Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 3 / b

Table of contents
- Solution b

Solution

Choose any finite generating set

T

of the finitely generated group

H

. Starting from a finite generating set

S_{0}

of

G

S = S_{0} \cup T .

Every

T

-word in

H

of length at most

r

is also an

S

-word in

G

of length at most

r

. The inclusion of the corresponding word-metric balls is injective, so the growth function of a finitely generated group satisfies

β_{H, T} (r) \leq β_{G, S} (r) (r \geq 0) .

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