Pro-\(p\) groups with a finite number of ends.
For a pro-\(p\) group \(G\), the number of ends \(e(G)\) is defined as \(e(G)=1-d_0+d_1\), where \(d_i=\dim_{\mathbb{F}_p}H^i(G,\mathbb{F}_p[\![G]\!])\) for \(i=0,1\). The adequacy of this definition is confirmed by the obtained pro-\(p\) analogs of results on the number of ends of abstract groups. In particular, it is proved that, as in the abstract case, a pro-\(p\) group \(G\) can have only 0, 1, 2, or infinitely many ends. This is a consequence of a stronger result: let \(G\) be an infinite pro-\(p\) group; if the \(G\)-module \(H^1(G,\mathbb{F}_p[\![G]\!])\) has a nonzero finite submodule, then \(H^i(G,\mathbb{F}_p[\![G]\!])\cong\mathbb{F}_p\). It is easy to see that \(e(G)=0\) if and only if \(G\) is a finite \(p\)-group. A classification theorem for pro-\(p\) groups with two ends is given. Moreover, a sufficient condition for a pro-\(p\) group to have precisely one end is obtained (if \(G\) has an infinite finitely generated normal subgroup of infinite index, then \(e(G)=1\)).
scientific article; zbMATH DE number 2177660