arXiv subjects

Lukas Vandeputte

Publications and source records attributed to Lukas Vandeputte.

7 recordsLinked to original sources

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↗

arXiv2026

Residual Finiteness Growth in Virtually Nilpotent Groups

The residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ of a finitely generated group $G$ is a function that gives the smallest value of the index $[G:N]$ with $N$ a normal subgroup not containing a non-trivial element $g$, in function of the word norm of that element $g$. It has been studied for several classes of finitely generated groups, including free groups, linear groups and virtually abelian groups. This paper shows that if $G$ is virtually nilpotent, then $\text{RF}_G = \log^δ$ for some $δ\in \mathbb{N}\cup\{0\}$, with moreover an explicit formula for $δ$ in terms of Lie algebras. This implies in particular that it is an invariant of the complex Mal'cev completion, leading to the application that residual finiteness growth is a profinite invariant for virtually nilpotent groups.

math.GR↗

arXiv2026

Residual finiteness properties of some of Halls groups

In this article we study a class of central extensions of $\mathbb{Z}\wr\mathbb{Z}$, as first described by Hall. On the one hand, we consider groups of this type with cyclic centre, our construction yields a rich class of groups. In particular we obtain a group that is conjugacy separable with solvable word problem but unsolvable conjugacy problem, we obtain a group with large conjugacy separability growth but small conjugator length function and residual finiteness growth, and we also obtain both a class of groups that for most functions $f:\mathbb{N}\rightarrow\mathbb{N}$ larger then $n^3$, contain a group $G$ such that the conjugator length of $G$ is given by $f$, as well as a group where the conjugator length is superlinear but subquadratic. On the other hand we also consider a different group with larger centre. This is the first example of a group where the residual finiteness growth is faster than any polynomial but slower than any exponential.

math.GR↗

arXiv2025

Twisted Conjugacy Growth of the Generalised Heisenberg Groups

We determine the twisted conjugacy growth function for automorphisms on generalised Heisenberg groups. In particular we demonstrate for these groups that up to a natural equivalence this function is either given by $n^k$ or $n^k\ln(n)$ for some constant $k$. We give a precise description for which of these functions we obtain in terms of dimensions and ranks on the upper and lower central series. In particular we obtain that the twisted conjugacy growth is always bounded by the conjugacy growth.

math.GR↗

arXiv2025

Independence of the conjugacy problem and conjugacy separability

We construct a class of finitely generated groups which have arbitrarily large conjugacy separability function, but in which the conjugacy problem can be solved in polynomial time, demonstrating that the McKinsey algorithm for the conjugacy problem can have complexity which lies arbitrarily far from the optimum.

math.GR↗

arXiv2024

Arbitrary residual finiteness and conjugacy separability growth

In a recent paper, Henry Bradford showed that all sufficiently fast growing functions appear as the residual finiteness growth function of some group. In this paper we show that the groups there constructed are conjugacy separable and that their conjugacy separability growth is equal to the residual finiteness growth. It follows that all sufficiently fast growing functions appear as the conjugacy separability growth function of some group. We extend this construction to a new class of groups such that given functions $f_1,f_2$ under the same constraints and satisfying $f_2\geq f_1$, we can find a group such that the residual finiteness growth is given by $f_1$ and the conjugacy separability growth by $f_2$, showing that the residual finiteness growth and conjugacy separability growth behave independently and can lie arbitrarily far apart.

math.GR↗

arXiv2024

Effective conjugacy separability of virtually abelian groups

A natural question for groups $H$ is which data can be detected in its finite quotients. A subset $X \subset H$ is called separable if for all $h\in H \setminus X$, there exists an epimorphism $φ$ to a finite group $Q$ such that $φ(h)\notinφ(X)$. More specifically, a group is said to be conjugacy separable if every conjugacy class is separable. It is known that many classes of groups are conjugacy separable, including virtually free and polycyclic groups. The minimal order of the quotient $Q$, in terms of the complexity of the conjugacy classes under consideration, is captured by the conjugacy separability function $\mathrm{Conj}_H: \mathbb{N} \to \mathbb{N}$. This function is in general ill understood, in fact the only large class of groups for which it is known exactly are the abelian groups. Indeed, in this case $\mathrm{Conj}_H$ is equal to the residual finiteness function, that is the size of quotients needed to separate singletons, and thus logarithmic if the group is infinite. Recent work has described the residual finiteness function for the class of virtually abelian groups, which gives a lower bound for the conjugacy separability function. The main result of this paper is a characterization of $\mathrm{Conj}_H$ for every virtually abelian group $H$. If the corresponding extension is associated with an irreducible representation over $\mathbb{Q}$, we demonstrate that we obtain the same function as the residual finiteness function. However, if the representation is not irreducible, we find an expression that is in some cases strictly larger, which we illustrate with several examples.

math.GR↗