6
votes
1answer
186 views

Topological groups defined by completely disconnected subgroups

Can you define a group topology on a group by specifying which subgroups should be discrete with respect to that topology (where a subgroup $S$ of $G$ is discrete if each $s\in S$ has an open ...
15
votes
1answer
449 views

Is a left topological group which is a manifold a topological group?

Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...
11
votes
2answers
367 views

subsets of groups which have to be closed no matter what

One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?
2
votes
0answers
95 views

Hall's paper on the profinite groups and Andre Weils “voisinage” notion

I am reading through a classical paper A Topology for Free Groups and Related Groups by Marshall Hall Jr. in which profinite groups are defined for the first time. There he defines on p. 129: ...
9
votes
0answers
245 views

Topology of marked groups for different number of generators

A $k$-marked groups is a pair $(G,S)$ where $G$ is a group and $S$ is an ordered set of $k$ generators of $G$. Each such pair can be identified with a normal subgroup of the free group $F_k$ of rank ...
7
votes
0answers
165 views

Which topological spaces are coset spaces of locally compact groups?

What is a topological characterization of the class of spaces that have the form $G/H$ for a locally compact, Hausdorff group $G$ and a closed subgroup $H$ ? Such a space $X=G/H$ necessarily ...
2
votes
1answer
99 views

Syndetically separated topological groups

I am looking for examples for a certain kind of topological groups: Definition: A topological group G is called syndetically separated if for every compact subset $K \subseteq G \setminus \{1\}$ ...
14
votes
0answers
448 views

Are amenable groups topologizable?

I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is ...
1
vote
0answers
123 views

Intersection of cocompact closed normal subgroups

Let $G$ be a locally compact Hausdorff topological group. Definition A closed normal subgroup $H \unlhd G$ is called cocompact if $G/H$ is compact with respect to the quotient topology. Note ...
11
votes
2answers
334 views

Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE]

And what else can be said, if so? (Original math.SE post) In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. ...
4
votes
0answers
111 views

Possible homogeneity of infinite dimensional Sierpinski carpet analogues?

Start with the Hilbert cube $H=I^\omega$, thinking of its coordinates as written in ternary expansion. Construct subsets $S_n$ by removing points from $H$ if for any $m$, at least $n$ of the ...
12
votes
1answer
880 views

(Closures of sets of) operations in topological groups.

Let $G$ be a topological group. For each $n \in \mathbb{Z}$, consider the continuous functions $f_{n} \colon G \to G : x \mapsto x^{n}$, and set $F := \{f_{n} \mid n \in \mathbb{Z}\}$. Is there a ...
3
votes
1answer
312 views

Action on a compact group

If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?
0
votes
0answers
250 views

classifying continuous functions on the $ax+b$ subgroup of $GL(2,\mathbb R)$

Let $G$ be 'ax+b' topological group i.e subgroup of $GL(2,\mathbb R)$ containing $2\times 2 $ matrices of type $\{\left(\begin{matrix} a&b\\\0&1\end{matrix}\right): a\neq 0\; a,b\in \mathbb ...
0
votes
1answer
240 views

homeomorphism of topological group

Let G be a topological group and a be an element of order 2 in G. Further suppose the element a does not belong to center of G. Then is it true that only homeomorphism f of G such that $f(ax)=f(x)a$ ...
3
votes
2answers
248 views

When does a LCA group not contain a (closed) infinite cyclic subgroup?

If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does not contain a closed (and hence discrete in the ...
6
votes
2answers
401 views

Distinct, non-homeomorphic, profinite topologies on a given abstract group ?

Just a silly little question which arose in connection with infinite Galois groups and their Krull topology:- can a given abstract group be endowed with distinct, non-homeomorphic, profinite ...
12
votes
1answer
662 views

Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
12
votes
7answers
1k views

Superfluous definitions

It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative. For if a and b are elements of R, and writing + for the group operation then applying ...
7
votes
2answers
429 views

Computational cost of converting between 3-manifold presentations

Given a 3-manifold presented as a triangulation, a Heegaard splitting, or a Dehn surgery, what is the computational cost of converting to the other two presentations? I would like Heegaard splittings ...
2
votes
2answers
924 views

Irreducible unitary representations of locally compact groups

Let $G$ be a locally compact group and let $\mu$ be a left Haar measure. We know that $\mu$ is unique up to a scalar in $\mathbf{R}_{>0}$. I don't know so much about unitary representations of ...
5
votes
0answers
455 views

Profinite topologies

We can define two topologies on a group $G$ by considering all normal subgroups of finite index (resp. of index a finite power of $p$ - where $p$ is a prime) as basis of $1\in G$. My questions: Under ...
10
votes
2answers
848 views

pro-discrete = locally compact and open normal subgroups have trivial intersection?

EDIT: After talking to some experts on the subject, I have concluded that a) the answer is not obvious or well-known for locally compact groups in general, b) the answer should be 'no' and I have some ...
10
votes
9answers
1k views

Proving the impossibility of an embedding of categories

A number of topological invariants take the form of functors $\mathscr{T}\to\mathscr{G}$, where $\mathscr{T}$ is the category of all topological spaces and continuous functions, and $\mathscr{G}$ is ...
11
votes
2answers
706 views

covers of $Z^\infty$

Is it possible to cover $Z^\infty$ (the infinite direct sum of $Z$'s with the $l_1$-metric) by a finite set of collections of subsets $U^0,...,U^n$ such that each collection $U^i$ consists of ...
3
votes
1answer
232 views

Topological simplicity and dense subgroups

Let $G$ be a (topologically) simple Hausdorff topological group. Let $H$ be a dense subgroup of $G$. Now throw away the topology. What restrictions are known on the structure of $H$ as an abstract ...
9
votes
2answers
632 views

Is there a way to see a topological group as the “Cayley graph” of its “infinitesimal generators”?

At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most ...
-2
votes
1answer
458 views

Countable open subgroup

In a Hausdorff topological group, how can I show that every infinite topological group has a countable open subgroup?
10
votes
2answers
476 views

Is there a non-trivial topological group structure of $\mathbb{Z}$?

More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?
5
votes
2answers
1k views

Quotient of a Hausdorff topological group by a closed subgroup

Sorry if this question is below the level of this site: I've read that the quotient of a Hausdorff topological group by a closed subgroup is again Hausdorff. I've thought about it but can't seem to ...
20
votes
8answers
2k views

Is there a compact group of countably infinite cardinality?

Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now. Is there a compact (Hausdorff, or even T1) (topological) ...
2
votes
1answer
235 views

Hausdorff Derived Series

There is a short section in the book Locally Compact Groups by Markus Stroppel (Chapter B7) on the notion of a "Hausdorff Solvable Group", which he defines as a topological group with a descending ...
19
votes
6answers
923 views

Explanation for E_8's torsion

To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is ...