Questions tagged [outer-automorphisms]
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31 questions
2
votes
1
answer
137
views
Indicability of $\mathrm{Out}(F_n)$
A group $G$ is said to be indicable if it surjects onto $\mathbb{Z}$.
If $n=1$: $\mathrm{Out}(F_1)=\mathbb{Z}/2\mathbb{Z}$ and no finite group surjects onto an infinite group.
If $n\geq 4$: $\...
- 911
4
votes
1
answer
318
views
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 ...
- 815
5
votes
1
answer
183
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 ...
- 3,001
9
votes
1
answer
377
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 ...
- 93
8
votes
1
answer
354
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 ...
- 163
2
votes
1
answer
554
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]\...
- 257
4
votes
1
answer
522
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 ...
- 175
3
votes
1
answer
137
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 \...
- 1,033
5
votes
0
answers
172
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 ...
- 8,529
8
votes
3
answers
424
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 $W(F_4)$?
- 2,773
7
votes
1
answer
569
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 \...
- 473
9
votes
0
answers
372
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 ...
- 27.2k
3
votes
1
answer
164
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)}}{:...
- 379
3
votes
1
answer
175
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 ...
- 257
4
votes
1
answer
338
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 $\...
- 257
7
votes
0
answers
319
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 \...
- 255
5
votes
0
answers
192
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 ...
- 1,996
8
votes
2
answers
923
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 ...
- 7,527
3
votes
1
answer
223
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 ...
- 3,186
3
votes
1
answer
91
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 $\...
- 457
7
votes
1
answer
523
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}(...
- 275
14
votes
4
answers
697
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) = \...
- 12.8k
3
votes
1
answer
207
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 ...
- 1,555
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 ...
- 493
1
vote
1
answer
227
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
151
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(...
- 339
7
votes
0
answers
311
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 ...
- 171
4
votes
0
answers
193
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
724
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 ...
- 618
9
votes
2
answers
598
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) \...
- 1,180
6
votes
1
answer
518
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 ...
- 28.1k