Questions tagged [profinite-groups]
The profinite-groups tag has no usage guidance.
302
questions
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Why are free groups residually finite?
Why is it that every nontrivial word in a free group (it's easy to reduce to the case of, say, two generators) has a nontrivial image in some finite group? Equivalently, why is the natural map from a ...
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votes
3
answers
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Why are profinite topologies important?
I hope this is not too vague of a question. Stone duality implies that the category Pro(FinSet) is equivalent to the category of Stone spaces (compact, Hausdorff, totally disconnected, topological ...
- 64.1k
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votes
4
answers
5k
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A profinite group which is not its own profinite completion?
Is there a profinite group $G$ which is not its own profinite completion?
Surely not, I thought. But upon looking into it, I found that there is a special name given to a $G$ which is its own ...
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votes
1
answer
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Nonabelian topological fundamental group of a conjugate variety
Let $X$ be a pointed algebraic variety over the field of complex numbers $\mathbb{C}$.
Let $\pi_1^{\rm top}(X)$ and $\pi_1^{\mathrm{\acute{e}t}}(X)$ denote the topological and the étale fundamental ...
- 13.4k
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votes
1
answer
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Galois Group as a Sheaf
I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois ...
- 14.9k
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votes
2
answers
2k
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Profinite groups as étale fundamental groups
Does every profinite group arise as the étale fundamental group of a connected scheme?
Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme?
...
23
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3
answers
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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-...
- 6,567
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votes
4
answers
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Homomorphism from $\hat{\mathbb{Z}}$ to $\mathbb{Z}$
I expect this question has a very simple answer.
We all know from primary school that there are no non-trivial continuous homomorphisms from $\hat{\mathbb{Z}}$ to $\mathbb{Z}$. What if we forget ...
- 4,155
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votes
4
answers
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What is the virtue of profinite groups as mathematical objects?
In my own research I use profinite groups quite frequently (for Galois groups and etale fundamental groups). However my use of them amounts to book-keeping: I only care about finite levels (finite ...
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2
answers
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Without choice, can every homomorphism from a profinite group to a finite group be continuous?
In ZFC, some homomorphisms from profinite groups to finite groups are discontinuous. For instance, see the examples in this question. However, all three constructions given use consequences of the ...
- 131k
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Is every compact topological ring a profinite ring?
There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
- 2,190
18
votes
2
answers
1k
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The Riemann zeta function and Haar measure on the profinite integers
In an answer to a question on MU about the Riemann zeta function, I sketched a proof that the probability distribution on $\mathbb{N}$ which assigns $n$ the probability
$$\frac{ \frac{1}{n^s} }{\zeta(...
- 114k
17
votes
3
answers
2k
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Finitely generated Galois groups
It is well-known that for a given natural number $n$ there is only finite number of extensions of $\mathbb Q_p$ of degree $n$. This result appears in many introductory books on algebraic number theory....
- 1,378
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3
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An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?
In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the ...
16
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0
answers
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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 ...
- 2,841
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votes
1
answer
634
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Linear embeddings of nilpotent pro-$p$ groups
Is it true that every finitely generated (topologically) torsion-free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
- 617
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1
answer
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Relations between the cohomology of discrete groups and of profinite groups
Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...
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Is the absolute Galois group of the rationals Hopfian?
Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
- 11.2k
14
votes
1
answer
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"Concretely" writing down elements in a free profinite group
Let $r$ be a natural number. The elements of the free group $F_r$ on $r$ generators have a nice concrete description as "words" in the $r$ generators (and their inverses). I'd like to know if there is ...
- 25.1k
13
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3
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883
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Example of reflective subcategory of (Groups) whose reflector doesn't preserve finite products
A subcategory $D$ of a category $C$ is called reflective, if the embedding $D \hookrightarrow C$ has a left adjoint $L:C \to D$. The left adjoint $L$ is called the reflector. If the category $C$ is ...
- 1,312
13
votes
1
answer
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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 ...
- 6,567
13
votes
1
answer
640
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Avoiding countable subgroups of a group homeomorphic to the Cantor space
Update: Further work with Adam (who answers below) and Piotr led to a rather satisfactory result about the problem that motivated the problem below, see our recent paper
The Haar Measure Problem. In ...
- 2,978
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votes
0
answers
541
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Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?
Let $R$ be a commutative ring, and, for $n\ge0$,
${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series
$u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which
$a_0\in R^\times$ and $u(x)\equiv x\pmod{x^...
- 4,048
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votes
4
answers
2k
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Elements of infinite order in a profinite group
Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general?
A start for (A): we can ask the same question ...
- 10.9k
12
votes
1
answer
759
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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 ...
- 470
12
votes
1
answer
419
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Applications of Lubotzky's linearity theorem?
Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL_n(K)$, where $K$ is a field of ...
- 25.6k
12
votes
0
answers
365
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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(...
- 40.2k
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0
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A question concerning model theory of groups
Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...
- 4,111
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2
answers
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Two Definitions of "Character" of topological groups
When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows:
Let $G$ be a topological group. A ...
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votes
1
answer
899
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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 $\...
- 497
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1
answer
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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 ...
- 6,567
11
votes
1
answer
800
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Dessins d'enfants and absolute Galois group
I would like to know what is the recent progress about the group homomorphism
$$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$
$\mathrm{Gal}(\overline{\mathbf{...
- 1,603
10
votes
3
answers
782
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History of profinite groups, when was it first mentioned? What was the original definition?
Searching left me hanging. One of my professors told me the definition using the topological properties was the first one but I cannot find any resources. Is that true? If not, how was it originally ...
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3
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Profinite completion of a semidirect product
If we have two finitely generated residually finite groups $G$ and $H$, is there are relation between
the profinite completions $\hat{G},\hat{H}$ and the profinite completion of a semidirect
product ...
- 1,368
10
votes
1
answer
431
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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 $GL_2(\widehat{\mathbb{Z}})$, ...
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2
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Exotic automorphisms of the fundamental group of a curve?
A while back, Jordan S. Ellenberg brought the following problem to my attention.
If $G$ is a residually finite group, let $\widehat G$ be its profinite completion.
Let $S$ be a closed surface of ...
- 10.5k
10
votes
1
answer
652
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Homomorphic images of a Cartesian product of finite groups
What can be said about the class of groups which can be represented as a homomorphic image of an (infinite) Cartesian product (=unrestricted direct product) of finite groups? What would be simple ...
- 2,774
10
votes
2
answers
381
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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 ...
- 11.2k
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0
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A uniform bound for a "true" non-congruence subgroup
Before stating my question, let me recall the Congruence Subgroup Property/Problem: Given simply connected absolutely and almost simple algebraic group $G$ with fixed realization as a matrix group one ...
- 638
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4
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Topological examples of profinite groups
I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first ...
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When is the profinite completion a pro-$p$ group?
My research area is mainly pro-$p$ groups and profinite groups. However, in the last few year I became also interested in discrete groups. Therefore, it seems to me a natural problem to look for ...
- 5,198
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0
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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, ...
- 2,137
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5
answers
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Does a group have a unique pro-p topology?
If $G$ is a residually $p$ group and $G_i$ ANY filtration (i.e. $G_i\subset G_{i-1}$ and $\cap G_i=e$) of normal
$p$-power index subgroups, is the corresponding filtration the usual pro-$p$ ...
- 2,567
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votes
2
answers
738
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Avoiding countable subgroups of general uncountable groups
The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...
- 2,978
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votes
3
answers
687
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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 ...
- 201
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2
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A metabelian quotient of a free group
I don't know much about free groups (excepted the very basics), and the following question may be trivial, although it isn't to me.
Let $F$ be a free group with $n$ generators $x_1,\dots,x_n$. ...
- 25.6k
8
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1
answer
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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 $\hat{G}$ ...
- 11.2k
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votes
1
answer
454
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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 ...
user145520
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votes
1
answer
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index of a closed subgroup of a profinite group
In the book "profinite groups, arithmetic, and geometry" of Shatz, the index $(G:H)$ of a closed subgroup $H$ of a profinite group $G$ is defined to be the supernatural number $lcm\big((G/U):(H/(H\cap ...
- 287
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1
answer
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Is there a left-orderable profinite group?
Is there a nontrivial profinite group $G$ with a binary transitive relation $<$ such that
$x<y$ implies $x\neq y$, and for any different $x,y \in G$ either $x < y$ or $y < x$ and such ...
- 11.2k