Tenant put up a fence on rental property. They are moving out. Can they take the fence with them.

What was the agreement with the landlord when they put it up.

Will they return the property to perfect condition, fill up the post holes, replant the grass ?

It can be argued both ways in law, a fence once in he ground is assumed part of the property,

But if they renter leaves it, the landlord could also change him against the deposit to remove it, unless it was specificlly allowed in writing.

So a lot more details on when it was put up, permission to put up and what type of fense

If they return the property to the exact condition it was in then they can take it. If not, then they can be charged for restoring the property.

In some places, capital improvements become the property of the property owner

Anything like a tree planted by a tenant automoatically becomes a "fixture" to the land. I don't see the fence being anything other than a "fixture" to the land and the tenant should not be able to take it with them. Same as with flowers and plants. Once in the ground they belong to the landlord.