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The structure of limit groups over hyperbolic groups

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Abstract

Let Γ be a torsion-free hyperbolic group. We study Γ-limit groups which, unlike the fundamental case in which Γ is free, may not be finitely presentable or geometrically tractable. We define model Γ-limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary Γ-limit group L, we canonically construct a strict resolution of a model Γ-limit group, which encodes all homomorphisms L → Γ that factor through the given resolution. We propose this as the correct framework in which to study Γ-limit groups algorithmically. We enumerate all Γ-limit groups in this framework.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan St., Chicago, IL, 60607-7045, USA

    Daniel Groves

  2. Centre for Mathematical Sciences, DPMMS, Wilberforce Road, Cambridge, CB3 0WB, UK

    Henry Wilton

Authors
  1. Daniel Groves
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  2. Henry Wilton
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Correspondence to Daniel Groves.

Additional information

The work of the first author was supported by the National Science Foundation and by a grant from the Simons Foundation (#342049 to Daniel Groves).

The second author was supported by the EPSRC.

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Groves, D., Wilton, H. The structure of limit groups over hyperbolic groups. Isr. J. Math. 226, 119–176 (2018). https://doi.org/10.1007/s11856-018-1692-2

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  • DOI: https://doi.org/10.1007/s11856-018-1692-2

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