Hatcher-Algebraic-Topology · Problem 6

Homology

Mathematics — Graph Theory · graphs-of-groups

Problem

Show that for a graph of groups all of whose edge homomorphisms are injective maps

\mathbb{Z} \to \mathbb{Z}

, we can choose

K\Gamma

to have universal cover a product

T \times \mathbb{R}

with

T

a tree. Work out in detail the case that the graph of groups is the infinite sequence

\mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{3} \mathbb{Z} \xrightarrow{4} \mathbb{Z} \to \cdots

where the map

\mathbb{Z} \xrightarrow{n} \mathbb{Z}

is multiplication by

n

. Show that

\pi_1(K\Gamma)

is isomorphic to

\mathbb{Q}

in this case. How would one modify this example to get

\pi_1(K\Gamma)

isomorphic to the subgroup of

\mathbb{Q}

consisting of rational numbers with denominator a power of 2?