These growths are called suckers, and they can be common both on living trees (some species sucker more than others, especially if damaged or stressed) and recently-removed trees. Suckers arise directly from living roots, which on a tree that was cut down, might persist for several years after main tree removal. Suckering is also common on healthy, living crapemyrtle.

You have two options -- physical removal or chemical treatment (herbicide). Both will probably need to be repeated until successful. Herbicide that is systemic, like glyphosate, should eventually work, as it kills roots after absorption into the foliage. The catch is that the amount of roots left behind supporting this relatively small amount of sucker growth (remember, it was supporting an entire canopy before the trunk was cut) is large, so there is lot of stored energy for growth supporting the re-emergence of suckers each time some succumb to herbicide. Therefore, you might be re-treating new suckers for some time until they go away. Hard to tell how long this process will take, but it might last into next year at the worst, as an educated guess. It may depend on how healthy the root system was for the tree removed, since you said it was a large specimen.

When using herbicide on plants with very water-repellent, waxy leaves, you might need to add a surfactant. Also called spreader-stickers, these additives help the spray adhere to the leaf long enough to be better absorbed. Always verify the herbicide chosen allows for the use of a surfactant, though, as some formulations might already include one, or may be incompatible with one. These products have their own usage label and tend to be sold alongside pesticides in garden centers, since the two are often used together.

The chemical-free approach is to just keep removing any sucker growth that appears as promptly and thoroughly as you can. This serves to starve the roots, since each time sprouts form, they must use-up some of the root energy stores, and if you deny them the ability to photosynthesize by removing them promptly, you are keeping the roots from replenishing that energy for growth. Like using a battery without recharging it, eventually, this process will use-up enough root reserves that new suckers will become more sporadic and then just cease to form as the roots die out and start to decay. Here too, it's hard to guess how long that will take, but at least is saves you the expense of using herbicide and the risks that come along with it (having nearby plants be damaged by unintended drift of the spray, for example).

Miri

Thanks for your help!

So when you wrote:

"When using herbicide on plants with very water-repellent, waxy leaves"..., did you mean that a crape myrtle has very water-repellent, waxy leaves? And does Round Up work well on those leaves, or should I try another brand?

I'm first going to keep cutting the leaves and then I will spray the small stalks that go underground.

Thanks so much!

Crapemyrtle foliage is moderately waxy, yes; some plants, like Euonymus, are waxier, so it's all relative. (All plant foliage has some wax on it.) Roundup is a brand name, and not all formulations under that name use the same ingredients. We have not tested a specific formulation of Roundup, but glyphosate, the active ingredient in many types of Roundup, is a systemic ingredient used to kill the roots of perennial and woody plants. Triclopyr is another common systemic ingredient, but it carries greater risks to accidentally exposing other plants. The main advantage of using triclopyr might be that it doesn't affect grasses to the extent that it affects broadleaf plants, so if suckers were growing up through a lawn, you might be able to spray them with triclopyr without worrying about lawn damage (as always, consult the label to be certain, as every product differs), versus having all foliage contacted by a glyphosate spray affected more-or-less equally. There are brands that use glyphosate in their products that do not have the Roundup brand name; it's up to you which you choose to use.

Miri

You're welcome!