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216 questions
4
votes
1
answer
130
views
Reduced representatives of products of conjugates of a cyclically reduced word in a free group
Let $F_n$ be the free group with fixed set of free generators $\{x_1,x_2,\dots,x_n\}$. Whenever $w\in F_n$, write $L(w)$ to denote the set of $i\in\{1,2,\dots,n\}$ such that $x_i$ appears as a letter ...
- 7,670
20
votes
2
answers
759
views
Injective endomorphisms of the free group $F_2$
Denote by $F_2= \langle x,y \rangle$ the free group on two generators. For every $w(x,y) \in F_2$ which is not contained in the subgroup generated by $x$, consider the (injective) endomorphism $\...
- 26.1k
0
votes
0
answers
30
views
Generalization of quasi direct product
I stumbled upon the following group action and wanted to know if there is any known terminology or known properties.
Let $F_3(u,v,A)$ be the free group on three letters $u,v$ and $A$ and let
$F_2(x,y)$...
- 1
3
votes
0
answers
126
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Other known 5-groups of rank 2 and exponent 5?
I am interested in the known details of the other finite quotients of the free Burnside groups $B(m,n)$ of rank $m$ and exponent $n$. GAP's ANUPQ package has for example
...
12
votes
1
answer
381
views
Roots of subgroups of free groups
My question, briefly, is "does the following hold":
Lemma. Let $H$ be a subgroup of a free group $F$. Suppose $h\in H$ is not a primitive element of $H$ (i.e. not contained in a free basis ...
- 3,093
1
vote
0
answers
105
views
Simple free product of groups with malnormal amalgamated subgroups
Let $G$ be a group. A subgroup $M$ of $G$ is said to be malnormal if for each $g \in G \setminus{M}$, $M^g \cap M = \{1\}$.
I was reading R. Camm: Simple free products, in which there is an example of ...
- 111
1
vote
0
answers
92
views
Average Whitehead minimized length
Let $w$ be a word uniformly sampled from the cyclically reduced word in the free group on $r$ elements.
I'm looking for the expected length of the Whitehead minimization of $w$.
I don't need a precise ...
- 321
1
vote
1
answer
239
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Reference request: Open problems about finite free products of finite groups
I'm working with finite free products of finite groups, i.e. a group $G$ given by $$G = F_1 \ast \ldots \ast F_n$$ where each $F_i$ is finite.
Do you know of any open problems as well as references ...
- 175
3
votes
1
answer
199
views
Variant of the coupon collector problem on free groups
I have the following variant of the birthday problem / coupon collector problem:
Let $w$ be a reduced word sampled uniformly from the set of reduced words of length $n$ in the free group $F_r$. As $n$...
- 321
4
votes
0
answers
117
views
Automorphism of a torsion-free abelian group
Let $p$ be an odd prime, and let $R_p$ be the additive group of all rational numbers whose denominator is a power of $p$. It is well known that the direct product $R_p\times R_p$ contains subgroups $H$...
- 177
1
vote
1
answer
230
views
Separating cosets in a descending sequence of subgroups
Let $G = G_1 \supset G_2 \supset G_3 \supset \dotsb$ be a descending sequence of finite index subgroups of a finitely generated residually finite group $G$. Let $K\subset G$ be a subgroup.
Let's say ...
- 7,745
16
votes
2
answers
740
views
$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$
In a paper I found the following result:
$$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$$
However, they got the result as a corollary of a ...
- 1,079
2
votes
1
answer
161
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 $n\geq 4$: $\...
- 1,079
4
votes
1
answer
219
views
Equation in the conjugacy class of a free group
I will pose the question in the form in which it originally appeared to me:
Let $a,b,c,d$ be different letters in a finite alphabet $\mathcal{Z}$. Let $Q$ and $R$ be finite words with letters from $\...
- 111
2
votes
0
answers
135
views
Test words in free profinite groups
Let $G$ be a group. An element $g \in G$ is said to be a test element if any endomorphism $\phi$ of $G$ such that $\phi(g) = g$ is an automorphism. The free group $F_2$ of rank $2$ is generated by $...
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