### Abstract

We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and construct a finite-index embedding of the fundamental group into the Coxeter group whose Davis complex is the universal cover of the polygonal complex. All of the groups which we embed have minimal index among torsion-free subgroups, and therefore are maximal among torsion-free subgroups.

Original language | English (US) |
---|---|

Pages (from-to) | 121-143 |

Number of pages | 23 |

Journal | Geometriae Dedicata |

Volume | 193 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Geometriae Dedicata*,

*193*(1), 121-143. https://doi.org/10.1007/s10711-017-0258-5

}

*Geometriae Dedicata*, vol. 193, no. 1, pp. 121-143. https://doi.org/10.1007/s10711-017-0258-5

**Maximal torsion-free subgroups of certain lattices of hyperbolic buildings and Davis complexes.** / Norledge, William; Thomas, Anne; Vdovina, Alina.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Maximal torsion-free subgroups of certain lattices of hyperbolic buildings and Davis complexes

AU - Norledge, William

AU - Thomas, Anne

AU - Vdovina, Alina

PY - 2018/4/1

Y1 - 2018/4/1

N2 - We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and construct a finite-index embedding of the fundamental group into the Coxeter group whose Davis complex is the universal cover of the polygonal complex. All of the groups which we embed have minimal index among torsion-free subgroups, and therefore are maximal among torsion-free subgroups.

AB - We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and construct a finite-index embedding of the fundamental group into the Coxeter group whose Davis complex is the universal cover of the polygonal complex. All of the groups which we embed have minimal index among torsion-free subgroups, and therefore are maximal among torsion-free subgroups.

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U2 - 10.1007/s10711-017-0258-5

DO - 10.1007/s10711-017-0258-5

M3 - Article

AN - SCOPUS:85020242768

VL - 193

SP - 121

EP - 143

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

IS - 1

ER -