2001 Volvo V70 Battery going dead

First thing you need to do is get the battery fully charged and make sure that the battery holds the charge. If it does, and the low voltage comes on only after the vehicle warms up, then the voltage regultor is going bad. Everything is computer control on this vehicle so when voltage starts to get around 9 volts or less, the electronic throttle stops responding and then other warning lights come on because the other control units lacks the 12 volts minimum it needs to operate properly. If they would to scan all the control modules they would see all kinds of fault codes related to low battery voltage. Now with 113k miles on your vehicle you may consider replacing the whole alternator. If it were my car I would only replace the voltage regulator but only because I know how to get to it, I can get the part at cost and I am doing my own labor. If you were my customer I would let you decide. 9 out of 10 times the voltage regulator will solve the problem but you do get that ONE that needs the alternator as well.

Have the battery charged up and then have them do a load test. If it fails the load test then you have a bad battery. If it is ok then when the charge light comes on, check to see if the alternator is charging. If it is not replace it. The voltage regulator comes with the alternator.