= Infinite index lowers a polynomial growth upper bound
Let $G$ be a <finitely generated group> whose word balls have cardinality at most $C(1+r)^d$, and let $K$ be a finitely generated infinite-index <subgroup>. Then $K$ has <polynomial growth of a group> of degree at most $d-1$. To prove this, let $S,T$ be finite generating sets of $G,K$, and bound the $S$-length of every $T$-generator by $M$. The infinite locally finite <Schreier graph> of left cosets of $K$ has at least $r+1$ vertices within radius $r$. Choose representatives $g_j$ of those cosets with $|g_j|_S\le r$. The sets $g_jB_T(r)$ are disjoint and lie in $B_S((M+1)r)$. Therefore $(r+1)|B_T(r)|\le C(1+(M+1)r)^d$, which gives the claimed bound. In particular an infinite-index finitely generated subgroup cannot occur if the right side would force its nonempty word balls to shrink to zero cardinality.
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