JournalsggdVol. 12, No. 1pp. 289–358

Thompson groups for systems of groups, and their finiteness properties

  • Stefan Witzel

    Universität Bielefeld, Germany
  • Matthew C.B. Zaremsky

    University of Albany (SUNY), USA
Thompson groups for systems of groups, and their finiteness properties cover
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Abstract

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups FF, VV, VbrV_{\mathrm{br}} and FbrF_{\mathrm{br}} arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pure braid groups.

We give new examples of families of groups that admit a cloning system and study how the finiteness properties of the resulting generalized Thompson group depend on those of the original groups. The main new examples here include upper triangular matrix groups, mock reflection groups, and loop braid groups. For generalized Thompson groups of upper triangular matrix groups over rings of SS-integers of global function fields, we develop new methods for (dis-)proving finiteness properties, and show that the finiteness length of the generalized Thompson group is exactly the limit inferior of the finiteness lengths of the groups in the family.

Cite this article

Stefan Witzel, Matthew C.B. Zaremsky, Thompson groups for systems of groups, and their finiteness properties. Groups Geom. Dyn. 12 (2018), no. 1, pp. 289–358

DOI 10.4171/GGD/444