TITLE: Why additional constraint is need for this two groups to be isomorphic?
QUESTION [6 upvotes]: I'm reading AMS's book Papers on Topology, which collects Poincare's papers on topology.
However, the first paper stops me.
In the paper, he considered the group generated by transformations in $\mathbb{R}^3$, the generators are:
$(x,y,z)\rightarrow (x+1,y,z)$
$(x,y,z)\rightarrow (x,y+1,z)$
$(x,y,z)\rightarrow(\alpha x+\beta y,\gamma x+\delta y, z+1)$
where $\alpha\delta-\beta\gamma=1$.
Obviously, the third transformation is a rotation in $XY$-plane and one move in $Z$.
We denote the group generated above by $(\alpha,\beta,\gamma,\delta)$.
Then Poincare claims that two groups $(\alpha,\beta,\gamma,\delta)$ and $(\alpha',\beta',\gamma',\delta')$ cannot be isomorphism unless the two transformation in $\mathbb{R}^2$:
$(x,y)\rightarrow(\alpha x+\beta y,\gamma x+\delta y)$
$(x,y)\rightarrow(\alpha' x+\beta' y,\gamma' x+\delta' y)$
are conjugate of each other by a linear transformation with integer coefficient.
I'm confused by this claim and can not find why.
I think if we denote the three transformation $\sigma_1,\sigma_2,\sigma_3$ and $\sigma_1',\sigma_2',\sigma_3'$, we can just form an isomorphism: $\phi(\sigma_i)=\sigma_i'$.
Why the additional constraint is needed?

REPLY [3 votes]: Since you are interested in topology, you would do well to read Peter Scott's beautiful paper from 1983, called "The geometries of 3-manifolds". The manifolds you are asking about are Solvmanifolds (at least in the case @Mark Sapir is discussing in his answer), and are discussed in detail starting on page 470 of that paper (including an answer to your question), but you would do well to read the whole thing.