Questions tagged [outer-automorphisms]

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3
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1answer
77 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)}}{:...
1
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1answer
74 views

Outer automorphism group of $F(G)$

By a nice helpful comment in my last question, I see that if $\Phi(G)=1$ then ${\rm Out}(F(G))$($\cong G/F(G))$) is isomorphic to a direct product of ${\rm GL}(n_i,p_i)$. Actually, I’m digesting an ...
3
votes
1answer
152 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
1answer
258 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
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0answers
289 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
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0answers
125 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
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2answers
404 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
1answer
156 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
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1answer
77 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 $\...
6
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1answer
266 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
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4answers
462 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
1answer
142 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 ...
5
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1answer
794 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
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1answer
175 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
1answer
123 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
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0answers
227 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
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0answers
179 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
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2answers
618 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
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2answers
542 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) \...
5
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1answer
428 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 ...