[CliffordK (OP)]

I was just thinking tonight.

Valles Marineris is about 7km deep.

What would be the effect of making a valley on Mars, say 50 km deep?  100 km deep?  Would there be a problem cutting through the crust into the mantel?

Orient it directly East/West (as much as possible) to get the full length of daytime sunshine.  Mars has an axial tilt of about 25°, so at the equator, if one sloped the sides at 25°, one should get full sunlight to the valley floor.

How deep would one have to trench to get a decent atmospheric pressure (say ½ ATM), in open air? 
Geothermal temperature may help some too....  assuming it could be maintained for centuries or millenia, and wasn't TOO HOT.

One would still need supplemental oxygen and Nitrogen, or other inert gas.  But, at least the pressure wouldn't put one in an immediate danger to life.  And, perhaps also have a more temperate environment, especially at night.

A lack of a cold trap would be a big problem, so any evaporated water would likely get lost, and convection would likely force atmospheric circulation.

Perhaps this is all easier said than done, considering the deepest open pit mine built on Earth in the last century is about 1.2km deep, or a little less than a mile deep.

One might conclude that it would be easier to just build a greenhouse. 

[yor_on]

It certainly is a undertaking worthy of Jules Verne

Do it as a hardcore SF Clifford.

[Bored chemist]

"What would one get from making an artificial valley on Mars? "
An Arts Council grant?