All Questions
14 questions
4
votes
0
answers
97
views
Characterization of Vilenkin group
It is shown in [1, Section 1] by C.W. Onneweer that every infinite compact, metrizable, zero-dimensional commutative group is a Vilenkin group. My question is does this implication also hold if we ...
- 85
9
votes
1
answer
223
views
$p$-adic analytic pro-$p$ group satisfies a pro-$p$ identity?
Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity
$w$ if for every homomorphism $ f:...
- 667
2
votes
0
answers
106
views
Hereditarily just-infinite pro-$2$ groups
An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...
- 667
1
vote
0
answers
98
views
Existence of countable dense normal subgroups of global Galois group
Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...
- 863
1
vote
0
answers
120
views
Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups
While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...
- 1,013
1
vote
1
answer
95
views
Quotients of pro-$p$ groups linear over a complete Noetherian local ring
Let $R$ be complete Noetherian local ring with finite residue field $\mathbb{F}$ of characteristic $ p $. We say that a pro-$p$ group $G$ is linear over $R$ if it is isomorphic to a closed subgroup of ...
- 863
4
votes
0
answers
103
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 ...
- 329
4
votes
1
answer
381
views
Are braid groups conjugacy separable?
I would like to re-ask a question that was raised in the comments here:
Normal subgroups of braid groups
Namely, a group $G$ is called conjugacy separable, if it holds that for every two elements $x,...
- 5,039
4
votes
0
answers
135
views
Improvements of the Reidemeister-Schreier index formula for particular classes of groups
I have a couple of questions regarding possible improvements of the Reidemeister-Schreier index formula: let $G$ be a $d$-generated group and let $H$ be a subgroup of $G$, then
$$d(H) \le (d-1) \...
- 116
5
votes
1
answer
329
views
A hyperbolic group with a small profinite completion
Is there a finitely generated non-elementary word hyperbolic group the profinite completion of which is known (or conjectured) to be rather restricted, that is: abelian, pro-$p$, virtually prosolvable,...
- 11.3k
3
votes
0
answers
209
views
Growth of the number of generators in hyperbolic groups
Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index.
I would like to know if one ...
- 11.3k
0
votes
0
answers
192
views
Thin profinite groups - nonabelian analogues of p-adic integers
Let $p$ be a prime number, $S = C_p$ a cyclic group of order $p$, $G = \mathbb{Z}_p$ the profinite additive group of $p$-adic integers. It is well known that all the closed nontrivial subgroups of $G$ ...
- 11.3k
10
votes
3
answers
855
views
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 ...
- 373
3
votes
0
answers
144
views
Infinitely generated powerful pro-$p$ groups
A pro-$p$ group of finite subgroup rank has an open subgroup $P$ that is uniformly powerful, meaning that $[P,P]$ is contained in the group generated by $2p$-th powers in $P$, and raising elements to ...
- 4,728