homomorphism from the group of automorphisms of G to QI(G) is a monomorphism only if K(G) equals the centre Z(G) of G. The converse holds if K(G)=Z(G) is torsion free. We apply this criterion to many interesting classes of groups.
Let $G$ be a group. The central automorphism group $Aut_c(G)$ of $G$ is the centralizer of $Inn(G)$ the subgroup of $Aut(G)$ of inner automorphisms. There is a one to one map $ σ\mapsto h_σ$ from the set $Aut_c(G)$ onto the set $Hom(G,Z(G))$ of homomorphisms from $G$ onto its center, with $ h_σ(x)=x^{-1} σ(x)$. This map can be used to obtain informations about the size of $Aut_c(G)$, and also about its structure in some special cases. In this paper we see how to use it to obtain informations about the structure of $Aut_c(G)$ in the general case. The notion of the adjoint group of a ring is the…
It is well known that if a group G factorizes as G = NH where H\leq G and N is normal in G then the group structure of G is determined by the subgroups H and N, the intersection of N with H and how H acts on N with a homomorphism f : H -> Aut(N). Here we generalize the idea by creating extensions using the semi-automorphism group of N. We show that if G=NH is a Moufang loop, N is a normal subloop, and H=<u> is a finite cyclic group of order coprime to three then the binary operation of G depends only on the binary operation of N, the intersection of N with H, and how u permutes the elements of…
We construct a model for the space of automorphisms of a connected p-compact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a p-compact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper ``The classification of 2-compact groups'', where we prove the conjectured classification of 2-compact groups and determine their automorphism spaces.
We give a counterexample to a conjecture by Miasnikov, Ventura and Weil, stating that an extension of free groups is algebraic if and only if the corresponding morphism of their core graphs is onto, for every basis of the ambient group. In the course of the proof we present a partition of the set of homomorphisms between free groups which is of independent interest.
We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical {\em core} $\mathcal{J}$ associated to such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of $\mathcal{J}$, modulo certain infinitesimal automorphisms, is a locally compact group $\mathcal{G}$. The automorphism groups of models of the theory are related with $\mathcal{G}$, not in general via a homomorphism, but by a {\em …
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-Drutu-Sapir on the mapping class groups of ori…
It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group automorphisms that can be coherently extended along any outgoing homomorphism. One is thus motivated to define a notion of (categorical) inner automorphism in an arbitrary category, as an automorphism that can be coherently extended along any outgoing morphism, and the theory of such automorphisms forms part of the theory of covariant isotropy. In this paper, we prove that the categorical inner automorphisms in any category $\mathsf{Group}^{\mathcal{J}}$ of p…
For any natural n, we construct an aleph_n-free abelian groups which have few homomorphisms to Z . For this we use ``aleph_n-free (n+1)-dimensional black boxes''. The method is relevant to e.g. construction of aleph_n-free abelian groups with a prescribed endomorphism ring.
A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.
The paper is devoted to a study of certain fixed point properties, and their relatives, in the context of full automorphism groups of countable rooted trees. Namely, we study Serre's property (FA'), also called unsplittability, property (FA), the uncountable strong cofinality, and ample generics. We give a new proof of a theorem of Psaltis to the extent that automorphism groups of rooted trees are unsplittable, and show under what circumstances automorphism groups of rooted trees have uncountable strong cofinality (and thus property (FA).) Also, we prove a necessary and sufficient condition fo…
We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.
We compute the virtual cohomological dimension (VCD) 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 two-dimensional right-angled Artin group. In the case of a right-angled group with defining graph a tree, the bounds agree.
We prove that the automorphism group of an arbitrary non-abelian free group is complete. It generalizes the result by J.Dyer and E.Formanek (1975) stating the completeness of automorphism group of finitely generated free groups. Using the description of involutions in automorphism groups of free groups (J. Dyer, P. Scott, 1975) we obtain a group-theoretic characterization of inner automorphisms determined by primitive elements in the automorphism group of any non-abelian free group F. It follows that the subgroup Inn(F) is characteristic in Aut(F), and hence the latter one is complete.
We show that several classes of groups G of PL-homeomorphisms of the real line admit non-trivial homomorphisms from G to the additive group of reals that are fixed by every automorphism of G. The classes of groups enjoying the stated property include the generalisations of Thompson's group F studied by K. S. Brown, M. Stein, S. Cleary, and Bieri-Strebel but also the class of groups investigated by Bieri-Neumann-Strebel in [BNS87, Theorem 8.1]. It follows that every automorphism of a group in one of these classes has infinitely many associated twisted conjugacy classes.
In this paper, we initiate the study of palindromic automorphisms of groups that are free in some variety. More specifically, we define palindromic automorphisms of free nilpotent groups and show that the set of such automorphisms is a group. We find a generating set for the group of palindromic automorphisms of free nilpotent groups of step 2 and 3. In particular, we obtain a generating set for the group of central palindromic automorphisms of these groups. In the end, we determine central palindromic automorphisms of free nilpotent groups of step 3 which satisfy the necessary condition of Br…
In this paper, we study some properties of the outer automorphism group of free Burnside groups of large odd exponent. In particular, we prove that it contains free and free abelian subgroups.
We prove that if the order of a splitting automorphism of free Burnside group~$B(m,n)$ of odd period~$n\ge1003$ is a prime power, then the automorphism is inner. Thus, we give an affirmative answer to the question on the coincidence of splitting and inner automorphisms of free Burnside groups $B(m,n)$ for automorphisms of orders~$p^k$ ($p$ is a prime number). This question was posed in the Kourovka Notebook in 1990 (see 11th ed., Question 11.36.~b).
In this thesis I study the automorphism tower of free nilpotent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group. This allows me to lift the description of the derivation tower of the free nilpotent Lie algebra to obtain information about the automorphism tower of the free nilpotent group. The main result in this thesis states that the automorphism tower of the free nilpotent group $Γ(n,d)$ on $n$ …
