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$\begingroup$

Let $G$ be the group $$ \begin{pmatrix} 1 & a_{12} & a_{13} & a_{14}\\ 0 & 1 & a_{23} & a_{24}\\ 0 & 0 & 1 & a_{34}\\ 0 & 0 & 0 & 1 \end{pmatrix} $$ where $a_{ij}\in\mathbb R$ and $\Gamma=G\cap\mathrm{GL}_4\,\mathbb Z$. What is the set of injective homomorphisms $\Gamma\to G$?

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    Where do the $a_{ij}$ live? The reals?2012-07-24
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    Yes, sorry, added.2012-07-25
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    Hm, so what have you tried? You certainly get the natural inclusion and then you can look at the conjugates of that subgroup.2012-07-25
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    But will conjugates alone give all injective homomorphisms? For example, in dimension 3, the set of injective homomorphisms is the 6-dimensional space $GL_2\mathbb R\times\mathbb R^2$ with $G$ acting transitively on the $\mathbb R^2$ factor, but trivially on $GL_2\mathbb R$. Including and conjugating would thus only yield $\{\text{pt}\}\times\mathbb R^2$ and not the whole set...2012-07-25

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