What palms are these?

I have a large number of palms down the whole length of the house, these were planted before I bought the place around 10 years ago, not sure how long they have been there. Some are massive (2 storey house level) most are around 3-3.5m tall.

In a sticky situation with the neighbors who live downhill from us and the shared fence. It’s leaning over on their side in one particular spot… it’s about 25-30m shared fence. Their land is about a foot below ours for most of the property. They have a retaining wall on their side but it just looks like pine sleepers, I can’t tell how well it’s built.

We both want to replace the fence, all good, but they are saying the palms will continue to cause it damage so want them all gone (if close enough to the fence) before we do it.

I don’t necessarily want to do that, they are quite beautiful and established and some are massive. As I said, only one particular spot is falling and if their concerns were valid I would assume the rest of the fence would be damaged already. You can see how close some of these are to the fence. Photo 2 in particular, these are the largest ones and fence is fine there.

I’m trying to learn as much as I can about them and the root systems before I reply. Any pointers with the type will help me to research the root systems and determine how to move forward. Thanks.