9
votes

Accepted

### Shortest almost trivial element of free group

Repeating from the comments section:
This (natural and beautiful) question was previously asked and answered on this site. See Collapsible group words. It also appeared recently on math.se.
The ...

9
votes

Accepted

### Is every automorphism of $\mathrm{Aut}^+(F_2)$ induced by conjugation inside $\mathrm{Aut}(F_2)$?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}$If I have chased through the literature correctly, I think the answer to your question is "yes". Specifically:
Dyer–Formanek–...

9
votes

Accepted

### What are double groups mathematically?

As far as I can tell, a double group is a double cover of a group. Specifically, if $G \subset \operatorname{SO}(n)$ is a group acting by rotations of $n$-dimensional space, its double group is the ...

9
votes

### Regular orbits for automorphisms of finite simple groups

As pointed out by Michael Giudici the answer is given by a result of Horoševskiĭ. Here is a proof following the paper by Horoševskiĭ.
Lemma: Let $\phi$ be an automorphism of $G$ with $|\phi|$ ...

9
votes

### Regular orbits for automorphisms of finite simple groups

By a result of Horoševskiĭ you can never find such an automorphism, that is all automorphisms of finite simple groups have a regular orbit.

8
votes

### What are the Schur indices of irreducible representations of $\operatorname{SL}(2,p)$?

If $p \equiv 1$ mod $4$ then all faithful irreducibles have Schur index two, and Schur indicator $-1$. If $p \equiv 3$ mod $4$, then most do, but there are two irreducibles of dimension $(p+1)/2$ with ...

7
votes

### Is any representation of a finite group defined over the algebraic integers?

I just stumbled across this ancient question, and I want to point out my notes here that prove the result in question. The proof is basically the same as moonface's accepted answer, but with two ...

5
votes

### Relations between relations in the positive braid monoid

My conjectured list generates $\pi_1$. This proof is basically the one in Lemma 3 and Proposition 4 of Tits' paper that Sam Hopkins referenced in the comments, plus some facts from Garside theory.
...

4
votes

### Regular orbits for automorphisms of finite simple groups

If I am reading your question correctly, then I think $A_{5}$ is an example where this fails. The automorphism group is isomorphic to $S_{5}$. The only elements of composite order in the automorphism ...

4
votes

Accepted

### Are all "almost projective" groups free?

Yes.
As noted by YCor in the comments, both for the category of groups and the category of finitely generated grups if we take $G$ to be a free group (with the rank equal to the minimal size of a ...

Community wiki

3
votes

### Relation between Floyd and Gromov boundaries of hyperbolic groups

For any Floyd function $f$ (as in Karlsson's paper) not decaying exponentially too fast, the Gromov and Floyd compactifications indeed coincide. In fact, there is a more general result for relatively ...

3
votes

Accepted

### Fixed points free automorphisms of Teichmüller spaces

Yes. In a bit more detail: if the Teichmüller space has positive dimension then the given topological surface admits a pseudo-Anosov homeomorphism. (This is an exercise, but perhaps a non-trivial one,...

2
votes

### What are the Schur indices of irreducible representations of $\operatorname{SL}(2,p)$?

This is not a complete answer, but shows how this can be calculated from the character table (at least in this case) without using the usual Frobenius-Schur formula $\nu(\chi) = \frac{1}{|G|} \left( \...

2
votes

Accepted

### If $F$ is a prosoluble subgroup of a free profinite product $\amalg G_i$ and $F \cap G_i^g$ is pro-$p$, is also $F$ pro-$p$?

Let $I = \{1,2\}$ and let $G_1 = G_2$ be groups of order $p=2$. Their free product is $G = \langle \delta, \varepsilon |\, \delta^2= \varepsilon^2=1\rangle$, which is $\langle \tau, \varepsilon |\, \...

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