# Tagged Questions

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### The set of subgroups of $F_2$

This question came up in our algebraic topology class and our Professor didn't know the answer. I also couldn't find an answer so far. What is the cardinality of the set of subgroups of $F_2$? ...
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### Wild automorphisms of profinite groups

Is there a profinite group $G$, a continuous automorphism $\alpha$ of $G$ and a topologically finitely generated closed subgroup $H \leq G$ such that $\alpha(H) \lneq H$ ? Note that if an ...
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### Fully residually free groups and completion

Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?
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### Thickening graphs to get honest actions

Let $X$ be a finite graph. Its fundamental group is the free group $F_n$ on (say) $n$ generators. Let further an automorphism $\phi$ of $F_n$ be given. It is not true in general that this ...
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### Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...
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### Collections in direct products and freeness

I am looking for references about the following type of questions: Let $G$ and $H$ be two groups, let $(g_i:i\in I)\subset G$ and $(h_i:i\in I)\subset H$ be collections of group elements, and ...
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### Products of subgroups of a free group

Let $F$ be a free group, and let $A,B \leq F$ be two subgroups such that $AB$ contains a nontrivial normal subgroup of $F$. Must either $A$ or $B$ contain a nontrivial normal subgroup of $F$? What if ...
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### products of conjugates in free groups

While trying to carry out some technical arguments in free groups, I have encountered the following problem, to which I don't know the answer. Let $F$ be a free group and let $g,a_1,\ldots,a_n \in F$....
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### Number of trivializations of a trivial word in the free group

Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...
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### History of Tarski's problems on free groups

As is known, Tarski posed his questions about first-order theories of non-abelian free groups around 1945. However, the questions were not published in his papers or books. What is the original ...
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### Must nonunit in group algebra of free group generate proper two-sided ideal?

Let $F$ be a free group and $k$ be a field. If $x$ is an element of the group algebra $k[F]$ that is not a unit (equivalently, that is not a nonzero scalar multiple of an element of $F$), must the 2-...
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### Linear groups which don't contain products of free groups

Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...
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### Do free profinite groups satisfy Howson's theorem?

Let $F$ be a free profinite group, and let $A,B \leq F$ be finitely generated closed subgroups. Must $A \cap B$ be finitely generated?
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### Schreier's formula and supersolvable groups

A finitely generated profinite group $G$ is said to satisfy Schreier's formula if for every open subgroup $L \leq_o G$ we have $d(L) = (d(G)-1)[G:L] + 1$. Here $d$ stands for the smallest cardinality ...
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### Can a profinite completion be free pro-p?

Is there a prime number $p$ and a finitely generated residually finite group whose profinite completion is a free pro-$p$ group on a nonempty finite set? Thanks to YCor we see that we cannot take the ...
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### Conjugates and infinite index subgroups of free groups

Here I am asking for an analogue of Generating infinite index subgroups of a free group Let $F$ be a nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of ...
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### 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 ...
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### Is a free group a product of f.g subgroups of infinite index?

Let $F$ be a free group, and let $H,K \leq F$ be finitely generated subgroups of infinite index in $F$. Is it possible that for the set of products we have $HK = F$ ?
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### Is the free abstract group residually of rank d > 2?

Let $d \geq 2$ be an integer, and let $\mathcal{F}_d$ be the family of finite groups such that $G \in \mathcal{F}_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements. Is ...
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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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### Exhausting a free pro-p group

Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$. Let $p$...
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### Action of the pure braid group on the commutator subgroup of a free group

Let $P=P_n$ be the pure braid group on $n$ strands and $F=F_n$ the free group on $n$ generators. I'm interested in a nice description of the action of $P$ on the derived subgroup $F'$ which somehow ...
Let $F$ be a free group on a finite set $X$, and let $M$ be a finitely generated subgroup. Marshall Hall's theorem states that $M$ is closed in the profinite topology on $F$. That is, $M$ is the ...
Let $F$ be a nonabelian free profinite group, $H \leq_c F$ finitely generated with $[F:H] = \infty$. Must there be some $\{1\} \neq N \lhd_c F$ such that $N \cap H = \{1\}$?