What is a word for what gladiators do?

Gladiators gladiate. Yes, it is a word and it is mentioned as a back-formation from gladiator in Wiktionary. It is a noun-to-verb derivation, originally from the Latin noun gladiator ("swordsman"), from gladius ("sword"). (However, it is also used in botany and means "sword-shaped").

Gladiation is also in Wiktionary and defined as a combat between gladiators. These derivations don't appear in most authoritative dictionaries but they are used as a neologism.

We no longer go to the Colosseum to watch gladiators gladiate each other into oblivion. We haven't since Constantine the Great made such contests illegal in A.D. 325.

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If you want a common word, you can consider swordfight. (It is also related to the origin of gladiator: gladius "sword").

You didn't prefer long phrases but another option is gladiatorial fight which is the most precise term.

Combat seems good:

noun

[MASS NOUN]
1 Fighting between armed forces:

verb

1.1 archaic Engage in a fight with; oppose in battle:

The combat of gladiators in the colosseum satisfied the blood lust of Romans.

From Gibbon's The History of the Decline and Fall of the Roman Empire

This severe reformer shews no more indulgence to a tragedy of Euripides, than to a combat of gladiators

From Dunford's Rome: The Mini Rough Guide:

Gladiatorial combat as a Roman tradition was a direct import from the Etruscans, who thought it seemly to sacrifice a few prisoners of war or slaves at the funeral games of an important person.

Since combat is of Latin origin, it seems particularly approptiate:

combat (n.)

1560s, from Middle French combat (16c.),
from combattre
(see combat (v.)).

1560s, from Middle French combat (16c.),
from Old French combattre (12c.),
from Late Latin combattere,
from Latin com- "with" (each other) (see com-) + battuere "to beat, fight"
(see batter (v.)).

"strike repeatedly, beat violently and rapidly," early 14c.,
from Old French batre "to beat, strike" (11c., Modern French battre "to beat, to strike"),
from Latin battuere "to beat, strike," an old word in Latin, but almost certainly borrowed from Gaulish,
from PIE root * bhau- "to strike"
(cognates: Welsh bathu "beat;" Old English beadu "battle," beatan "to beat," bytl "hammer, mallet").

Wikipedia uses the word combat in reference to the activity of gladiators:

In the earliest munera, death was considered the proper outcome of combat. [Emphasis added]

Fight
verb 1. take part in a violent struggle involving the exchange of physical blows or the use of weapons.

The best single-word summary that comes to mind of what gladiators do would be

slaughter
2. the savage killing of a person

^{TFD Online}