On the space of ends of countable groups

The notion of ends of group is possibly the earliest geometric property of 
finitely generated groups, introduced by Freudenthal in 1930. It was 
readily established that every such group has 0, 1, 2, or infinitely many 
ends, and in the latter case is a Cantor space. Stallings in the late 60s 
characterized infinitely-ended finitely generated groups in terms of 
group splittings. In 1950, Specker found a natural way to define the space 
of ends for arbitrary groups, and proved that for an infinitely generated 
group, the space of ends is either a singleton or is infinite. We study 
the topology of this space. We prove that it has no isolated point, and 
that for some infinitely generated countable groups it is a Cantor space, 
while for some other cases it is non-metrizable.