Finiteness Properties of Fibre Products over Virtually Nilpotent Quotients
Abstract
The n-(n+1)-(n+2) theorem, recently established by Cohen and Shusterman, says that if two groups of type Fn+1 map onto a common quotient Q of type Fn+2 and one of the two kernels is of type Fn, then the associated fibre product is of type Fn+1. We prove a stronger, symmetric variant of this theorem when Q is virtually nilpotent. More precisely, for i∈{1,2}, let 1→ Ni→ Γioversetsmallπi→ Q→ 1 be short exact sequences of groups and let P={(γ1,γ2)∈Γ1×Γ2: π1(γ1)=π2(γ2)} be their fibre product. Let k,l,m,n∈ℕ0, assume that N1 and N2 are of type FPk and FPl, respectively, and that Γ1 and Γ2 are of type Fm and Fn, respectively. We prove that P is of type Fmin{k+l+1,m,n}. The same formula holds with the homological finiteness properties FPr in place of Fr. This provides an alternative proof of the homotopical and homological Virtual Surjections Theorem, which gives a finiteness criterion for subgroups of direct products in terms of their embedding in the ambient product.
Related
- Homological finiteness properties of wreath products
- Quantitative finiteness for maximal complex totally geodesic submanifolds of non-arithmetic ball quotients
- On the residual finiteness of the non-abelian exterior square of wreath products and the Grigorchuk group
- Essential quotients of the cohomology of products of symmetric and alternating groups
- Kronecker Products, Polarity Quotients and Large Graph Constructions
BibTeX
@misc{kuckuck2026finiteness,
title = {{Finiteness Properties of Fibre Products over Virtually Nilpotent Quotients}},
author = {Benno Kuckuck},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08911},
eprint = {2609.08911},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08911},
}
RIS
TY - UNPB
TI - Finiteness Properties of Fibre Products over Virtually Nilpotent Quotients
AU - Benno Kuckuck
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08911
UR - https://arxiv.org/abs/2609.08911
AB - The n-(n+1)-(n+2) theorem, recently established by Cohen and Shusterman, says that if two groups of type F_n+1 map onto a common quotient Q of type F_n+2 and one of the two kernels is of type F_n, then the associated fibre product is of type F_n+1. We prove a stronger, symmetric variant of this theorem when Q is virtually nilpotent. More precisely, for i∈{1,2}, let 1→ N_i→ Γ_ioversetsmallπ_i→ Q→ 1 be short exact sequences of groups and let P={(γ_1,γ_2)∈Γ_1×Γ_2: π_1(γ_1)=π_2(γ_2)} be their fibre product. Let k,l,m,n∈ℕ_0, assume that N_1 and N_2 are of type FP_k and FP_l, respectively, and that Γ_1 and Γ_2 are of type F_m and F_n, respectively. We prove that P is of type F_min{k+l+1,m,n}. The same formula holds with the homological finiteness properties FP_r in place of F_r. This provides an alternative proof of the homotopical and homological Virtual Surjections Theorem, which gives a finiteness criterion for subgroups of direct products in terms of their embedding in the ambient product.
ER -