Singular Loci of SL2(C)-Representation Varieties of Non-Hopfian Baumslag--Solitar Groups
Abstract
We study the singular loci of the reduced SL2(CC)-representation varieties Rm,n of non-Hopfian Baumslag--Solitar groups B(m,n)=⟨ a,b| b−1am b=an⟩. We determine their dimensions and profile functions, which list the numbers of irreducible components in descending order of dimension. The components have dimensions three or five. Their reduced singular loci have the explicit description Sing Rm,n ={(ε I,Y): εm=εn, (tr Y)2=((m+n)2)/mn}, where ε∈{±1} and Y∈SL2(CC). Each is a disjoint union of μ copies of the smooth affine quadric Q={(x,y,z)∈CC3:x2+yz=1}, where μ=2 for m−n odd and μ=4 for m−n even. Thus its complex algebraic isomorphism type depends only on this parity. Each quadric is homotopy equivalent to S2, so the singular locus has integral homology ZZμ in degrees zero and two and zero in all other degrees. An algebraic stratification gives its compactly supported E-polynomial μ(q2+q), with q=uv. At every singular point, Rm,n is locally analytically the product of a smooth surface and a nodal curve.
Related
- On Small Doubling in Right-Ordered Groups and Baumslag-Solitar Groups-II
- Baumslag-Solitar Subgroups Obstruct Model Companions for Fields with Group Actions
- On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and SLn(Z)
- The measure equivalence classification of Baumslag-Solitar groups
- Singular Loci of SL2(C)-Representation Varieties of Torus Knot Groups
BibTeX
@misc{liriano2026singular,
title = {{Singular Loci of SL2(C)-Representation Varieties of Non-Hopfian Baumslag--Solitar Groups}},
author = {S. Liriano},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03402},
eprint = {2611.03402},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03402},
}
RIS
TY - UNPB
TI - Singular Loci of SL2(C)-Representation Varieties of Non-Hopfian Baumslag--Solitar Groups
AU - S. Liriano
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03402
UR - https://arxiv.org/abs/2611.03402
AB - We study the singular loci of the reduced SL_2(CC)-representation varieties R_m,n of non-Hopfian Baumslag--Solitar groups B(m,n)=⟨ a,b| b^−1a^m b=a^n⟩. We determine their dimensions and profile functions, which list the numbers of irreducible components in descending order of dimension. The components have dimensions three or five. Their reduced singular loci have the explicit description Sing R_m,n ={(ε I,Y): ε^m=ε^n, (tr Y)^2=((m+n)^2)/mn}, where ε∈{±1} and Y∈SL_2(CC). Each is a disjoint union of μ copies of the smooth affine quadric Q={(x,y,z)∈CC^3:x^2+yz=1}, where μ=2 for m−n odd and μ=4 for m−n even. Thus its complex algebraic isomorphism type depends only on this parity. Each quadric is homotopy equivalent to S^2, so the singular locus has integral homology ZZ^μ in degrees zero and two and zero in all other degrees. An algebraic stratification gives its compactly supported E-polynomial μ(q^2+q), with q=uv. At every singular point, R_m,n is locally analytically the product of a smooth surface and a nodal curve.
ER -