All Questions
11 questions
2
votes
0
answers
124
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Salvaging Howson's theorem for free profinite groups
This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...
- 11.3k
3
votes
1
answer
189
views
Measuring products of finitely generated subgroups of free groups
Let $F$ be a finitely generated free group, $H_1, \dots, H_n$ finitely generated subgroups of infinite index in $F$, and $\epsilon > 0$. Must there be an epimorphism to a finite group $\phi \colon ...
- 11.3k
10
votes
2
answers
383
views
Can a positive measure subset of a free group be nowhere dense?
Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have ...
- 11.3k
5
votes
0
answers
189
views
Is there a nontrivial profinite word which is trivial in any group with at most d generators?
Let $F$ be a free profinite group of rank $\aleph_0$, and let $d \in \mathbb{N}$.
Let $N_d \lhd_c F$ be the intersection of all open normal subgroups $L \lhd_o F$ for which $F/L$ can be generated by ...
- 11.3k
2
votes
1
answer
306
views
An epimorphism into a profinite group
Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...
- 11.3k
1
vote
0
answers
134
views
Dense free subgroups
Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?
- 11.3k
1
vote
0
answers
82
views
A bound on the size of the center
Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
- 11.3k
2
votes
1
answer
286
views
Making a profinite group free
Let $F$ be a free profinite group, $G$ a profinite group. Suppose that the free profinite product $F \amalg G$ is a free profinite group. Must $G$ be a free profinite group?
For abstract groups the ...
- 11.3k
4
votes
1
answer
455
views
Generators of Sylow subgroups
Is there a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for each finite supersolvable group $G$, and a Sylow subgroup $S \leq G$ we have $d(S) \leq f(d(G))$?
Here $d(H)$ denotes the ...
- 11.3k
5
votes
0
answers
406
views
Profinite groups, completions, and Schreier's formula
Let $G$ be a finitely generated profinite group, and $H \leq_o G$. We say that $H$ satisfies Schreier's formula in $G$ if $d(H) - 1 = (d(G)-1)[G:H]$. We say that $G$ satisfies Schreier's formula if ...
- 11.3k
5
votes
1
answer
453
views
Normal Subgroup Growth
Let $F$ be a free group on $d$ generators.
Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$?
Explicitly, for each natural number $...
- 11.3k