4
votes
2answers
80 views

Realizing $\mathbb{Z}/n \mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$ as an isometry group

Every finite group arises as the isometry group of a subspace of an euclidean space (Albertsona, Boutin, Realizing Finite Groups in Euclidean Space). What are natural examples of spaces realizing the ...
2
votes
2answers
64 views

Why does the additive subgroup of $\mathbb{R}$ generated by $1$ and $\sqrt{2}$ contain arbitrary small elements? [duplicate]

Let $G\subset \mathbb{R}$ be the additive subgroup of $(\mathbb{R},+)$ defined by $G=\mathbb{Z}+\sqrt{2}\mathbb{Z}$. I want to prove that for every $\epsilon>0$ there exists an element ...
0
votes
0answers
112 views

Is it group or not? [closed]

$r$ is a metric of space $L = R^3$. Does G - multiplicity of transformations of $L$ ( for each $g\in G$ exist $n\in\mathbb Z$ $: r(g(x),g(y))=2^n r(x,y)$ for each $x,y\in L$) form a group? How to ...
1
vote
1answer
72 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 ...
1
vote
0answers
47 views

Statistics for random permutations

Let $S_n$ be the symmetric group on $n$ elements, let $d$ be the Cayley distance, and let $m$ be a Haar measure on $S_n$. Let $s$ denote a random permutation with respect to $m$, i.e., $s$ is an ...
11
votes
1answer
138 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 ...
5
votes
3answers
314 views

Equivalence to properly discontinuous action

Let $X$ be a metric space and let $G$ be a group of homeomorphisms $X \to X$ acting on $X$. We say $G$'s action is properly discontinuous in case for every $x \in X$ and compact $K \subseteq X$, there ...
5
votes
2answers
188 views

Group theory with analysis

I've studied group theory upto isomorphism. Topics include : Lagrange's theorem, Normal subgroups, Quotient groups, Isomorphism theorems. I too have done metric spaces and real analysis properly. ...
5
votes
0answers
198 views

Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
0
votes
0answers
64 views

How to show $f$ is an isometry onto the $n$-gon?

The proof of $|D_{2n}|=2n$ is given in Dummit-Foote text as: Now I was trying to verify the statement "there is a symmetry which sends vertex 1 into position i" rigorously. ...
0
votes
1answer
22 views

Elements of $D_{2n}$ in terms of isometries

In course of studying Dihedral Group I'm having trouble to get what exactly the elements of $D_{2n}$ are. According to the Dummit-Foote texts For each $n∈\mathbb Z^+,n≥3$ let $D_{2n}$ be the set ...
1
vote
2answers
76 views

Open Subgroup of (R,+)

Let G be an open subgroup of (R,+) Show that G=R. Note: I've tried taking an interior point of G. Can Archimedian Property be used?
4
votes
3answers
173 views

metric property in a group

Can we define a metric function on a group $G$? Please give examples other than $\mathbb{R}$. Actually most groups have elements in discrete manner. It sounds vague but I can't be more precise.
-2
votes
1answer
86 views

set of terminating decimals

Let $T\subset\mathbb Q$ be the set of all positive rational numbers that can be represented by a terminating decimal (in base 10), that is, a decimal whose tail consists of an infinite sequence of ...
5
votes
2answers
151 views

why we want to use grassmannian space?

I wonder what's the special about grassmannian space? Why we want to use this space? On wikipedia, it says: "By giving a collection of subspaces of some vector space a topological structure, it is ...
3
votes
2answers
59 views

Group, metric, completion

Let $G$ be a group, $(G, \rho)$ - metric space, $p: G \rightarrow \mathbb{R}_+$ such that $p(x)=0 \iff x=e_G, \ \ p(x^{-1})=p(x), \ \ p(xy)\le p(x)+p(y), \ \ p(xy)=p(yx)$ Now let $\rho ...
0
votes
1answer
57 views

Is there an associative (monoid) operation on $\mathbb{R}_{\geq 0}$ which is also a metric?

Is there an associative operation $\star \colon \mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} $ wich also happens to be a metric on $\mathbb{R}_{\geq 0}$? Furthermore is there ...
3
votes
1answer
228 views

Quasi-Isometry [Geometric Group Theory]

How can I prove that if $S,S'$ are two different finite generating sets of a group $G$ , then the metric spaces induced by the "word metric" are quasi-isometric? The definition of quasi-isometry is: ...
6
votes
0answers
157 views

Virtually cyclic groups

Let $G$ be a group with finite generating set $A$, define the distance $$d(g,h)=\mathrm{min}\{n:gh^{-1}=a_1^{\varepsilon_1}\dots a_n^{\varepsilon_n}, a_i\in A,\varepsilon_i=\pm1\}.$$ Define the ball ...
11
votes
0answers
215 views

$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity

Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq ...