On the area of the complement of the invariant component of certain \(b\)-groups and on sequences of terminal regular \(b\)-groups (Q1585435)

It is known that terminal regular \(b\)-groups are dense in the Bers boundary \(\partial T(G)\) of Teichmüller space. The author considers a convergent sequence \(\{\varphi_n\}^\infty_{n-1}\) of terminal regular \(b\)-groups in \(\partial T(G)\) and seeks to deduce properties of the limit group \(\varphi\) from the properties of the sequence. In particular, he considers a complicated asymptotic trace condition which he calls condition (a), and also a natural area condition, condition (b), which says that the Euclidean area of the complement of the invariant component of \(\varphi_n\) tends to 0 as \(n\) approaches infinity. He then proves: (i) conditions (a) and (b) together are sufficient in order that the limit group \(\varphi\) be totally degenerate; (ii) if \(\varphi\) is totally degenerate without accidental parabolics, then any sequence \(\varphi_n\) of terminal regular \(b\)-groups converging to \(\varphi\) must satisfy conditions (a) and (b); (iii) if \(\varphi\) is totally degenerate with accidental parabolics, then there is a sequence \(\varphi_n\) converging to \(\varphi\) which satisfies (b) but not (a); (iv) if \(\varphi\in\partial T(G)\) is arbitrary (totally degenerate or not), then there is a sequence \(\varphi_n\) converging to \(\varphi\) which satisfies (b); (v) if \(\dim(T(G))\geq 1\), then there exist convergent sequences \(\varphi_n\) satisfying (a) (but not (b)) whose limit group \(\varphi\) is not totally degenerate.