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Let $G$ be a group with presentation $$\langle a,t| t^{-1}at = a^{2}\rangle$$

I want to find a normal form for $G$. That is, a collection of words on the alphabet $\{a,t\}$ such that they represent the elements of $G$ in a unique way. Things that I know:

·) Every element of $G$ can be represented as $t^{n}a^{k}t^{-m}$ for $n,m,k \in \mathbb{Z}$ with $n, m \geq 0$.

·) There's a well-defined monomorphism

$$ \begin{array}{rcl} G & \hookrightarrow & GL_{2}(\mathbb{Q})\\ a & \longmapsto & \begin{pmatrix} 1 & 1\\ 0 & 1\\ \end{pmatrix} =: A\\ t & \longmapsto & \begin{pmatrix} \dfrac{1}{2} & 0\\ 0 & 1\\ \end{pmatrix} =: T\\ \end{array} $$ Then, $t^{n}a^{k}t^{-m} = t^{n'}a^{k'}t^{-m'}$ iff $T^{n}A^{k}T^{-m} = T^{n'}A^{k'}T^{-m'}$, which is equivalent to say $$m - n = m^{'} - n^{'} \text{ and } \dfrac{k}{2^{n}} = \dfrac{k^{'}}{2^{n^{'}}}$$ From this, how can I get a list of diferent words and conclude that every word in $G$ is one of these words, i.e. remove the redundant words and find the normal form?

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You need to add a criterion to insure uniqueness. For example, consider the "choice" function $\phi$ such that for any word $w \in G$ : $$ \phi(w) = \min\{n \geq 0 ~|~ \exists m\geq 0, k\in \mathbb{Z} : t^n a^k t^{-m} = w\} $$

This function is well defined from your first bullet point. Any word has a unique reduced word attached to it : if $w = t^n a^k t^{-m}$ then we also have $w = t^{\phi(w)}a^{k2^{\phi(w)-n}}t^{n-m-\phi(w)}$ which is a form that does not depend on $n$, $m$ or $k$ from your second bullet point.

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