Depending on the law that governs your lease, which in turn depends on the type of lease and a couple other factors (you didn't mention) , a LL is response to take steps to handle pest infestations. If a LL could tie the infestation to the tenant--not merely claim that dirty dishes equals mice--the LL could then pursue the tenant for the fees. A LL in such a situation COULD be liable to the tenant for failing to act and COULD be held accountable for damages and the tenant COULD terminate the lease (would need to be a pretty well documented and extreme case).
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