It partly depends on what kind of bamboo it is.
Many bamboos are well behaved and don't tend to spread. Some spread very slowly.
There are some, of which Sasa Palmatum is the best example, that are very agressive and invasive. Sasa Palmatum has some of the largest leaves, which look like paddles, so some people are tempted to grow it. It can only be grown safely with great care. To grow in a garden adjacent to another garden is extremely anti-social. It's possible they didn't realise how dangerous the plant is.
We don't know what type of bamboo you have.
If it is Sasa, you could try digging it up, but it will only come back, from the other side.
The only real solution is to kill the part of the plant on your land, possibly using glyphosate. That might result in damage, possibly fatal, to the plant on your neighbour's side of the garden. I don't know what the legal position is, for trying to kill a plant, which actually belongs to your neighbour.
If it is one of the more acceptable varieties of bamboo, then you might be able to control it simply by cutting it down.
You really need to speak to your neighbour, and agree on some plan between you.
That's just my view.