2
votes
2answers
31 views

Cocompact group actions have cobounded orbits

Assume $X$ is a complete, locally compact, geodesic metric space (in particular, $X$ has closed balls are compact, by the Hopf-Rinow Theorem). Assume $G$ acts isometrically on $X$. We say $Q\subset X$ ...
2
votes
1answer
101 views

Modern mathematics for dummies

I have poor university mathematical education, but Math fascinates me, so I decided to educate myself for a bit. I know there is a dozen modern mathematical fields I know nearly nothing about like ...
3
votes
0answers
37 views

Fundamental group of 7-gon with labelling scheme $abaaab^{-1}a^{-1}$

I want to calculate the fundamental group of of a $7$-gon with labelling scheme $abaaab^{-1}a^{-1}$. I will call the quotient space $X$ with reference point $x_0$. This is what I tried: If you have ...
2
votes
1answer
28 views

What does it mean for a group to act cocompactly by isometries on a topological space $X$?

What does it mean for a group to act cocompactly by isometries on a topological space $X$? I know if $X$ is a topological group, and $A$ a subspace, then $A$ is cocompact iff $X/A$ is compact. Not ...
1
vote
2answers
53 views

Examples of continuous non-transitive group actions

In studying topology, I encountered this problem: Let $S$ be a topological space and let $G$ be a topological group acting continuously on $S$ (group action as $G \times S \to S$ map is continuous). ...
0
votes
0answers
66 views

Canonical topology on standard groups?

I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete ...
0
votes
0answers
32 views

Choosing a canonical fundamental domain

I have a set of equations that partitions a certain space into equivalent regions. For a given point $p$ contained in region $R_1$, there are equivalence relations giving its equivalent position in ...
1
vote
1answer
52 views

Residually Finite group $\Rightarrow$ Totally disconnected

How can I prove that a residually finite group $G$ is totally disconnected? I considered the topology generatad by $\{Ng\}_{N\in\eta,\;g\in G}$ where $\eta=\{N\unlhd G \;, |G:N|<\infty\}$ and I ...
1
vote
1answer
52 views

Simple question on topological groups

Why is $\{1\}$ closed in a totally disconnected topological group?
3
votes
1answer
70 views

The abelian group of homeomorphism

Let $G$ be a subgroup of the group of homeomorphisms on the circle, and we suppose $G$ is abelian, if every element of $G$ has a fixed point on the circle, does it imply that $G$ has a common fixed ...
26
votes
5answers
471 views

Can $S^2$ be turned into a topological group?

I know that $S^1$ and $S^3$ can be turned into topological groups by considering complex multiplication and quaternion multiplication respectively, but I don't know how to prove or disprove that $S^2$ ...
1
vote
1answer
33 views

not regular group homomorphism

Let $G = GL(1,\mathbb{C})$, and let $\theta: G \rightarrow G$ defined by $\theta(z) \mapsto \bar{z}$. Show that $\theta$ is a group homomorphism that is not regular. Perhaps my struggle comes from ...
2
votes
0answers
52 views

isomorphism of algebric torus

I'm trying to prove the following: Let $D_n=(\mathbb{C}^{\times})^n$ (an algebric torus of rank $n$). Assuming $D_k$ is isomorphic to $D_n$ as an algebric group. Prove that $k=n$. So far, I managed ...
6
votes
1answer
86 views

Profinite topology of a Group

Let $G$ be a group. Consider now the set of all (left for instance) cosets in $G$ of subgroups of finite index. This set is a base for a topology in $G$. I found somewhere that if $G$ is residually ...
3
votes
1answer
52 views

Constructing The Cayley Graph and quasi-isometry to $\mathbb{Z}$

If we have a group $G$ defined by: $G=\langle a,b\mid b^2=1\rangle$ then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ...
0
votes
1answer
40 views

Show that $\mathbb{R}$ is not quasi-isometric to $[0, \infty )$ [closed]

Struggling to find a contradiction, any help appreciated, thanks!
6
votes
1answer
61 views

Baumslag–Solitar $B(1,2)$ is not hyperbolic

I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat. The Cayley graph can be ...
2
votes
1answer
58 views

Finite cyclic subgroups of the homeomorphisms of the real numbers

This inquiry was inspired by this post: The automorphism group of the real line with standard topology. While trying to come up with some interesting examples, I attempted to figure out the finite ...
4
votes
2answers
110 views

Fundamental group of the Poincaré Homology Sphere

I'm working on the Poincaré Homology Sphere $P_3$ and would like to compute it's Homology $H_1$ and fundamental group. I would like to identify it's fundamental group with the binary icosahedral group ...
1
vote
2answers
43 views

Topologically dense subgroup

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
1
vote
0answers
43 views

How can we characterize all topological groups given $G$?

The idea is that all topologies on G (not necessarily making it a topo group) can be completely specified by a set of functions $F = \{f: G \to G\}$ if you form a basis for the topology like: $B = ...
2
votes
0answers
53 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
1
vote
0answers
48 views

An example of a topological space which is a group, but is not a “topological group”

Is there any example of a topological space $X$ which has a group structure, but the maps $(x,y)\mapsto xy$ and $x\mapsto x^{-1}$ are not continuous?
10
votes
1answer
124 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
1
vote
1answer
29 views

Can anything be said for the topology of a topological monoid?

A topological group is one in which the group operations (the multiplication and inverse) are continuous, or equivalently as a group object in $\mathbf{Top}$. They are uniformisable and hence are ...
11
votes
2answers
153 views

Non-isomorphic Group Structures on a Topological Group

Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ...
1
vote
1answer
55 views

Constructing Topological Groups [closed]

In general, is there a way to construct topological groups? That is, given two topological groups $X$ and $Y,$ can I construct a topological group $Z$ using $X$ and $Y$ in said construction? I have ...
1
vote
1answer
74 views

Every metric space has a $(1, 1)$-net

I'm trying to show that every metric space $X$ has a $(1, 1)$-net but struggling - surely $(1, 1)$ is just arbitrary and I've run out of obvious subgroups of $X$ to play with. Any help plz! Here a ...
9
votes
1answer
115 views

if we set topology on a group like that, is it important?

Let $G$ be a group and $\omega$ be set of all subgroup of $G$. Since $\omega$ is closed under intersection, it is trivial to check that $\omega$ satisfies to conditions to be a base. Thus,Let $T$ be ...
2
votes
1answer
90 views

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact.

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact. This seems pretty straight forward but how will I be able to prove this? I saw this ...
3
votes
2answers
189 views

What topological group is $\mathbb R/\mathbb Z$?

The integers $\mathbb Z$ are a normal subgroup of $(\mathbb R, +)$. The quotient $\mathbb R/\mathbb Z$ is a familiar topological group; what is it? I've found elsewhere on the internet that it is ...
4
votes
2answers
89 views

Is a subgroup of a topological group a topological group?

I'm trying to solve the problem from Munkres: Let H be a subspace of G. (Where G is a topological group). Show that if H is a subgroup of G, then both H and H closure are topological groups. Is it ...
0
votes
0answers
65 views

How to prove (S1, *) is a topological group?

I'm trying to prove that (S1, *) is a topological group. Where S1 = {z complex : |z| = 1}. So I want to show that S1 x S1 -> S1: (z,w) -> z*w and S1 -> S1 (z) = z^(-1). I'm not really sure where to ...
5
votes
2answers
559 views

Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
1
vote
1answer
53 views

Quotient group $G/G_0$ in Group Topology

I'm stuck on this (apparently) simple thing: If $G$ is a topological group and $G_0$ is the connected component of $G$ containing the identity then $G/G_0$ is discrete if and only if $G_0$ is open. ...
5
votes
2answers
125 views

Infinite Galois Theory

In infinite Galois theory for the one-one correspondence as in the finite case, one needs to introduce Krull topology. What is the intuition behind defining such topology ?
2
votes
1answer
54 views

Combining the axioms of a topological group

According to Wikipedia, a topological group $G$ is a topological space and a group, such that the functions $$(x,y) \mapsto x\cdot y\\x\mapsto x^{-1} $$are continuous. Is the single requirement that ...
1
vote
0answers
25 views

Existence of slices for the action of a subgroup

Assume that a group $G$ acts on a space $M$ in such a way that there exists a slice at a point $m \in M$. Let $H \subseteq G$ be a subgroup. Under which additional assumptions (if there are any) can ...
6
votes
0answers
143 views

Orbit space of a free, proper G-action principal bundle

Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such ...
0
votes
0answers
46 views

Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
0
votes
0answers
51 views

Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
3
votes
2answers
71 views

Topological Group $G$ totally disconnected $\Rightarrow$ $G$ hausdorff?

On Wikipedia, I read that a topological group is necessarily Hausdorff if it is totally disconnected. Is that true? I read it on this page: http://en.wikipedia.org/wiki/Totally_disconnected_group ...
2
votes
2answers
200 views

HNN extensions as fundamental groups

I have heard that the Seifert–van Kampen theorem allows us to view HNN extensions as fundamental groups of suitably constructed spaces. I can understand the analogous statement for amalgamated free ...
11
votes
1answer
143 views

Proving that a metric space is a group

I'm stuck on this relatively hard problem. Let $G$ be a non-empty set, $d$ a distance on $G$ and $\cdot$ an associative operation on $G$ $\cdot$ is such that $$\forall a \in G , \forall x \in G ...
4
votes
2answers
89 views

“Uniform groups” (similar to topological groups)?

Why have I heard about topological groups, but nothing about "uniform groups" (uniform spaces endowed with a group)?
0
votes
0answers
34 views

Chromatic classes of vertices of a polyhedron

For a convex polyhedron, how do I figure out all possible proper chromatic classes of its vertices (so that all vertices that are assigned the same color constitute a separate class, and no two ...
1
vote
1answer
99 views

Can a locally compact group with closed singleton be countable but not discrete?

Problem: Prove if a locally compact group $(G,*)$ contains a closed singleton then it must be either discrete or uncountable Proof Given: Assuming $G$ is countable we can write $G = \displaystyle ...
4
votes
0answers
109 views

How many compatible group structures does a topological space admit?

Suppose I have a topological space $(G,\tau)$ and am interested in whether there exists a topological group $(G,*,\tau)$. In other words, can we assign a binary operation $*$ to $(G,\tau)$ which ...
1
vote
0answers
48 views

Visually apealing holologous transformation of a given contour

There is this problem which roughly says: You want to put a framed picture onto the wall with a cord to the picture frame. The cord is a single one, and both ends are attached to the frame. ...
5
votes
0answers
162 views

Dense uncountable proper subgroup of $(\mathbb{R},+)$

Probably someone had asked this question on StackExchange, but can one construct a dense uncountable proper subgroup of $(\mathbb{R},+)$?