# Help to understand the geodesics in $BS(1, 2)$

I would like to understand the sets of geodesics in $$BS(1, 2)$$, which is described in https://arxiv.org/pdf/1908.05321.pdf, Proposition 3 (page 3).

Write $$G=B S(1, 2)=\left\langle a, t \mid t a t^{-1}=a^2\right\rangle$$ Let $$\mathbb{Z}_2=\left\{x \in \mathbb{Q} \mid k^e x \in \mathbb{Z}\right.$$ for some $$\left.e \in \mathbb{Z}\right\}$$ and consider the semidirect product $$\mathbb{Z}_2 \rtimes \mathbb{Z}$$, where the action of $$\mathbb{Z}$$ on $$\mathbb{Z}_2$$ is multiplication by $$2$$. Then $$B S(1, 2) \cong \mathbb{Z}_2 \rtimes \mathbb{Z}$$.

Proposition 3. Let $$G=B S(1,2)$$. The set of words in the following forms comprise a set of unique geodesic representatives for the elements in $$\mathbb{Z}_2$$.

2a. $$\left\{\epsilon, a^{ \pm 1}, a^{ \pm 2}, a^{ \pm 3}\right\}$$

2b. $$\left\{a^{x_0} t a^{x_1} t \cdots a^{x_d} t^{-d}|d \geq 1,| x_d \mid \in\{2,3\}, A\right\}$$

2c. $$\left\{t^{-b} a^{x_0} t a^{x_1} \cdots t a^{x_d} t^{-c}\left|b, c, d \geq 1, d=b+c, x_0 \neq 0,\right| x_d \mid \in\{2,3\}, A\right\}$$

2d. $$\left\{t^{-d} a^{x_0} t \cdots t a^{x_d} \mid d \geq 1, x_0 \neq 0, A\right\}$$

Here, $$A$$ signifies the conditions $$\left|x_i\right| \leq 1$$ for $$i, if $$x_{i-1} \neq 0$$ then $$x_i=0$$ for $$i, if $$x_d>0$$ then $$x_{d-1} \geq 0$$, and if $$x_d<0$$ then $$x_{d-1} \leq 0$$.

If I understand the condition $$A$$ correctly, $$w= a^{-1}ta^{0}ta^2 t^{-2}$$ is an element in the set described in (2b), but $$w$$ can also be represented by $$a^1 t a^3 t^{-1}$$ which has a shorter length. Did I misunderstand something here?

Thank you for reading. (I have copied this question from StackExchange)

• It's so strange to write $\mathbf{Z}_2$ to mean $\mathbf{Z}[1/2]$. The notation $\mathbf{Z}_2$ is very widely used to mean the $2$-adics, in which all primes are invertible except precisely 2...
– YCor
Commented Oct 6, 2023 at 14:04
• You are right, the second word has a shorter length than the first word and they represent the same group element. Maybe you should contact the authors by email. I consider that a better etiquette than posting here anyway. This question about potential errors in their paper might not be noticed by the authors. Commented Oct 6, 2023 at 14:40