Tenant can sue the property owner.

You own the condo, HOA ownes the common area. 

If tenant slip and fall in the common area, they should sue the HOA.  In fact any guest can sue HOA if they slip and fall in the common area, doesn't have to be a tenant. 

Management company may or may not be involved.   It's like an owner hired someone to clean the snow, but when someone falls, it's ultimately owner's responsibility, not the snow removal company.