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Big Torelli groups: generation and commensuration. (English) Zbl 1516.57024

Summary: For any surface \(\Sigma\) of infinite topological type, we study the Torelli subgroup \(\mathcal J(\Sigma)\) of the mapping class group MCG \((\Sigma)\), whose elements are those mapping classes that act trivially on the homology of \(\Sigma \). Our first result asserts that \({\mathcal J}(\Sigma)\) is topologically generated by the subgroup of MCG \((\Sigma)\) consisting of those elements in the Torelli group which have compact support. Next, we prove the abstract commensurator group of \({\mathcal J}(\Sigma)\) coincides with MCG \((\Sigma)\). This extends the results for finite-type surfaces [B. Farb and N. V. Ivanov, Math. Res. Lett. 12, No. 2–3, 293–301 (2005; Zbl 1073.57012); T. E. Brendle and D. Margalit, Geom. Topol. 8, 1361–1384 (2004; Zbl 1079.57017); ibid. 12, No. 1, 97–101 (2008; Zbl 1128.57303); Y. Kida, J. Math. Soc. Japan 63, No. 2, 363–417 (2011; Zbl 1378.57027)] to the setting of infinite-type surfaces.

MSC:

57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
57M50 General geometric structures on low-dimensional manifolds
20F65 Geometric group theory
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