A few years ago, one of my geese was trying to get to a tasty dandelion leaf on the other side of the chicken wire fence. She apparently got her bill caught in there, and instead of maneuvering it out, panicked. When I came home and found her, she had gotten loose but sustained about an inch of ripped bill and was bleeding.
It was a Saturday, so I took her to a local emergency vet clinic, which quickly cauterized the wound and stopped the bleeding. She healed up wonderfully.
If your bird still has a split spot in the bill, it may need cauterizing. Just as some human wounds only need neosporin and others are so severe they need stitches, it is the same with our birds. Or cauterizing, as the case may be.
If the bill heals but with a split spot, dirt and mud will get into the split eventually, and the goose could get infection very easily. Any infection in the head area is a particular concern, because it can enter the brain and kill them.
I've had my five geese behind a chicken wire fence for fifteen years, and this was the only incident I ever had of one getting injured on it. I think it was a fluke. But I do know how important it is to only graze geese in areas where they can't get themselves into trouble. One kind of trouble, besides those mentioned by other posters here, is tiny sharp objects, like nails, staples, or sharp-edged wood bits. They can ingest these, and it can kill them. When I have handymen work in my yard, I give them a tin can to put all their debris in so it doesn't wind up in the yard as an invitation to curious geese.
