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Questions tagged [profinite-groups]

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8
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0answers
187 views

Colimit of continuous cohomology over subgroups

Suppose $G$ is a profinite group, in fact in the applications I'm interested in it would be a $p$-adic analytic group similar to $GL_{n}(\mathbb{Z}_{p})$. Say $M$ is a profinite $G$-representation, ...
5
votes
0answers
95 views

Are double cosets of cyclic subgroups separable in a special linear group?

Let $A,B \in \mathrm{SL}_3(\mathbb{Z})$. Set $$S = \langle A \rangle \cdot \langle B \rangle = \{A^mB^n : m,n \in \mathbb{Z}\}.$$ Is $S$ closed in the profinite topology on $\mathrm{SL}_3(\mathbb{...
2
votes
1answer
82 views

For a pro-p, profinite group, abelianization being finitely generated is the same as being topologically finitely generated

I remember reading (without proof) that for $\Gamma$ a profinite, pro-$p$ group, the following are equivalent: 1) Every open subgroup $\Gamma_0$ is topologically finitely generated. 2) The ...
5
votes
1answer
180 views

Linear representation of the free metabelian / 2-step nilpotent profinite groups on 2 generators

Let G be the free profinite group on 2 generators, $A=G/[G,[G,G]],B=G/[[G,G],[G,G]]$, then what is the structure of the groups $A$ and $B$? I heard that $A$ is isomorphic to the group of such ($3\...
3
votes
1answer
177 views

Every group of totally disconnected type is locally profinite?

Let $G$ be a Hausdorff topological group in which every point has a neighborhood basis of open compact neighborhoods. Let's call this a group of totally disconnected (td)-type. On the other hand, we ...
5
votes
0answers
94 views

Duality between coalgebras and (pseudocompact) algebras - uniqueness

The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below. Let $k$ be a field. The ...
9
votes
0answers
218 views

Continuous cohomology of a profinite group is not a delta functor

Let $G$ be a profinite group, then there is a general notion of continuous cohomology groups $H^n_{\text{cont}}(G, M)$ for any topological $G$-module $M$ (I require topological $G$-modules to be ...
11
votes
1answer
270 views

Profinite completion of finitely presented groups

Let $G$ be a finitely presented group, $\widehat{G}$ be the profinite completion of $G$, and $f: G\rightarrow \widehat{G}$ be the natural map. My question is: Is there an example of $G$ for which $\...
0
votes
0answers
71 views

Profinite groups with finite torsion

Let $G$ be a profinite abelian group such that for every $x\in G$ and every $n\in\mathbb Z$ the preimage of $x$ under the multiplication by $n$ map is finite. Does it follow that the torsion subgroup ...
5
votes
1answer
109 views

Dense abstract free subgroups in a free profinite group

Let $\langle a, b \rangle = F_2$ be a two-generator free group and $\hat{F_2}$ be its profinite completion. Is there an element $c\in \hat{F_2}$ such that $\langle a, b, c\rangle \le \hat{F_2}$ is ...
4
votes
1answer
184 views

What are the LCA groups that are the Pontryagin dual of a locally profinite abelian group?

For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian): finite groups $\leftrightarrow$ ...
2
votes
1answer
108 views

maximal pro-l-quotients of absolute Galois groups

Let $K$ be a field, preferably a function field of a variety $X$ over $\overline{\mathbb{F}}_p$. I am looking for an answer or existing literature on the following question: What is known about the ...
7
votes
0answers
124 views

Torsion in a tensor product over a group ring

Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra. Is it true ...
5
votes
0answers
81 views

Maximal subgroups of infinite index and profinite completion

Preliminary remark: I'm mainly interested in an answer (or link to ressources) in the specific context of the first Grigorchuk group, but I believe that it may be of some interest to state the ...
2
votes
0answers
82 views

Profinite closure of characteristic subgroup

Let $F$ be a free group of finite rank, and $K\subset F$ a finite index characteristic subgroup. Let $\hat{F}$ be the profinite completion of $F$ (i.e. a free profinite group of same rank), and $\bar{...
3
votes
1answer
99 views

Freeness of a quotient group

Let $p$ be a prime. Let $G=\langle x,y\rangle^{\textrm{pro-}p}$ be the pro-$p$ completion of the free group $\langle x,y\rangle$ generated by symbols $x$ and $y$. Define $G_{n+1}=[G,G_n]$ and $G_1=G$ (...
3
votes
0answers
128 views

A question about continuous group cohomology

Let $G$ be a profinite topological group, $M$ a discrete $G$-module. If $M$ is "P", is every $H^i_{\rm cont}(G,M)$ also "P"? or at least is it a subgroup/subquotient of an abelian group that is "P"? ...
1
vote
1answer
144 views

Is there an elementary reason for why $SL_2(\mathbb{F}_p)$ for $p>5$ does not embed into $SL_2(\mathbb{Z}_p[w])?$

This is an exercise from Serre's book on Galois cohomology. Let $p>5$ and consider the groups $SL_2(\mathbb{F}_p)$ and $SL_2(\mathbb{Z}_p[w])$ where $w$ is a primitive $p$th root of unity. Is ...
4
votes
1answer
165 views

No lifts in an exact sequence of profinite groups?

In pg. 24 of his book on Galois cohomology, Serre gives the following exercise: "Give an example of an extension $1 \to P \to E \to G \to 1$ of profinite groups with the following properties: (i) $...
7
votes
0answers
204 views

On an inequality concerning the strict cohomological dimension of a profinite group

This is an exercise from Serre’s book on Galois cohomology. Let $G$ be a profinite group and $H$ a normal closed subgroup and suppose that the cohomological dimension at the prime $p$ of $G/H$ is ...
1
vote
1answer
228 views

The Unit Group of $\mathbb{Z}_p$

Let $\mathbb{Z}_p$ the ring of $p$-adic numbers. It's known that the multiplicative unit group $\mathbb{Z}_p ^\times$ can be set theoretically described as $\bigcup _{1 \le a \le p-1} a+ p\mathbb{Z}_p$...
4
votes
0answers
114 views

subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose mod-2 image is the symmetric group

Let $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ be a closed subgroup of the symplectic group over the $2$-adic integers whose image under the mod-$2$ homomorphism $\pi : \mathrm{Sp}_{2g}(\mathbb{Z}_2)...
2
votes
0answers
53 views

Representations theory of Groups with compact quotient

Let $G$ be a locally profinite group and $H$ a closed normal subgroup of $G$, with $G/H$ a profinite group. If $\rho$ is an irreducible (smooth) representation of $G$ what can we say about the ...
1
vote
0answers
169 views

On groups with finite pro-$p$ completion for all primes $p$

Say that a group has Property X if its pro-$p$-completion is finite for every prime $p$. For instance, every perfect group has Property X. Is there a finitely generated, residually finite group $G$ ...
7
votes
0answers
232 views

When does p-profinite completion commutes with maps from a $p$-finite space?

background Let $\mathcal{S}$ be the ($\infty$-)category of spaces and $\mathcal{S}_{p-\text{finite}}$ the full subcategory spanned by the $p$-finite spaces (that is, the spaces with finitely many ...
8
votes
1answer
346 views

Every profinite group is a quotient of a profinite free group by a normal subgroup that is free profinite?

It is well known that any group is a quotient a free group by a normal subgroup that is free. More precisely if $G$ is a group the exists a short exact sequence of groups $$1\rightarrow F^{'}\...
2
votes
1answer
147 views

Profinite extension of a Lie group

Let $H,G,K$ be three topological groups, we say that $G$ is an extension of $K$ by $H$ if the following short sequence $$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0$$ is exact. (If $H$ is a ...
11
votes
1answer
211 views

Are there open subgroups of $SL_2(\widehat{\mathbb{Z}})$ which are $GL_2(\widehat{\mathbb{Z}})$-conjugate, but not $SL_2$-conjugate?

I apologize if this is too obvious, but I figure it must have a quick answer. Are there open subgroups $\Gamma\le SL_2(\widehat{\mathbb{Z}})$ which are conjugate in $GL_2(\widehat{\mathbb{Z}})$, but ...
2
votes
0answers
101 views

Problem with a proof of Wilson's 'Profinite groups'

(Crossposted on StackExchange Mathematics: https://math.stackexchange.com/questions/2391626/problem-with-a-proof-of-wilsons-profinite-groups) I need help with the proof of Proposition (3.1.3) given ...
9
votes
1answer
322 views

Does $GL_2(\widehat{\mathbb{Z}})$ contain a dense finitely generated subgroup?

It's well known that $SL_2(\widehat{\mathbb{Z}})$ contains $SL_2(\mathbb{Z})$ as a dense and finitely generated subgroup. However, $GL_2(\mathbb{Z})$ is not dense in $GL_2(\widehat{\mathbb{Z}})$, ...
4
votes
1answer
121 views

What is the probability of generating a given procyclic subgroup in $\mathrm{Gal}(\bar{K}/K)$?

This question began as Why are procyclic subgroups of Galois groups of number fields free profinite?, which fizzled out, but which garnered some helpful comments from YCor. Let $K$ be a field, take $\...
5
votes
0answers
139 views

Why are procyclic subgroups of Galois groups of number fields free profinite?

On p832 of Coombes, Harbater - Hurwitz familes and arithmetic Galois groups, the following is claimed: Let $K$ be a number field, take $1 \neq \omega \in \mathrm{Gal}(\bar{\mathbb{Q}}/K)$, and let $...
1
vote
0answers
203 views

Inverse limits and first isomorphism theorem for compact topological groups

This question was originally asked on MathSE here. I have a problem with Proposition (1.2.1) from J. Wilson's book 'Profinite Groups' The proposition is the following: Let $(G, \varphi_i : G \to ...
5
votes
0answers
264 views

Subgroups and quotients of an abelian pro-finite group

It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$. I'm wondering whether there is a counterpart for profinite groups. For example is it true ...
12
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0answers
210 views

Does each compact topological group admit a discontinuous homomorphism to a Polish group?

A compact topological group $G$ is called Van der Waerden if each homomorphism $h:G\to K$ to a compact topological group is continuous. By a classical result of Van der Waerden (1933) the groups $SO(...
7
votes
1answer
251 views

Is there a residually finite non-elementary hyperbolic group whose profinite completion is boundedly generated?

Is there a residually finite hyperbolic group $G$ that is not virtually cyclic, such that there exists finitely many procyclic closed subgroups $C_1, \dots, C_n$ of the profinite completion $\hat{G}$ ...
2
votes
0answers
78 views

Automorphisms of a free topological product

Let $G$, $G_1$, $G_2$ be Hausdorff topological groups. I am mainly interested in the case when those groups are profinite. Let $G$ act continuously on $G_1$ and $G_2$ via continuous automorphisms, i.e....
4
votes
0answers
91 views

Do the “Nielsen” IA-automorphisms of a profinite free group $\widehat{F}$ of rank 2 form a normal subgroup of $\mathrm{Aut}(\widehat{F})$?

Let $F$ be the discrete free group of rank 2, and let $\widehat{F}$ be its profinite completion, equipped with an embedding $$i : F\hookrightarrow\widehat{F}$$ By a result of Asada, this embedding ...
2
votes
1answer
241 views

Is $SL_n(\mathbb{Z}_p)$ virtually torsion free?

If so, is there a way to conclude this from Malcev's theorem? In general, what is known about virtually torsion freeness of non-finitely generated linear groups?
4
votes
2answers
239 views

subgroups of $\prod_p C_p$

Consider the group $G:=\prod_p C_p$ where the product is taken over all primes, endowed with the product topology. I'm trying to classify the compact subgroups of $G$. Is there any subgroups of $G$ ...
0
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0answers
94 views

short exact sequence of profinite groups

Let $A\rightarrow B\rightarrow B/A$ be a short exact sequence of topological groups. Is it true that if there exists a continuous function $B/A\rightarrow B$ (of underlying spaces) such that the ...
22
votes
3answers
956 views

Is $\widehat{\mathbb{Z}}[[t]]\cong\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]]$?

Let $\widehat{\mathbb{Z}}[[\widehat{\mathbb{Z}}]] := \varprojlim_{n,m}(\mathbb{Z}/n)[x]/(x^m-1)$ be the complete group algebra of the profinite free group of rank 1. In Corollary 5.9.2 of Ribes-...
10
votes
1answer
331 views

Difference between the completed group algebra and the profinite completion of a group ring

Let $G$ be a reasonably nice group, say residually finite if need be. We may consider the group algebra $\mathbb{Z}[G]$. Let $\widehat{\mathbb{Z}[G]} := \varprojlim_I\mathbb{Z}[G]/I$ be the ...
2
votes
2answers
278 views

space of closed subgroups of profinite group

I am looking for a reference on the space $\mathcal{Sub}(G)$ of closed subgroups of a profinite group $G$, which naturally has the structure of a profinite topological space: Because the collection ...
1
vote
0answers
141 views

lie algebra associated to a profinite group

is there a natural way to associate a Lie algebra (over real numbers or other field) to a profinite group in a functorial way ? I.e. I'm looking for a functor $L: \mathbf{ProfinGroups}\rightarrow \...
0
votes
1answer
169 views

Subgroup of free profinite group is free profinite?

The question is already in the title. It is known that any subgroup of a free group is free. My question is: Is a closed subgroup of a free profinite group is again a free profinite group ?
3
votes
1answer
196 views

induced isomorphism in continuous cohomology

Suppose that we have a morphism between profinite groups $f: G_{1}\rightarrow G_{2}$ such that $f^{\ast}:H_{cont}^{\ast}(G_{2},A)\rightarrow H_{cont}^{\ast}(G_{1},A) $ is an isomorphism for any finite ...
6
votes
1answer
431 views

Finite Homomorphic images of infinite products of finite solvable groups

I conjecture that: Every Finite Homomorphic image of an infinite (with arbitrary cardinality) product of finite solvable groups is solvable -- or at least Not a simple (non-abelian) group. I can ...
3
votes
0answers
74 views

Analogues of relative property $(\tau)$ for Schreier graphs

Suppose I have an expanding family of Schreier graphs $Z_n=\text{Sch}(G_n,S_n,X_n)$ of groups $G_n=\underbrace{G\wr\ldots\wr G}_{\text{$n$ times}}$ acting on sets $S_n=S^n$ by generating sets $X_n$, ...
3
votes
1answer
162 views

When does a characteristic subgroup remain characteristic in the profinite completion?

Suppose you have a group $F$, and a characteristic subgroup $K\le F$. Under what conditions on $F$ and $K$ is the closure $\overline{K}$ inside the profinite completion $\widehat{F}$ also ...