ZDV Controller reported IFR traffic descended from their assigned altitude to below the MVA in an area of limited communications that resulted in a CFTT event.

Narrative

Aircraft X entered my airspace nearly along the sector 21/15 boundary. This boundary also generally follows an 080/070 MIA (Minimum IFR Altitude) line. Aircraft X checked in at 080; I read them the altimeter - no problems. A few minutes later; Aircraft X made a request that sounded like they wanted 075 but I questioned it due to the quality of the transmission and height of the MIA they were in. I asked them to repeat the request and they again said requesting 075. I immediately told them that the MIA in their area was 080; at which point their data-block began flashing MSAW - still saying 080. I told them to maintain 080 and read them the 'low altitude alert'; phraseology in which I may not have said perfectly. The pilot advised they were just descending to avoid a cloud and were climbing back up. At this point; their Mode C showed 072. I advised that if they needed lower a left turn would take them to a 070 MIA. I called the adjacent sector and requested control. The pilot climbed and maintained 080 for the remainder of the flight in my airspace. At no point did I authorize deviations or a descent from 080. It being a Aircraft X; that pilot/trainer may have been under the assumption that they were VFR which would explain some of the confusion with one of the pilots. I could have verified that the pilot was IFR equipped and capable as well as verified if the pilot has visual with the ground. The pilot remained calm leading me to think that they may have thought it was a VFR flight. All indicators told me they were IFR and so they were treated as such.