All Questions
7 questions with no upvoted or accepted answers
5
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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 ...
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5
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406
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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 ...
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3
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92
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Symmetric group in terms of block permutations
For $i+j+k=N$, consider the permutation $\Pi_{i,j,k}\in S_N$, which keeps the numbers $0,\ldots,i-1$ fixed, and exchanges the numbers $i,\ldots,i+j-1$ with the numbers $i+j,\ldots,i+j+k-1$.
$$\Pi_{i,j,...
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2
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147
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Rank gradient in free products amalgamating a finite subgroup
Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$.
Must $A *_C B$ have positive rank gradient?
See Which 3-manifolds have ...
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2
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Salvaging Howson's theorem for free profinite groups
This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...
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1
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Dense free subgroups
Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?
- 11.3k
1
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A bound on the size of the center
Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$
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