The answer depends on how the deed is titled.
I will assume that you are tenants in common. Absent a statement in the deed assigning percentages of ownership, she owns 50% of the property. Tenants in common are generally not liable for improvements made by the other tenant. Similarly, tenants in common are not liable for rent to the other tenant in common because one lives in the home - unless the tenant in common has been refused the right to also live there. On the other hand, tenants in common are generally liable for their half of the carrying costs of the home which would include the mortgage and property taxes.
Buying her out for $10K might be a good deal.
If she refuses to convey the property to you or to settle for a reasonable amount, you would have to file a partition action seeking a court to sell the property and for an accounting of the contributions and expenses paid by the tenants in common. That will cost you much more than the $10,000.
I am not a GA attorney, laws vary from state to state, therefore you should always consult a local attorney.
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She owns half the house. If all she wants is $10K that is likely a great deal. Have a lawyer draft a proper contract and deed so this doesn't bite you later.
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