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- TL;DR Summary
- I want to show that group acts freely and discreetly on IR^2 but I am failing to see what the group looks like and I can't describe the orbits

So before I start I technically do now that the group I am dealing with is just a representation of the Klein bottle but I am not supposed to use that as a fact because the goal of the problem is to derive that information.

Problem:

Let G be a group of with two generators a and b such that aba = b. Show that formulas ## a(x,y) = (x,y+1) ## and ## b(x,y) = (x+1,1-y) ## determine a free and discrete action of G on ## \mathbb{R}^2 ##

So the problem I have is that I don't know what the relation of the generators gives me except that all strings of form ## aba ## are reduced to "b" and similarly for the inverses. I can see that any part of an orbit obtained by just applying "a" or "b" is going to be discrete but I don't know how to translate it to an arbitrary member of the group. I had an idea of showing that the orbit is discrete because both actions of generators can be represented as translations over the integer lattice for ## (x,y) ## and (x,-y) ## respectively, however this comes to bite me in the ass later as I need to show that there is no identification on the interior of the unit square so I really need to find a different way to calculate the orbit.

This is a HW problem so please don't give me full solutions immediately I am just in need of a hint or a nudge in a right direction

Problem:

Let G be a group of with two generators a and b such that aba = b. Show that formulas ## a(x,y) = (x,y+1) ## and ## b(x,y) = (x+1,1-y) ## determine a free and discrete action of G on ## \mathbb{R}^2 ##

So the problem I have is that I don't know what the relation of the generators gives me except that all strings of form ## aba ## are reduced to "b" and similarly for the inverses. I can see that any part of an orbit obtained by just applying "a" or "b" is going to be discrete but I don't know how to translate it to an arbitrary member of the group. I had an idea of showing that the orbit is discrete because both actions of generators can be represented as translations over the integer lattice for ## (x,y) ## and (x,-y) ## respectively, however this comes to bite me in the ass later as I need to show that there is no identification on the interior of the unit square so I really need to find a different way to calculate the orbit.

This is a HW problem so please don't give me full solutions immediately I am just in need of a hint or a nudge in a right direction