Questions tagged [outer-automorphisms]

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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 ...
Andi Bauer's user avatar
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9 votes
1 answer
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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 ...
Sarah's user avatar
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1 answer
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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 ...
AlekseiG's user avatar
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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]\...
1123581321's user avatar
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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 ...
XYSquared's user avatar
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1 answer
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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 \...
NWMT's user avatar
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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 ...
Sean Lawton's user avatar
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3 answers
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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 $W(F_4)$?
Daniel Sebald's user avatar
7 votes
1 answer
486 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 \...
Ben Heidenreich's user avatar
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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 ...
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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)}}{:...
Groups's user avatar
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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 ...
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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 $\...
1123581321's user avatar
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315 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 \...
ctst's user avatar
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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 ...
Rylee Lyman's user avatar
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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 ...
stupid_question_bot's user avatar
3 votes
1 answer
208 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 ...
Andrei Smolensky's user avatar
3 votes
1 answer
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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 $\...
Antoine's user avatar
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$\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}(...
Student's user avatar
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14 votes
4 answers
665 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) = \...
Kevin Walker's user avatar
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3 votes
1 answer
200 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 ...
Drike's user avatar
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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 ...
Javier's user avatar
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1 answer
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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 $G$ ...
Samuel Schlesinger's user avatar
4 votes
1 answer
144 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(...
user75691's user avatar
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0 answers
293 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 ...
Michal Ferov's user avatar
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0 answers
189 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]...
S. Mohsen Ghoraishi's user avatar
10 votes
2 answers
713 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 ...
Alessandro Sisto's user avatar
9 votes
2 answers
585 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) \...
Dr Shello's user avatar
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6 votes
1 answer
510 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 ...
Noah Snyder's user avatar
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