In the paper Anti-de Sitter space, squashed and stretched Bengtsson and Sandin introduce the Lorentzian analogue of the squashed 3-sphere. After looking up Berger spheres, it seems what is meant with "squashing" in the case of $S^3$ is the following:

Take the Lie algebra $L$ spanned by $z_1$,$z_2$ and $z_3$, which fulfill the necessary relations so $L$ can generate $S^3$. Then take the quotient group $B(\beta) = G/H$ of the group $G = S^3 \bigoplus \mathbb{R}$ by the one-parameter subgroup $H$ generated by (e.g.) $\alpha z_1 + \beta z_2$, where $\alpha^2 + \beta^2 = 1$, $z_1 \in L$ and $z_4$ a left invariant vector field tangent to $\mathbb{R}$. $B(\beta)$ is then called a Berger sphere. I interpret this as "squashing" $S^3$ along $\mathbb{R}$. Why is the condition $\alpha z_1 + \beta z_2$ necessary?

Coming back to the mentioned paper, the authors state that Taub-NUT cosmologies can be described by squashing $S^3$ along Hopf fibres, i.e. $S^1$. This confuses me as the topology is just $S^3 \times \mathbb{R}$, is there a covering space involved?

In the Lorentzian analogue of squashed $S^3$, $AdS_3$ has the topology of $\mathbb{R}^2\times S^1$ and $z_4$ can be either spacelike ($z_4^i z_{4,i} = 1$) or timelike ($z_4^i z_{4,i} = -1$). Consequently there are two ways of of squashing, but along which fibre does one squash in this case? Furthermore, what is meant with "Stretching"?

EDIT:

So I figured out at least the "stretching" part: For example at one stage the introduce the metric of $AdS_3$, squashed/stretched along a spacelike fibre: \begin{align} \mathrm{d}s^2_\lambda = \frac{1}{4}\left(-\mathrm{cosh}^2(\omega)\mathrm{d}\tau^2 + \mathrm{d}\omega^2 + \lambda^2 \left(\mathrm{d}\sigma + \mathrm{sinh}(\omega)\mathrm{d}\tau\right)\right)\,, \end{align} where $\lambda$ is a real stretching parameter. They simply call the case $\lambda^2 < 1$ "squashing" and $\lambda^2 > 1$ "stretching".

Furthermore, the Hopf-fibres, along which one squashes $AdS_3$, are the geodesic congruences of the spacetime, i.e. $\partial_t$ and $\partial_{\phi}$, which both are compactified do to identifications like e.g. $\phi \sim \phi + 2 \pi$.

The two standing questions are:

- Along which fibres is one squashing $S^3$ to get Taub-NUT space?
- Is in the case of the Berger sphere the condition $\alpha^2 + \beta^2 = 1$ necessary as a identification condition, if not why is it necessary?
- Where are the Hopf fibres in the calculation, along which we are squashing?