Tree stump

Nope! Sorry, they'll probably grow like mad. I would suggest digging down as far as you can, all round the stump, to expose the main roots, then cut through them with a chainsaw (or pull-saw if they are smaller), then pull out the stump.

As a conservation volunteer, this is a problem I face on a regular basis when building woodland footpaths.
It depends on the species and condition of the tree, as to whether or not it is likely to sprout again from the roots. If it is a broadleafed tree, then it is coppiceable - that means that shoots may spring up from the old stump. Suckers may also spring up from side roots, up to quite a fair distance away. (Cherries do this a lot, as do many fruit trees.)
I hate to break the bad news to you, but lack of light does not stop this process. It is gravitropic - after a disruption to the root/trunk energy exchange system, any root with energy will respond by pushing a tendril up against gravity, from whatever depth it is at - you are basically hoping that the root will run out of energy before leaves can provide input.
If it is a sycamore or an ash, newly cut down, it is a certainty that the stump will sprout new growth, and the solution is really just to remove as much of the stump to as great a depth as possible. With them you really want to remove it all, they are damned hard to kill. Yes, this is probably where the saying "being stumped" comes from - it isn't easy !
You can get machines called stump grinders, but I believe hiring them is expensive, and the alternative is to pay for a tree surgeon to use one - or use a grubbing mattock to do it yourself. Loppers or a pruning saw may also help.
A grubbing mattock is basically a pick, but it has an axe blade on one side of the head, and on the other side it has a wide adze blade, like an azada but more heavyweight. The correct technique is to dig a trench out round the stump, chopping through the side roots as you go and scooping out the soil, until you have created a pit on one side deep enough to start levering the stump into. As you do this (a normal pick or a pinchbar will do well, be very careful not to damage your back) you will expose the roots under the trunk, and once they are cut it is usually just a case of how do you get the damn thing up out of the hole. (Remove the earth from the roots and roll it, basically.)
If you are fit or lucky and it is a small stump of less than 50cm diameter, it may be under an hour's work. But was I you, I would assume three hours, and have a rest if you finish early.

Yups, dig it out. Said in less words than Monsieur Snohare

Thanks for those replies everyone. Its just as I feared. Will just have to do it the hard way.