Bamboo is a very invasive plant. It was brought here in the nineteenth century from Asia and it’s an opportunistic plant. Some species are spreaders, some are clumpers. The clumpers are usually OK, but the spreaders spread and they’re very difficult to dig out because their roots are extremely strong – they’re like ropes and they’re a lot of them. When the roots can’t be got at, the best thing to do is to cut them back back and spray the stumps with a weed killer. If you’re an organic gardener, keep cutting them back physically until they give up.
If they’re next door to you and invading your land, you can build a trench, maybe at least half a metre deep. Then put a thick layer of material like industrial vinyl flooring, or special bamboo control membrane. That will hopefully keep the roots back on their own site.