Questions tagged [profinite-groups]

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6
votes
1answer
154 views

Top cohomology of profinite Poincaré duality group

The paper "Cohomology of p-adic analytic groups" by Symonds and Weigel is considered one of the main references for continuous cohomology of profinite groups. There is a passage I do not ...
3
votes
1answer
154 views

Absolute Galois group with unique closed non-open subgroup

Is there an absolute Galois group that is not a subgroup of $\hat{\mathbb{Z}}$ and that has one and only one closed non-open subgroup?
7
votes
2answers
360 views

Definition of a profinite category

When studying objects like profinite groups, profinite spaces and profinite rings, I have noticed that some properties just remain the same. For example they will always be inductive limits of some ...
1
vote
0answers
44 views

Pro-p completion of a quotient of $U/w(U)$ is virtually nilpotent for a finitely generated free group $U$

Let $w$ be a word of a free group. Assume that $H/\overline{w(H)}$ is virtually nilpotent for every finitely generated pro-$p$ group $H$. Let $U$ be a finitely generated free group and $T$ the maximal ...
2
votes
0answers
91 views

Finitely generated subgroups of the absolute Galois group

Consider the absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. It seems to me that, in general, providing an "explicit" description of the elements of this ...
4
votes
0answers
81 views

Bound of word width in compact $p$-adic analytic group

A theorem proved by A. Jaikin-Zapirain in On the verbal width of finitely generated pro-$p$ groups says that: If $G$ is a compact $p$-adic analytic group, then every word $w$ of a free group $F$ has ...
2
votes
1answer
203 views

Fast algorithm for computing $\sum_m (n \mod m)/m!$

I'm interested in quickly computing an embedding of the profinite integers $\widehat{\mathbb{Z}}$ into the unit interval $\left[0,1\right]$. $\widehat{\mathbb{Z}}$ can be represented as compatible ...
4
votes
0answers
97 views

Rational cohomology cohomology of $p$-adic analytic groups

It is a result of Lazard that given $G$ a compact $p$-adic analytic group then we have an isomorphism \begin{equation} H^*(G; \mathbb{Q}_p) \cong H^*(T_eG; \mathbb{Q}_p) \end{equation} where $T_eG$ is ...
4
votes
2answers
162 views

Why is every nilpotent-by-finite finitely generated pro-p-group always $p$-adic analytic

I'm studying the paper "On the verbal width of finitely generated pro-p groups" by Andrei Jaikin-Zapirain (link at ProjectEuclid) and I cannot see a claim made in a proof. I don't know if ...
0
votes
0answers
58 views

Mapping property of $p$-Sylow groups of profinite groups

Let $G$ be an abelian profinite groups. Then we have the Sylow group decomposition $$G\cong \prod_p G_p.$$ In the case of finite groups, we have $ \prod_p G_p\cong \bigoplus_p G_p$ and thus $$\text{...
2
votes
0answers
83 views

kernel and cokernel of corestriction map in cohomology of a profinite group

Let $G$ be a profinite group, $N$ a normal open subgroup and $A$ a discrete $G$-module. We have a corestriction map $cor: H^1(N, A)_{G/N} \to H^1(G, A)$. Are there any results on the kernel and ...
2
votes
0answers
49 views

On exactness of associating smooth representation-functor $(\,)^\infty$

Let $G$ be a locally profinite group, e.g. reductive group over $\mathbb{Q}_p$. For a (abstract) representation $(\pi,V)$ of $G$ and $K\subset G$ compact open subgroup denote by $V^K\subset V$ the $\...
4
votes
0answers
142 views

Countability of conjugacy classes in profinite groups

In the MOF question [1] it was asked if $G$ is a second-countable profinite group with uncountably many subgroups, does it follow that it has uncountably many closed subgroups modulo conjugacy? A ...
3
votes
1answer
221 views

Spaces of closed subgroups of a profinite group up to conjugacy

$\DeclareMathOperator{\Sub}{\operatorname{Sub}}$ Let $G$ be a profinite group and consider the space $\Sub(G)$ of closed subgroups of $G$ equipped with the profinite topology. That is, we have $G = \...
4
votes
0answers
98 views

What do the eigenvalues of a random element of $\mathbb Z_\ell[\Gamma]$ look like?

Let $\Gamma = \varprojlim \Gamma_n$ be a profinite group with $\Gamma_n$ finite quotients. For concreteness, let us fix $\Gamma_n = \operatorname{PGL}_2(\mathbb Z/\ell^n)$ so $\Gamma = \operatorname{...
3
votes
0answers
69 views

Colimits in cohomology of profinite arithmetic groups

Let $G\subset \operatorname{GL}_n$ be a linear algebraic group over $\mathbb{Q}$ and let $\Gamma\subset G\cap \operatorname{GL}_n(\mathbb{Z})$ be an arithmeric subgroup without torsion. Using a result ...
3
votes
0answers
100 views

Rational cohomology of p-adic general linear groups

I wanted to compute the cohomology ring $H^*(GL_n(\mathbb{Z}_p); \mathbb{Q}_p)$ (with $p$ fixed prime as usual). I found some incomplete notes stating that the computation should go as follows. First ...
1
vote
1answer
40 views

Infinite pro-$p$ group of finite solvable length and finite coclass

I was reading about infinite pro-$p$ groups of finite coclass from the book "The Structure of Groups of Prime Power Order" by Leedham-Green and McKay. I asked this question in math....
4
votes
0answers
155 views

Galois representation with infinite image but finite image everywhere locally

Fix a prime $l$. Let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_n(\mathbb{Q}_l)$ be a semisimple continuous representation. Assume $\phi$ has finite image when restricted to $\mathrm{...
5
votes
1answer
213 views

Sylow subgroups of abelian profinite groups

If $G$ is a finite abelian group, then we have a decomposition $$G\cong \prod_{p} G(p)$$ where $G(p)$ is the $p$-Sylow subgroup of $G$. This product makes sense as for all but finitely many primes $p$,...
2
votes
0answers
104 views

Outer Galois representations and Tate modules of Jacobian varieties

Let $X$ be a proper smooth curve over a field $k$. Then we have an exact sequence of profinite groups \begin{equation*} 1 \to \pi_1(X_{\overline k}) \to \pi_1(X) \to G_k \to 1, \end{equation*} ...
8
votes
0answers
195 views

Finitely generated commutative rings with the same profinite completion

Let $R_1$ and $R_2$ be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: $\widehat{R_1}\cong \widehat{R_2}$. Suppose that $R_1$ is a domain. Does ...
8
votes
1answer
252 views

Commutator subgroup of the absolute Galois group - a closed subgroup

Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible that the commutator subgroup of the absolute Galois group of $K$ (considered as an abstract group) is a closed subgroup? This property ...
1
vote
0answers
107 views

How a profinite group can be obtained from its normal open subgroups?

Let $\Delta$ be a set, each element of which is a profinite group (2 distinct elements of $\Delta$ may be isomorphic). Under what conditions on $\Delta$, there exists a profinite group $G$ which has $\...
1
vote
1answer
199 views

Is there a free profinite abelian group on a profinite set?

Let $\mathit{Profinite}_{\mathrm{Ab}}$ be the category of profinite abelian groups, and let $\mathit{Profinite}_{\mathrm{Set}}$ be the category of profinite sets. Does the forgetful functor $$\mathit{...
4
votes
1answer
239 views

Measure of subsets of profinite groups

Let $G$ be an infinite profinite group, so $$G=\lim_{\longleftarrow}G/N$$ where $N$ runs through the open normal subgroups. I have two questions: Is $G$ of Haar measure zero in the compact group $\...
3
votes
1answer
278 views

Continuous function defined by measurable sets

Is the following slightly generalization of Corollary 20.17 in Hewitt and Ross Book (page 296) correct? Let $A$ be a subset of a profinite group $G$ ( compact, Hausdorff, totally disconnected ...
5
votes
2answers
284 views

Mackey theory in the setting of locally profinite groups

$\DeclareMathOperator\Hom{Hom}$Let $R$ be a commutative ring (not necessarily unital). Let $G$ be a finite group, and let $H_1, H_2$ be subgroups of $G$. Recall the following standard result [1, Thm. ...
11
votes
1answer
637 views

Does a (nice) centerless group always have a centerless profinite completion?

This is an extension of a question I asked here on Math.SE Assume that I have a finitely generated residually finite centerless group $G$. Is it true that the profinite completion $\hat{G}$ also has ...
1
vote
2answers
267 views

Meaning of epimorphism from full Galois group to some group

My problem has two parts: let $\;G:=\operatorname{Gal}(\overline{\Bbb Q}/\Bbb Q)\;$ be the full Galois group of the rationals and $\;K\;$ be some finite group, then: (1) Does having an epimorphism (...
5
votes
0answers
189 views

Applications of one of Serre's Theorems

This theorem is due to Serre: Let $G$ be a profinite group, $p$ prime. Assume that $G$ has no element of order $p$ and let $H \leq G$ be an open subgroup. Then $cd_p(G) = cd_p(H)$. Where $cd_p(...
3
votes
0answers
79 views

Splitting the canonical projection to the free pro-p group

Let $\widehat F(k)$ be the free profinite group on $k$ generators and let $p$ be a prime. Then there is a canonical projection $\pi\colon \widehat F(k)\to \widehat F_p(k)$ where $\widehat F_p(k)$ is ...
9
votes
0answers
249 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
115 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
146 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
197 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
227 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 ...
7
votes
0answers
204 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
418 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
347 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
109 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
168 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
327 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
170 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
246 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
138 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 ...
3
votes
0answers
127 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
105 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
227 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
161 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 ...

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