And when you push it, the UI, in fact, disappears.

But, so does the rest of the Viewer, and even the GUI, as your machine switches to the text console on virtual terminal 1.

ALT-F7 should take you back to the GUI, but you will find the viewer still showing the UI.

Some folks enable the Advanced Menu with CTRL-ALT-SHIFT-D, then use the Advanced -> Rendering -> Features -> UI menu to turn off the UI, film clips, then kill the SL viewer and relog in to get control back.
No need for that, anymore.
Before starting the SL Viewer, use xmodmap to change your keyboard mappings.
First, you need the keycode number for the F1 key.
xmodmap -pk | grep '(F1)'
67 0xffbe (F1) 0x1008fe01 (XF86_Switch_VT_1)
The 67 is the keycode we want.
The (XF86_Switch_VT_1) is the keymapping we need to override.
We want to tell X that an F1 is just an F1, not a magic request to switch to text console 1.
Before starting the SL Viewer, run:
xmodmap -e "keycode 67 = F1 F1"
This line means:
run xmodmap, and evaluate the expression "keysymbol 67 equals an F1 unshifted and F1 if shifted"
Presto, CTRL-ALT-F1 in the SL Viewer now HIDES and RESTORES the UI.

After exiting the Viewer, to restore normal Linux CTRL-ALT-F1 behavior, run:
xmodmap -e "keycode 67 = F1 XF86_Switch_VT_1"
And, now, CTRL-ALT-F1 works as usual to switch to text console 1.
Hope this helps!