Tree busting up sidewalk outside of our new rental house...am I responsible??

[AJ Romero]

My husband and I just closed on our first rental SFH today ... Yay!! We are now working to get it rent-ready...and one of the things that came up was a giant tree in the front yard whose root has completely buckled up the sidewalk in front of the house. I have no problem chopping down that tree before it destroys all of the sewer lines to the house but what the heck needs to be done with the sidewalk? Do I have to call the city? If so, who pays for this repair? If someone tripped and fell out there could they sue us?

Thanks for any feedback,

AJ