When spectators and combatants in Ancient Rome sought the ruling of the Emperor on whether a defeated gladiator should live or die, how did the Emperor signal that the man was to be killed?

I think it was a thumbs down

If I remember correctly (I was but a mere youth at the time), the sign for "Let him live" was thumbs down, indicating that the sword (or weapon) should be struck in the dirt. To indicate death, I think the thumb went to the chest, specifically, the heart, to indicate where to put the weapon.

Thumbs up for death ! We have long used thumbs up to signify approval,or happiness, or that something was fine, but we've been wrong The thumb was, as DG says, used to represent a sword. So when it was 'unsheathed' and pointing up it was ready to strike down and kill the victim . To show approval and to save the loser from death the gesture was to push the thumb of one hand into the closed fist of the other, so 'sheathing the sword' or 'pollo compresso favor iudicabatur' as they would have said: 'good will is the verdict, by the thumb being kept in'.

We should blame a painter for our error being so common. The French C19 artist Gerome painted 'Pollice verso'. It shows a gladiator about to be killed while the emperor stretches out his hands, thumbs down, to indicate execution. Unfortunately, Gerome was a good artist but a poor classicist. He understood 'pollice verso' as 'by thumb turned down' In fact it means only 'by turned thumb' and 'verso', 'turned', here would be taken to mean 'turned up'.(That is, the thumb would not be inverted )

Once again color me impressed at a question (and answer) that is superb. ~Jim