# Questions tagged [outer-automorphisms]

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30
questions

4
votes

1
answer

299
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### Can $\text{Aut}(G)$ be extended to contain $G$?

Let $G$ be a group (finite, say) with center $Z$. The automorphism group $\text{Aut}(G)$ sits in a short exact sequence
$$ 1 \to G/Z \to \text{Aut}(G) \to \text{Out}(G) \to 1. $$
So when $Z\neq 1$, as ...

5
votes

1
answer

173
views

### Explicit formula for general group extension in terms of cartesian product set

According to Wikipedia and ncat lab general group extensions
$$N\rightarrow G\rightarrow Q$$
are classified by a group homomorphism
$$\rho: Q\rightarrow \operatorname{Out}(N)$$
subject to a constraint ...

9
votes

1
answer

373
views

### Morse theory on outer space via the lengths of finitely many conjugacy classes

Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected ...

8
votes

1
answer

352
views

### Outer automorphisms of finitely generated linear groups

Is there an example of a finitely generated subgroup $\Gamma \subset \mathrm{GL}_n(\mathbb{C})$ such that the group of outer automorphisms $\mathrm{Out}(\Gamma)$ contains finite subgroups of unbounded ...

2
votes

1
answer

549
views

### Growth rate of an outer automorphism of a free product

$\DeclareMathOperator\Out{Out}$Let $G=G_1\ast\cdots\ast G_k\ast F_p$ be a Grushko decomposition of a finitely generated group $G$, $\mathcal{O}$ the outer space relative to this decomposition, $[\phi]\...

4
votes

1
answer

495
views

### What is the outer automorphism group of the Lie group $\text{SL}_2(\mathbb{R})$ as an abstract group?

I hope to ask what the outer automorphism group of the Lie group $\text{SL}_2(\mathbb{R})$ is, just as an abstract group. It seems like Dieudonné's paper On the automorphisms of the classical groups ...

3
votes

1
answer

136
views

### Pairs of elements in $F_n$ with distinct translation lengths

Let $F_n$ be a free group of rank n and consider all possible non-degenerate length functions on $F_n$.
Could I be directed to a reference that give two non-trivial non-power-conjugate elements $g,h \...

5
votes

0
answers

161
views

### Finitely generated nilpotent groups with hyperbolic automorphisms

$\DeclareMathOperator\Out{Out}\DeclareMathOperator\GL{GL}$
Let $G$ be a finitely generated nilpotent group.
We call an automorphism of $G$ hyperbolic if the induced automorphism of the free part of ...

8
votes

3
answers

419
views

### Does $O(4,\mathbb{Q})$ have an exceptional outer automorphism?

Does the orthogonal group $O(4,\mathbb{Q})$ have an exceptional outer automorphism analogous to that of its subgroup, the Coxeter/Weyl group $W(F_4)$?

7
votes

1
answer

537
views

### Does Aut(G) → Out(G) always split for a compact, connected Lie group G?

The outer automorphism group of a topological group $G$ is constructed by the short exact sequence
$$
1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \...

9
votes

0
answers

360
views

### Groups with trivial outer automorphism group and prescribed center?

Given an arbitrary abelian group $A$, can we find a group $G$ such that
$\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)=1$, and
$Z(G)\simeq A$?
Why is this interesting? Given a group $G$, we have ...

3
votes

1
answer

158
views

### Is the Singer cycle preserved by field automorphisms and graph automorphisms?

Let $T=\operatorname{PSL}_n(q)$ with $n$ a prime number. Then the $\mathscr{C}_3$ subgroup $M=\langle x\rangle{:}\langle\sigma\rangle$ of $T$ is isomorphic to $\mathbb{Z}_{\frac{q^n-1}{(q-1)(n,q-1)}}{:...

3
votes

1
answer

174
views

### Asymmetry of outer space - injectivity radius

I'd like to ask a question on "Asymmetry of Outer Space" by Yael Algom-Kfir & Mladen Bestvina.
In Example $2$, page $4$ it says "Note that in this case the asymmetry can be explained by the fact ...

4
votes

1
answer

328
views

### outer automorphism classification

I am trying to understand Bestvina's "A Bers-like proof of the existence of train tracks for free group automorphisms". I'm going to ask a probably trivial question ... Here we go:
The automorphism $\...

7
votes

0
answers

316
views

### How does Outer Space look like without a simplex?

Considering the simplicial structure of Culler and Vogtmanns Outer Space $CV_n$. The question is now:
Let $\Delta \subset CV_n$ be a closed simplex of dimension $3n-4$ or $3n-5$, how does $CV_n \...

5
votes

0
answers

187
views

### Is there an equivariant simplicial deformation retract of Teichmüller space?

Let $S_g$ be a surface of genus $g \ge 2$. By analogy with Teichmüller space for $S_g$, Culler and Vogtmann studied Outer Space $CV_n$, with points projective classes of marked metric graphs with ...

8
votes

2
answers

901
views

### For nonabelian finite simple $G$, does $Aut(G)$ have a unique subgroup isomorphic to $G$?

If $G$ is a nonabelian finite simple group, $Aut(G)$ certainly contains a subgroup isomorphic to $G$, namely $Inn(G)$. Must this be the only subgroup of $Aut(G)$ isomorphic to $G$?
I can prove this ...

3
votes

1
answer

220
views

### Fixed points of the automorphisms of sporadic groups

Sporadic groups have very few outer automorphisms (in fact, $|\mathrm{Out}(G)|\leqslant2$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable ...

3
votes

1
answer

89
views

### A partition of the set of order 2 outer automorphisms of $SU(N)$

Let $N$ be an even integer, $N>2$. Let $E$ be the set of all outer automorphisms $\phi$ of $G = SU(N)$ which are of order 2, i.e. $\phi \circ \phi = \mathrm{id}_G$.
Choose a particular element $\...

7
votes

1
answer

495
views

### $\operatorname{Out}(F_n)$ is not linear for $n > 3$

The paper The Tits alternative for $\operatorname{Out}(F_n)$ I by Bestvina, Feighn and Handel and the paper Automorphisms of free groups and Outer space by Vogtmann both state that $\operatorname{Aut}(...

14
votes

4
answers

689
views

### Non-split Aut(G) $\to$ Out(G)?

I'm looking for examples of outer automorphisms of a finite group $G$ which do not lift to automorphisms (i.e. non-split quotient map $\mathrm{Aut}(G)\to \mathrm{Out}(G)$, where $\mathrm{Out}(G) = \...

3
votes

1
answer

205
views

### Extending an automorphism to an inner one

Let $D$ be a division ring. I have in mind the following result.
Theorem. For every automorphism $f$ of $D$, there is a division ring $E$ extending $D$ such that $f$ extends to an inner automorphism ...

7
votes

1
answer

2k
views

### Automorphisms of the Lie algebras $\mathfrak{sl}(2,R)$ and $\mathfrak{su}(2)$

I would like to know about the literature concerning the group of outer automorphisms of the Lie algebra $\mathfrak{sl}(2,R)$. This question is addressed in different places in a contradictory way. In ...

1
vote

1
answer

216
views

### What happens when you internalize outer automorphisms?

Given a finitely presented group $G = (Gen|Rel)$, we have a set of inner automorphisms $\{ \phi_a(x) = axa^{-1} | a \in G\}$. Defining the set of outer automorphisms to be those automorphisms of $G$ ...

4
votes

1
answer

147
views

### Genericity of irreducible automorphisms of free groups

I have seen in the literature that Irreducible outer automorphisms of a free group $F_n$ are "generic".
I would like to ask that if for example : it is true that for any generating set $X$ of $Out(...

7
votes

0
answers

303
views

### Outer automorphisms of direct product of residually finite groups

Let $A,B$ be finite generated residually finite groups such that $\mathop{Out}(A)$ and $\mathop{Out}(B)$ are residually finite.
Is $\mathop{Out}(A\times B)$ residually finite?
If not, what is the ...

4
votes

0
answers

191
views

### On a problem of Berkovich

What is the real history of the following problem proposed by Berkovich [Y. Berkovich, Z.Janko, Groups of prime power order. Volume 2, Expositions in Mathematics, 56, Walter de Gruyter, New York, 2011]...

10
votes

2
answers

721
views

### Centralizers of non-iwip elements of $Out(F_n)$

Does there exist an infinite order element $\phi\in Out(F_n)$, for some or all $n\geq 3$, which is not iwip but has finite index in its centralizer? How about an element such that all its non-zero ...

9
votes

2
answers

593
views

### Outer automorphisms of free groups into bigger free groups

This may be very either very simple or very unknown, but here goes: Let $F_n$ be the free group on $n$ generators and $Out(F_n)$ its outer automorphism group.
Can embeddings $Out(F_n) \...

6
votes

1
answer

514
views

### What is Out(G-mod) for a finite group G?

Following the notation of Etingof-Nikshych-Ostrik what is Out(G-mod) for a finite group G?
That is what are all bimodule cateogries over the fusion category G-mod of complex G-modules which have the ...