# The rank of the intersection of subgroups of a free group

Let $H$ and $K$ be finitely generated subgroups of a free group $F$, and suppose that $H$ has finite index in $F$. Is it true that $rank(H \cap K)-1 \leq (rank(H)-1)(rank(K)-1)$?

• (actually, I missed the "finite-index" part, so ignore my deleted comment). Nov 3 '13 at 19:32
• @mary seva, it looks like your question is about to be closed (and/or migrated) as 'not research level'. In fact, I think it is graduate-student level, and hence acceptable on MO. But I suspect the formulation, which reads like a homework problem, has irritated the voters to close. Could you tell us how this problem arises in your research?
– HJRW
Nov 3 '13 at 19:39
• While this is certainly not a great question, why does it get THAT many downvotes? Nov 3 '13 at 19:50
• @StefanKohl I am guessing because it looks like homework. Nov 3 '13 at 20:00