give the axe vs lay off

give the axe and lay off both are verbs.

Word	Noun	Adjective	Verb	Adverb
give the axe	No	No	Yes	No
lay off	No	No	Yes	No

As verbs, lay off is a hyponym of give the axe; that is, lay off is a word with a more specific, narrower meaning than give the axe:

give the axe: terminate the employment of; discharge from an office or position
lay off: dismiss, usually for economic reasons

Other hyponyms of give the axe include retire, pension off, clean out, furlough, squeeze out, dismiss, drop, send away, send packing.

give the axe (verb)	lay off (verb)
terminate a relationship abruptly	dismiss, usually for economic reasons
terminate the employment of; discharge from an office or position	put an end to a state or an activity