Topological Baumslag Lemmas

Abstract

This chapter deals with one of the principal technical tools of the monograph, namely an extension of Baumslag’s Lemma. The following are some of the main ingredients in this chapter:

• 1. A Topological Baumslag Lemma, which gives sufficient conditions to guarantee nontriviality of a word

  $$\displaystyle w (t_1,\cdots , t_k) = g_1 \mu _1(t_1) \cdots g_k \mu _k (t_k) $$

  for large values of the parameters t _i. Here the μ _j(t _j)s are one-parameter subgroups of a continuous (possibly analytic) group. The proof is quite general and reminiscent of Tits’ proof (Tits, J Algebra 20(2):250–270, 1972) of the Tits’ alternative for discrete linear groups in that the underlying idea consists of a ping-pong argument.

• 2. The one-parameter subgroups are often (generalizations of) parabolic and hyperbolic one-parameter subgroups of $\operatorname {\mathrm {PSL}}_2(\mathbb {R})$. To handle elliptic subgroups we use a complexification and Zariski density trick by embedding $\operatorname {\mathrm {PSL}}_2(\mathbb {R})$ in $\operatorname {\mathrm {PSL}}_2(\mathbb {C})$ and reduce the elliptic case to the hyperbolic one (Lemma 3.4).