Topological Baumslag Lemmas

  • Sang-hyun Kim
  • Thomas Koberda
  • Mahan Mj
Part of the Lecture Notes in Mathematics book series (LNM, volume 2231)


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. 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 ti. Here the μj(tj)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. 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).



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sang-hyun Kim
    • 1
  • Thomas Koberda
    • 2
  • Mahan Mj
    • 3
  1. 1.School of MathematicsKorea Institute for Advanced StudySeoulRepublic of Korea
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  3. 3.School of MathematicsTata Institute of Fundamental ResearchMumbaiIndia

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