If you refuse to use herbicides then yes, cutting the bamboo back repeatedly will eventually kill it as it will receive no chlorophyll.
Something that may help is pushing the pH to an extreme that is unpleasant for the bamboo. If the pH is too extreme, the bamboo will no longer be able to absorb nutrients well.
- Bamboo like slightly acid soil -- around 6 on a scale of 1-14. See this GreenEarthBamboo.com's article on pH here.
- You do need to consider the needs of the other plants the bamboo is growing through, though.
The more you stress the bamboo, the faster it will die. If the surrounding plants will allow a more acid soil, then you can lower your soil's pH (make it more acidic) by using several products. These include:
- sphagnum peat,
- elemental sulfur,
- aluminum sulfate,
- iron sulfate,
- acidifying nitrogen, and
- organic mulches.
If the plants like it more alkaline, then pelletized lime will push it in the other direction. Go to this website for more info on soil acidity and ways to adjust it.
ALWAYS test your soil first. Usually your state co-operative extension will help you with this. Soil pH is one of the major stressors of lawns and plants and the most frequently ignored by gardeners and often landscapers.
For more information:
Read Richard Heller's Q&A "I'm interested in planting bamboo in my yard. How can I keep it from spreading?"