The universal cover of ${\rm SL}(2,\mathbb R)$

What is the $\widetilde{\rm SL}(2,\mathbb R)$?

${\rm SL}(2,\mathbb R)$ is the set of all $2 \times 2$-real matrices with determinant one. The space $\widetilde{\rm SL}(2,\mathbb R)$ is its universal cover. There are several ways to think about this space. ${\rm SL}(2,\mathbb R)$ is for instance the unit tangent bundle of the hyperbolic plane $\mathbb H^2$. This point of view gives $\widetilde{\rm SL}(2,\mathbb R)$ a structure of a twisted metric line bundle over $\mathbb H^2$, which can be thought as a hyperbolic analogue of the Hopf fibration.

Click on the button below to reveal a concrete model of $\widetilde{\rm SL}(2,\mathbb R)$.

${\rm SL}(2, \mathbb R)$ acts transitively on the unit tangent bundle of $\mathbb H^2$. Hence, so does its universal cover. The orbit map provides a natural equivariant projection from $\widetilde{\rm SL}(2, \mathbb R)$ to $\mathbb H^2$ whose fibers are homeomorphic to $\mathbb R$. It turns out that, topologically, $\widetilde{\rm SL}(2, \mathbb R)$ is homeomorphic to $\mathbb H^2 \times \mathbb R$, for which a possible model is $$ X = \left\{ (x,y,z,w) \in \mathbb R^4 \mid x^2 + y^2 - z^2 = -1, z> 0\right\}.$$ However, unlike in $\mathbb H^2 \times \mathbb E$ the metric is twisted. More precisely, we endow $X$ with an $\widetilde{\rm SL}(2, \mathbb R)$ invariant metric.

Some views of $\widetilde{\rm SL}(2,\mathbb R)$

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Command QWERTY keyboard AZERTY keyboard
Yaw left a q
Yaw right d d
Pitch up w z
Pitch down s s
Roll left q a
Roll right e e
Move forward arrow up arrow up
Move backward arrow down arrow down
Move to the left arrow left arrow left
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Move upwards ' ù
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Balls in $\widetilde{\rm SL}(2,\mathbb R)$
Balls in $\widetilde{\rm SL}(2,\mathbb R)$

A ball in the unit tangent bundle of a punctured torus

HD pictures of $\widetilde{\rm SL}(2,\mathbb R)$ can be found in the gallery

Features of $\widetilde{\rm SL}(2,\mathbb R)$