A
RUPRECHT-KARLS-UNIVERSITÄT HEIDELBERG
Mathematisches Institut
Vorlesung Geometrische Gruppentheorie
Heidelberg, 06.12.2013
Exercise sheet 8
Intermediate growth
To hand in by Friday December 13, 12:00
Exercise 1. Let G be a finitely generated infinite group, and Gm = G × G × ... × G, m times,
m ≥ 2. Prove that:
1. If G has polynomial growth, then also Gm has polynomial growth, but their growth types
are different.
2. If G has exponential growth, then also Gm has exponential growth, and they have the
same growth type.
Exercise 2. Let f : R≥0 → R≥0 be a generalised growth function, increasing and such that
f (x) ≥ x. Assume also that for some m > 1 we have f f m .
1. Prove that f is not of polynomial type.
2. Prove that there exists A, ν > 0 such
write explicitely the definition of the
choose a suitable k, k must be small
enough, then mk ≥ Axν for some A, ν
that f (x) exp(Axν ). (Hint: Consider log f , and
inequality f f m , iterated k times. For every x
enough, such that log f (x) ≥ mk . If k is also big
> 0 independent on x).
Exercise 3. Let T be an infinite binary tree, with root r. Recall that every vertex of the tree
corresponds to a word in {0,1}, the length of the word corresponding to the level of the vertex
(distance from the root r). For every vertex v, let Tv denote the subtree rooted at v, and let
iv denote the automorphism iv : Aut(T ) → Aut(Tv ) < Aut(T ). Let a ∈ Aut(T ) denote the
involution that exchanges T0 and T1 .
Let φ denote the group homomorphism Aut(T ) × Aut(T ) → Aut(T ) given by φ(τ0 , τ1 ) =
i0 (τ0 ) · i1 (τ1 ).
The group Z/2Z acts on Aut(T ) × Aut(T ) by exchanging the two factors. Consider the
semi-direct product A = (Aut(T ) × Aut(T )) o Z/2Z with reference to this action. Consider the
map Φ : A → Aut(T ) defined by: Φ(τ0 , τ1 , 0) = φ(τ0 , τ1 ), Φ(τ0 , τ1 , 1) = φ(τ0 , τ1 ) · a.
Prove that Φ is a group isomorphism.
Exercise 4. We use again the notations of exercise 3. For every vertex v let av denote the
automorphism iv (a). The vertex 1n 0 is the vertex 1 . . . 10, where 1 is repeated n times.
1. Prove that there exists 3 elements b, c, d ∈ Aut(T ) such that b = φ(a, c), c = φ(a, d),
d = φ(Id, b). These elements are described by the following formulae:
b = (a0 · a13 0 · a16 0 · . . . ) · (a10 · a14 0 · a17 0 · . . . )
c = (a0 · a13 0 · a16 0 · . . . ) · (a12 0 · a15 0 · a18 0 · . . . )
d = (a10 · a14 0 · a17 0 · . . . ) · (a12 0 · a15 0 · a18 0 · . . . )
2. Prove that the elements b, c, d have order 2, commute with each other and satisfy the
relation bcd = Id. Hence hb, c, di is isomorphic to (Z/2Z)2 .
3. Prove the following relations: (ad)4 = (ac)8 = (ab)16 = Id. Deduce that the subgroups
ha, bi , ha, ci , ha, di are finite.