Tagged Questions
3
votes
2answers
49 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
45 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
124 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
124 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
188 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 ...
