Fake Schottky groups acting on the 3-sphere
A Kleinian group \(\Gamma\) on \(S^n, n\geq 2\), is said to be fake Schottky if \(\Gamma\) satisfies the following conditions: (A) \(\Gamma\) is a free group of finite rank; (B) \(\Gamma\) contains no parabolic elements; (C) \(\Gamma\) is geometrically finite; (D) \(\Gamma\) is not a Schottky group. The Schottky groups are well known to satisfy conditions (A) - (C). Bg virtue of the Maskit theorem the conditions (A) and (B) are necessary and sufficient for the group \(\Gamma\) to be Schottky in dimension \(n=2\). Hence fake Schottky groups are absent in dimension 2. In the present article a fake Schottky group acting on \(S^3\) is constructed.
scientific article; zbMATH DE number 1023944