In which case, the wall is likely to be a party
wall and therefore you are jointly responsible for it in that it provides a
dual purpose. Indeed, a triple purpose.
It denotes the boundary, it retains your garden, and
finally, it stops your garden, falling in to next door. It is of benefit to
both of you.
If the neighbour disputes this, would be worthwhile
printing this thread and showing them.
You will have the benefit of it and they also have
a benefit of it, and under the doctrine of mutual benefit and burden, you both
have to contribute towards its repair.
If it falls down, you lose your garden and so do
I would check your house insurance and advise your
neighbour to do the same in case you are insured either for the rebuilding cost
if it is through subsidence or for the legal costs of arguing with next door.
If they refuse to take their cheque book out.
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The thread remains open for us to continue this exchange