Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0904.3902

Enumeration of lifts of commuting elements of a group

Michael Natapov·Vladimir Turaev
View original source ↗Read source document ↗Export metadata ↓

Also available from

Abstract

Given commuting elements a, b of a group G and a group epimorphism q : G' \to G with finite kernel, the set of commuting lifts of a, b to G' is finite (possibly, empty). The second named author obtained a formula for the number of such lifts in terms of representations of Ker q. We apply this formula to several group epimorphisms q with the same kernel. In particular, we analyze the case where Ker q = Q_8 is the quaternion group. We show that in this case the number in question is equal to 0, 8, 16, 24, or 40. We show that all these numbers are realized by some G,G',q,a,b.

Explore related subjects

math.GR

Keep this discovery

BibTeXRIS

Michael Natapov, Vladimir Turaev. 2009-04-24. Enumeration of lifts of commuting elements of a group. https://doi.org/10.1088/1126-6708/2009/09/032

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗
arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗
arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗