# isometry

1. ### Discrete subgroups of isometry group of euclidean space

We say that a subgroup S of G is discrete if and only if subset topology on S is discrete. For subgroups of isometry groups of euclidean space an equivalent condition is: intersection of the S-orbit of any x has finite intersection with any compact set. Why is there such the equivalent...
2. ### metric space Isometry

Notes: R^2_{+}={(x,y):y>0 and the metric riemannian g at any point (x,y) is given by g(u,v)=\frac{u\cdot v}{y^2}
3. ### Increasing sequence of an isometry compositions

Let f: [0,1] \rightarrow [0,1] be an isometry in a metric space ([0,1], d_{Eucl}). Suppose f(0)=0, \ \ \ x\in [0,1] Could you explain to me why the sequence (f^n(x))_{n\geq1} is monotone and why does it converge with respect to the usual topology of [0,1] and also with respect to d and, in...
4. ### An isometry onto a dense subset of equiv classes.

I believe I have correctly solved most of the exercise in a text I am reading, but one portion is unclear to me. (X,d) is a metric space and S is the set of Cauchy sequences in X. (XBAR,rho) is the metric space of equivalence classes of the sequences in S, where rho(xbar,ybar) = lim...