#  $R_{\infty}$–property for finitely generated torsion-free 2-step nilpotent
groups of small Hirsch length

Karel Dekimpe, Maarten Lathouwers
<EMAIL_ADDRESS><EMAIL_ADDRESS>Supported by
Methusalem grant METH/21/03 - long term structural funding of the Flemish
Government.Researcher funded by FWO PhD-fellowship fundamental research (file
number: 1102424N).

###### Abstract

In this paper we will show that finitely generated torsion-free 2-step
nilpotent groups of Hirsch length at most 6 do not have the
$R_{\infty}$–property, while there are examples of such groups of Hirsch
length 7 that do have the $R_{\infty}$–property.

##  1 Preliminaries

In this paper we study the twisted conjugacy relation on 2-step nilpotent
groups of small Hirsch length. To be precise, we will show that all of the
finitely generated, torsion-free 2-step nilpotent groups of Hirsch length at
most 6 admit an automorphism with finite Reidemeister number, while there are
examples of Hirsch length 7 that have the $R_{\infty}$–property. In this first
section, we will start by recalling some basic notions about finitely
generated torsion-free nilpotent groups and secondly about twisted conjugacy
and Reidemeister numbers, especially for these groups.

###  1.1 Hirsch length and $\mathcal{I}(n,m)$ groups

For any group $G$ we define inductively $\gamma_{1}(G):=G$ and
$\gamma_{i+1}(G):=[\gamma_{i}(G),G]$ (for $i\in{\wiskunde N}_{>0}$) to be the
lower central series of $G$. A group is c-step nilpotent if $\gamma_{c}(G)\neq
1$ and $\gamma_{c+1}(G)=1$. Any finitely generated nilpotent group $G$ admits
a series $1=G_{1}\lhd G_{2}\lhd\dots\lhd G_{s}=G$ with cyclic factors, i.e.
$G_{i+1}/G_{i}$ is cyclic (see for example [KM79, Theorem 17.2.2]). A group
having such a subnormal series with cyclic factors is called a polycyclic
group. The Hirsch length $h(G)$ of a polycyclic group $G$ is defined as the
number of infinite cyclic factors in such a series. One can show the following
properties regarding the Hirsch length.

######  Lemma 1.1 ([Seg83, page 16]).

If $G$ is a polycyclic group (e.g. a finitely generated nilpotent group), then
the Hirsch length is well-defined (i.e. independent of the chosen series).
Moreover, if we fix some $H\subseteq G$ and $N\lhd G$, then the following
holds:

  1. (i)

$h(H)\leq h(G)$

  2. (ii)

$h(H)=h(G)\>\Longleftrightarrow\>H\subseteq_{\text{fin}}G$ (i.e. $H$ is a
finite index subgroup of $G$)

  3. (iii)

$h(G)=h(N)+h(G/N)$

  4. (iv)

$h(G)=0\>\Longleftrightarrow\>G$ is finite

In this paper, we will consider 2-step nilpotent groups. For these groups, it
is well known that the commutator map is bilinear.

######  Lemma 1.2 ([MKS66]).

If $G$ is a $2$-step nilpotent group, then $[\>.\>,.\>]:G\times G\to G$ is
bilinear, i.e. for all $g_{1},g_{2},g_{1}^{\prime},g_{2}^{\prime}\in G$ it
holds that

$[g_{1}g_{2},g_{1}^{\prime}g_{2}^{\prime}]=[g_{1},g_{2}^{\prime}]\cdot[g_{2},g_{2}^{\prime}]\cdot[g_{1},g_{1}^{\prime}]\cdot[g_{2},g_{1}^{\prime}].$

The adapted lower central series $\sqrt{\gamma_{i}(G)}$ of $G$ is defined by
taking the isolator of the lower central series, i.e.
$\sqrt{\gamma_{i}(G)}:=\\{g\in G\>|\>\exists k\in{\wiskunde
N}_{>0}:\>g^{k}\in\gamma_{i}(G)\\}$. This series is central (i.e.
$[\sqrt{\gamma_{i}(G)},G]\subset\sqrt{\gamma_{i+1}(G)}$). It will terminate in
the trivial group if and only if $G$ is a torsion-free nilpotent group.
Moreover, the terms are characteristic subgroups of $G$. Hence, any
automorphism $\varphi\in{\rm Aut}(G)$ induces automorphisms
$\varphi_{i}\in{\rm Aut}(\sqrt{\gamma_{i}(G)}/\sqrt{\gamma_{i+1}(G)})$. For
finitely generated torsion-free nilpotent groups, the factors of the adapted
lower central series are free abelian. Thus, in this case, the induced
automorphisms $\varphi_{i}$ correspond to invertible integer matrices. We will
address the eigenvalues of these matrices by talking about the eigenvalues of
$\varphi_{i}$ (or more generally the eigenvalues of $\varphi$). We will use
the rank of the factors of the adapted lower central series to further divide
the class of finitely generated torsion-free 2-step nilpotent groups.

######  Definition 1.3.

Let $G$ be a finitely generated torsion-free 2-step nilpotent group. We say
that $G\in\mathcal{I}(n,m)$ (with $n,m\in{\wiskunde N}_{>0}$) if

$\frac{G}{\sqrt{\gamma_{2}(G)}}\cong{\wiskunde
Z}^{n}\quad\text{and}\quad\sqrt{\gamma_{2}(G)}\cong{\wiskunde Z}^{m}.$

Fix some $G\in\mathcal{I}(n,m)$. Note that
$h(G)=h(G/\sqrt{\gamma_{2}(G)})+h(\sqrt{\gamma_{2}(G)})=n+m$. Fix a
${\wiskunde Z}$-basis
$\\{g_{1}\sqrt{\gamma_{2}(G)},\dots,g_{n}\sqrt{\gamma_{2}(G)}\\}$ of
$G/\sqrt{\gamma_{2}(G)}$. Since $G$ is finitely generated nilpotent, it holds
that $\gamma_{2}(G)\subseteq_{\text{fin}}\sqrt{\gamma_{2}(G)}$ (by for example
[Bau71, Lemma 2.8]) and thus also $\gamma_{2}(G)\cong{\wiskunde Z}^{m}$. Note
that $\sqrt{\gamma_{2}(G)}\subset Z(G)$ (since the adapted lower central
series is a central series of $G$). Using Lemma 1.2 this implies that
$\gamma_{2}(G)$ is generated by $\\{[g_{i},g_{j}]\>|\>1\leq i<j\leq n\\}$.
Hence, we obtain that

$m=h(\gamma_{2}(G))\leq\\#\\{[g_{i},g_{j}]\>|\>1\leq i<j\leq n\\}\leq
n(n-1)/2.$

Thus when considering 2-step nilpotent groups of a fixed Hirsch length
$k\in{\wiskunde N}_{>0}$, we can distinguish them by using the classes
$\mathcal{I}(n,m)$ with $n,m\in{\wiskunde N}_{>0}$, $n+m=k$ and $m\leq
n(n-1)/2$.

###  1.2 Twisted conjugacy

Let $G$ be a group and $\varphi\in{\rm End}(G)$. We say that two elements
$a,b\in G$ are $\varphi$-conjugate or twisted conjugate if there exists some
$c\in G$ such that $a=cb\varphi(c)^{-1}$. Being twisted conjugate is an
equivalence relation on $G$. The number of equivalence classes is called the
Reidemeister number $R(\varphi)$ of $\varphi$ and the collection of all
Reidemeister numbers of automorphisms of $G$ is called the Reidemeister
spectrum ${\rm Spec}_{R}(G)$, i.e.

${\rm Spec}_{R}(G):=\\{R(\varphi)\>|\>\varphi\in{\rm
Aut}(G)\\}\subset{\wiskunde N}_{>0}\cup\\{\infty\\}.$

We say that $G$ has the $R_{\infty}$–property if ${\rm
Spec}_{R}(G)=\\{\infty\\}$.

In order to determine whether or not an automorphism has an infinite
Reidemeister number, we will make use of the next two (well-known) theorems.

######  Theorem 1.4 ([DG14, Lemma 2.2] and [Rom11, Corollary 4.2]).

Let $G$ be a finitely generated torsion-free $c$-step nilpotent group and
$\varphi\in{\rm Aut}(G)$, then the following are equivalent:

  1. (i)

$R(\varphi)=\infty$

  2. (ii)

There exists some $i=1,2,\dots,c$ such that $\varphi_{i}\in{\rm
Aut}\left(\frac{\sqrt{\gamma_{i}(G)}}{\sqrt{\gamma_{i+1}(G)}}\right)$ has $1$
as an eigenvalue.

  3. (iii)

There exists some $i=1,2,\dots,c$ such that $R(\varphi_{i})=\infty$.

######  Theorem 1.5 ([DGO21, Proposition 5] and [Rom11, Lemma 2.7]).

Let $G$ be a finitely generated nilpotent group. Let

$G=G_{1}\supseteq G_{2}\supseteq\dots\supseteq G_{s}\supseteq G_{s+1}=1$

be a central series of $G$ and $\varphi\in{\rm Aut}(G)$ such that the
following holds:

  1. 1.

All the factors $G_{i}/G_{i+1}$ (with $i=1,2,\dots,s$) are torsion-free.

  2. 2.

For all terms $G_{i}$ (with $i=1,2,\dots,s+1$) it holds that
$\varphi(G_{i})=G_{i}$.

Then it holds that:

$R(\varphi)=\prod_{i=1}^{s}R(\varphi_{i})$

where $\varphi_{i}:G_{i}/G_{i+1}\to G_{i}/G_{i+1}$ (with $i=1,2,\dots,s$) are
the induced automorphisms on the factor groups $G_{i}/G_{i+1}$.

In order to express the Reidemeister number, we will frequently use the
following notation

$|\>\cdot\>|_{\infty}:{\wiskunde Z}\to{\wiskunde
N}_{>0}\cup\\{\infty\\}:x\mapsto|x|_{\infty}:=\begin{cases}|x|&\text{ if
}x\neq 0\\\ \infty&\text{ if }x=0\end{cases}$

where $|\>\cdot\>|$ denotes the absolute value. Note that for a finitely
generated torsion-free nilpotent group, we can apply Theorem 1.5 to the
adapted lower central series of $G$. Combined with the well-known description
of the Reidemeister spectrum for free abelian groups of finite rank (see e.g.
[GW09]) we obtain the next result.

######  Lemma 1.6.

Let $G$ be a finitely generated torsion-free 2-step nilpotent group and
$\varphi\in{\rm Aut}(G)$, then

$R(\varphi)=R(\varphi_{1})R(\varphi_{2})=|\det(Id-\varphi_{1})|_{\infty}\>|\det(Id-\varphi_{2})|_{\infty}$

where $\varphi_{1}\in{\rm Aut}(G/\sqrt{\gamma_{2}(G)})$ and
$\varphi_{2}\in{\rm Aut}(\sqrt{\gamma_{2}(G)})$ are the induced automorphisms
on the factors of the adapted lower central series.

The next lemma allows us to more easily check that an endomorphism on a
finitely generated torsion-free nilpotent group is in fact an automorphism.

######  Proposition 1.7.

Let $G$ be a finitely generated torsion-free nilpotent group. If
$\varphi\in{\rm End}(G)$ and $\varphi_{1}\in{\rm
Aut}(G/\sqrt{\gamma_{2}(G)})$, then $\varphi\in{\rm Aut}(G)$.

###### Proof.

By (the proof of) Proposition 6.1.1 in [Sen23], we obtain the following
result:

If $\psi\in{\rm End}({\wiskunde Z}^{n})$ and $A\subset_{\text{fin}}{\wiskunde
Z}^{n}$ such that $\psi(A)=A$, then $\psi\in{\rm Aut}({\wiskunde Z}^{n})$. (1)

We use induction on the nilpotency class $c\in{\wiskunde N}_{>0}$ of $G$
together with result $(\ref{eq:claim})$ to prove the proposition. If $c=1$,
then there is nothing to prove, since in that case $\varphi=\varphi_{1}\in{\rm
Aut}(G)$.
Let $c>1$ and assume that the claim holds for finitely generated torsion-free
nilpotent groups of nilpotency class $\leq c-1$. Consider the group
$H:=G/\sqrt{\gamma_{c}(G)}$. Note that

$\gamma_{i}(H)=\frac{\gamma_{i}(G)\sqrt{\gamma_{c}(G)}}{\sqrt{\gamma_{c}(G)}}$

for all $i=1,\dots,c$. Hence, $H$ is a finitely generated torsion-free
nilpotent group of nilpotency class $\leq c-1$. Since $\varphi\in{\rm End}(G)$
and $\varphi(\sqrt{\gamma_{c}(G)})\subset\sqrt{\gamma_{c}(G)}$ (see for
example [Dek96, Lemma 1.1.2 (1)]) we can consider the induced morphism
$\overline{\varphi}\in{\rm End}(H)$. By [Dek96, Lemma 1.1.4] it holds that
$\sqrt[H]{\gamma_{2}(H)}=\sqrt[G]{\gamma_{2}(G)}/\sqrt[G]{\gamma_{c}(G)}$.
Hence, we obtain that

$\frac{H}{\sqrt[H]{\gamma_{2}(H)}}=\frac{H}{\sqrt[G]{\gamma_{2}(G)}/\sqrt[G]{\gamma_{c}(G)}}\cong\frac{G}{\sqrt[G]{\gamma_{2}(G)}}.$

We consider the induced morphism $\overline{\varphi}_{1}\in{\rm
End}(H/\sqrt{\gamma_{2}(H)})$. By the commutativity of the next diagram and
since $\varphi_{1}$ is an automorphism, also $\overline{\varphi}_{1}$ is an
automorphism.

${\frac{H}{\sqrt{\gamma_{2}(H)}}}$${\frac{H}{\sqrt{\gamma_{2}(H)}}}$${\frac{G}{\sqrt{\gamma_{2}(G)}}}$${\frac{G}{\sqrt{\gamma_{2}(G)}}}$$\scriptstyle{\overline{\varphi}_{1}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$${\circlearrowleft}$

Hence, by the induction hypothesis it follows that $\overline{\varphi}\in{\rm
Aut}(G/\sqrt{\gamma_{c}(G)})$.
Now we argue that $\varphi_{c}\in{\rm Aut}(\sqrt{\gamma_{c}(G)})$. Note that
$\gamma_{c}(G)\lhd_{\text{fin}}\sqrt{\gamma_{c}(G)}\cong{\wiskunde Z}^{n}$
(for some $n\in{\wiskunde N}_{>0}$) and
$\varphi(\gamma_{c}(G))\subset\gamma_{c}(G)$. Thus by result
$(\ref{eq:claim})$ it suffices to prove that
$\varphi(\gamma_{c}(G))=\gamma_{c}(G)$. Fix any $g_{1}\in G$ and
$g_{2}\in\gamma_{c-1}(G)$. Since $\overline{\varphi}$ is an automorphism,
there exists some $g_{1}^{\prime}\in G$ and $h_{1}\in\sqrt{\gamma_{c}(G)}$
such that $g_{1}=\varphi(g_{1}^{\prime})h_{1}$. Moreover, since
$\overline{\varphi}(\gamma_{c-1}(H))=\gamma_{c-1}(H)$ and
$g_{2}\sqrt{\gamma_{c}(G)}\in\gamma_{c-1}(H)$ there are
$g_{2}^{\prime}\in\gamma_{c-1}(G)\sqrt{\gamma_{c}(G)}$ and
$h_{2}\in\sqrt{\gamma_{c}(G)}$ such that $g_{2}=\varphi(g_{2}^{\prime})h_{2}$.
Since $G$ is a finitely generated torsion-free $c$-step nilpotent group, it
follows that $\sqrt{\gamma_{c}(G)}\subset Z(G)$. Hence, we obtain that

$[g_{1},g_{2}]=[\varphi(g_{1}^{\prime})h_{1},\varphi(g_{2}^{\prime})h_{2}]=[\varphi(g_{1}^{\prime}),\varphi(g_{2}^{\prime})]=\varphi([g_{1}^{\prime},g_{2}^{\prime}])\in\varphi([G,\gamma_{c-1}(G)\sqrt{\gamma_{c}(G)}])=\varphi(\gamma_{c}(G)).$

So we can conclude that $\varphi(\gamma_{c}(G))=\gamma_{c}(G)$ and thus by
result $(\ref{eq:claim})$ it follows that $\varphi_{c}\in{\rm
Aut}(\sqrt{\gamma_{c}(G)})$.
Any finitely generated torsion-free nilpotent group is Hopfian (i.e. any
epimorphism is an automorphism) and thus it suffices to prove that $\varphi$
is surjective. Fix any $g\in G$. Since $\overline{\varphi}$ is an
automorphism, there exists some $g^{\prime}\in G$ such that
$\varphi(g^{\prime})\sqrt{\gamma_{c}(G)}=g\sqrt{\gamma_{c}(G)}$. Using that
$\varphi_{c}\in{\rm Aut}(\sqrt{\gamma_{c}(G)})$ we can take some
$h\in\sqrt{\gamma_{c}(G)}$ such that
$g=\varphi(g^{\prime})\varphi(h)=\varphi(g^{\prime}h)$. Hence, $\varphi$ is a
surjective morphism and thus it is an automorphism. ∎

##  2 Reidemeister spectrum of groups in $\mathcal{I}(n,1)$

In this section, we determine the Reidemeister spectrum of groups in
$\mathcal{I}(n,1)$.

######  Proposition 2.1 ([Seg83, Proposition 5 (p. 265)]).

Let $n\in{\wiskunde N}_{>1}$ and $G\in\mathcal{I}(n,1)$, then $G$ is
isomorphic to exactly one of the following groups:

  1. (i)

If $n=2r$:

$G(d_{1},d_{2},\dots,d_{r}):=\Bigg{\langle}x_{1},\dots,x_{r},y_{1},\dots,y_{r},z\>\bBigg@{3}|\>\begin{array}[]{ll}[x_{i},y_{j}]=[x_{i},x_{j}]=[y_{i},y_{j}]=1&\text{
if }i\neq j\\\ [x_{i},y_{i}]=z^{d_{i}}&\text{ for all }i\\\ z\text{ is
central}&\end{array}\Bigg{\rangle}$

where $d_{1}|d_{2}|\dots|d_{r}$, each $d_{i}\in{\wiskunde N}$ and $d_{1}\neq
0$.

  2. (ii)

If $n=2r+1$:

$G(d_{1},d_{2},\dots,d_{r})\times{\wiskunde Z}$

where $d_{1}|d_{2}|\dots|d_{r}$, each $d_{i}\in{\wiskunde N}$ and $d_{1}\neq
0$.

Now we determine the Reidemeister spectrum of both classes of groups, starting
with the case of even $n$.

######  Theorem 2.2.

Let $n\in{\wiskunde N}_{>1}$ with $n=2r$ and $G\in\mathcal{I}(n,1)$, then

${\rm Spec}_{R}(G)=2{\wiskunde N}_{>0}\cup\\{\infty\\}.$

###### Proof.

By Proposition 2.1 $(i)$ we can assume that $G=G(d_{1},\dots,d_{r})$ with
$d_{1}|d_{2}|\dots|d_{r}$, each $d_{i}\in{\wiskunde N}$ and $d_{1}\neq 0$.
Note that $\sqrt{\gamma_{2}(G)}=\langle z\rangle\cong{\wiskunde Z}$.
Fix any automorphism $\varphi\in{\rm Aut}(G)$ wit $R(\varphi)<\infty$. By
Lemma 1.6 we know that $R(\varphi)=R(\varphi_{1})R(\varphi_{2})$. Since the
only finite Reidemeister number of ${\wiskunde Z}$ is 2 (see for example
[Rom11]), it holds that $R(\varphi_{2})=2$. Hence, $R(\varphi)\in 2{\wiskunde
N}_{>0}\cup\\{\infty\\}$.
Note that the identity automorphism has infinite Reidemeister number. Fix
$k_{1},\dots,k_{r}\in{\wiskunde N}_{>0}$ and define the map $\varphi$ on $G$
that is induced by

$\varphi(x_{i})=y_{i},\quad\varphi(y_{i})=x_{i}y_{i}^{-k_{i}},\quad\varphi(z)=z^{-1}.$

Using Lemma 1.2, we obtain that

$[\varphi(x_{i}),\varphi(y_{i})]=[y_{i},x_{i}y_{i}^{-k_{i}}]=[y_{i},x_{i}][y_{i},y_{i}^{-k_{i}}]=z^{-d_{i}}=\varphi(z)^{d_{i}}.$

One can easily check that the other relations are also preserved and thus
$\varphi\in{\rm End}(G)$. The matrix of $\varphi_{1}\in{\rm
Aut}(G/\sqrt{\gamma_{2}(G)})$ with respect to the ${\wiskunde Z}$-basis
$\\{x_{1}\sqrt{\gamma_{2}(G)},\dots,x_{r}\sqrt{\gamma_{2}(G)},y_{1}\sqrt{\gamma_{2}(G)},\dots,y_{r}\sqrt{\gamma_{2}(G)}\\}$
equals

$\begin{pmatrix}0&\mathds{1}_{r}\\\ \mathds{1}_{r}&-{\rm
diag}(k_{1},\dots,k_{r})\end{pmatrix}$

which is an invertible integer matrix. Hence, Proposition 1.7 tells us that
$\varphi\in{\rm Aut}(G)$. Note that $\varphi_{2}\in{\rm
Aut}(\sqrt{\gamma_{2}(G)})$ is the inversion map on ${\wiskunde Z}$ and thus
by Lemma 1.6 we obtain that

$R(\varphi)=R(\varphi_{1})R(\varphi_{2})=\left|\det\left(\mathds{1}_{n}-\begin{pmatrix}0&\mathds{1}_{r}\\\
\mathds{1}_{r}&-{\rm
diag}(k_{1},\dots,k_{r})\end{pmatrix}\right)\right|_{\infty}|\det(1-(-1))|_{\infty}=2k_{1}\dots
k_{r}.$

By taking for example $k_{2}=\dots=k_{r}=1$, we obtain all of $2{\wiskunde
N}_{>0}$ as Reidemeister number of an automorphism of $G$. Hence, we proved
that ${\rm Spec}_{R}(G)=2{\wiskunde N}_{>0}\cup\\{\infty\\}$. ∎

Now we consider the case where $n=2r+1$ is odd.

######  Theorem 2.3.

Let $n\in{\wiskunde N}_{>1}$ with $n=2r+1$ and $G\in\mathcal{I}(n,1)$. Fix
$d_{1},\dots,d_{r}\in{\wiskunde N}$ (with $d_{1}\neq 0$ and
$d_{1}|\dots|d_{r}$) such that $G\cong G(d_{1},\dots,d_{r})\times{\wiskunde
Z}$, then

${\rm Spec}_{R}(G)=\begin{cases}2{\wiskunde N}_{>0}\cup\\{\infty\\}&\text{ if
}d_{r}=0\\\ 4{\wiskunde N}_{>0}\cup\\{\infty\\}&\text{ if }d_{r}\neq
0\end{cases}.$

###### Proof.

Without loss of generality we assume that
$G=G(d_{1},\dots,d_{r})\times{\wiskunde Z}$ and we denote with $u$ a generator
of the ${\wiskunde Z}$-factor.
We first consider the case where $d_{r}=0$ (and thus, since $d_{1}\neq 0$,
$r>1$). Note that then

$\displaystyle G$
$\displaystyle=\Bigg{\langle}x_{1},\dots,x_{r},y_{1},\dots,y_{r},z,u\>\bBigg@{3}|\>\begin{array}[]{ll}[x_{i},y_{j}]=[x_{i},x_{j}]=[y_{i},y_{j}]=1&\text{
if }i\neq j\\\ [x_{i},y_{i}]=z^{d_{i}}&\text{ for all }i\leq r-1\\\
x_{r},y_{r},u,z\text{ is central}&\end{array}\Bigg{\rangle}$
$\displaystyle=G(d_{1},\dots,d_{r-1})\times{\wiskunde Z}^{3}$

where ${\wiskunde Z}^{3}$ is generated by $x_{r},y_{r}$ and $u$. Moreover,
$\sqrt{\gamma_{2}(G)}=\langle z\rangle$. Hence, completely similar as in the
proof of Theorem 2.2, we obtain that ${\rm Spec}_{R}(G)\subset 2{\wiskunde
N}_{>0}\cup\\{\infty\\}$. For the other inclusion, take any $k\in{\wiskunde
N}_{>0}$ arbitrary. By Theorem 2.2 we can fix some $\psi\in{\rm
Aut}(G(d_{1},\dots,d_{r-1}))$ such that $R(\psi)=2$. Since ${\rm
Spec}_{R}({\wiskunde Z}^{3})={\wiskunde N}_{>0}\cup\\{\infty\\}$ (see for
example [Rom11, Section 3]), we can fix some $\psi^{\prime}\in{\rm
Aut}({\wiskunde Z}^{3})$ such that $R(\psi^{\prime})=k$. By Proposition 2.4 in
[Sen21] it follows that the automorphism
$\varphi:=\psi\times\psi^{\prime}\in{\rm Aut}(G)$ has Reidemeister number
$R(\varphi)=R(\psi)R(\psi^{\prime})=2k$. Hence, we argued that ${\rm
Spec}_{R}(G)=2{\wiskunde N}_{>0}\cup\\{\infty\\}$.

We now consider the case $d_{r}\neq 0$ (and thus $d_{i}\neq 0$ for all $i$,
since $d_{1}|\dots|d_{r}$). Fix some automorphism $\varphi\in{\rm Aut}(G)$
with $R(\varphi)<\infty$. Since $\sqrt{\gamma_{2}(G)}=\langle z\rangle$ and
$R(\varphi_{2})<\infty$, it must hold that $\varphi(z)=z^{-1}$ and thus
$R(\varphi_{2})=2$. Note that since $d_{i}\neq 0$ for all $i$, it holds that
$Z(G)=\langle u,z\rangle\cong{\wiskunde Z}^{2}$. Since $Z(G)$ is a
characteristic subgroup and $\varphi|_{Z(G)}$ is an automorphism, it thus
follows that $\varphi(u)=u^{\alpha}z^{\beta}$ with $\alpha\in\\{-1,1\\}$ and
$\beta\in{\wiskunde Z}$. If we denote with $\overline{\varphi}\in{\rm
Aut}(G/Z(G))$ the induced automorphism, then we obtain by applying Theorem 1.5
to $G\supset Z(G)\supset 1$ that

$R(\varphi)=R(\varphi|_{Z(G)})R(\overline{\varphi})=R(\overline{\varphi})\left|\det\left(\mathds{1}_{2}-\begin{pmatrix}\alpha&0\\\
\beta&-1\end{pmatrix}\right)\right|_{\infty}=2|1-\alpha|_{\infty}R(\overline{\varphi}).$

Since $R(\varphi)<\infty$, we thus obtain that $\alpha=-1$ and
$R(\varphi)=4R(\overline{\varphi})\in 4{\wiskunde N}_{>0}\cup\\{\infty\\}$.
Note that the identity morphism has infinite Reidemeister number. Fix some
$k_{1},\dots,k_{r}\in{\wiskunde N}_{>0}$ and consider the map $\varphi$ on $G$
induced by

$\varphi(x_{i})=y_{i},\quad\varphi(y_{i})=x_{i}y_{i}^{-k_{i}},\quad\varphi(z)=z^{-1},\quad\varphi(u)=u^{-1}.$

One can check that $\varphi$ preserves the relations (using Lemma 1.2) and
thus it induces an endomorphism of $G$. The matrix of the induced map
$\varphi_{1}\in{\rm Aut}(G/\sqrt{\gamma_{2}(G)})$ with respect to the
${\wiskunde Z}$-basis
$\\{x_{1}\sqrt{\gamma_{2}(G)},\dots,x_{r}\sqrt{\gamma_{2}(G)},y_{1}\sqrt{\gamma_{2}(G)},\dots,y_{r}\sqrt{\gamma_{2}(G)},u\sqrt{\gamma_{2}(G)}\\}$
equals

$\begin{pmatrix}0&\mathds{1}_{r}&0\\\ \mathds{1}_{r}&-{\rm
diag}(k_{1},\dots,k_{r})&0\\\ 0&0&-1\end{pmatrix}$

which is an invertible integer matrix. Hence, Proposition 1.7 tells us that
$\varphi\in{\rm Aut}(G)$. By applying Lemma 1.6 we obtain that

$R(\varphi)=R(\varphi_{1})R(\varphi_{2})=\left|\det\left(\mathds{1}_{n}-\begin{pmatrix}0&\mathds{1}_{r}&0\\\
\mathds{1}_{r}&-{\rm diag}(k_{1},\dots,k_{r})&0\\\
0&0&-1\end{pmatrix}\right)\right|_{\infty}|1-(-1)|_{\infty}=4k_{1}\dots
k_{r}.$

We can now for example take $k_{2}=\dots=k_{r}=1$ to obtain an automorphism
$\varphi\in{\rm Aut}(G)$ with $R(\varphi)=4k_{1}$. Since $k_{1}\in{\wiskunde
N}_{>0}$ can be taken arbitrary, we proved that ${\rm Spec}_{R}(G)=4{\wiskunde
N}_{>0}\cup\\{\infty\\}$. ∎

##  3 Hirsch length 5 or less

Recall that if $G$ is a finitely generated torsion-free 2-step nilpotent
group, then $G\in I(n,m)$ for some $n,m>0$ with $h(G)=n+m$ and $m\leq
n(n-1)/2$. Hence, there do not exist such groups of Hirsch length 1 or 2 and
the only ones of Hirsch length 3 and 4 belong to $\mathcal{I}(2,1)$ and
$\mathcal{I}(3,1)$ respectively. By Theorem 2.2 and 2.3 we know that those of
Hirsch length 3 have Reidemeister spectrum $2{\wiskunde
N}_{>0}\cup\\{\infty\\}$ and those of Hirsch length 4 have Reidemeister
spectrum $4{\wiskunde N}_{>0}\cup\\{\infty\\}$. Again by Theorem 2.2, we
already know that all groups in $\mathcal{I}(4,1)$ have Reidemeister spectrum
$2{\wiskunde N}_{>0}\cup\\{\infty\\}$. Thus to understand the situation up to
Hirsch length 5, we only need to consider the groups of $\mathcal{I}(3,2)$.
The next lemma gives a presentation for any of those groups. In this section,
we shorten the notation of the presentations and omit all trivial commutators
of generators. E.g. in the statement of the following lemma, it is implicitly
understood that $[x_{2},x_{3}]=1$ and also $[x_{i},z_{j}]=1$ (for $i=1,2,3$
and $j=1,2$).

######  Lemma 3.1.

Let $G\in\mathcal{I}(3,2)$, then it can be presented via

$G=\langle
x_{1},x_{2},x_{3},z_{1},z_{2}\>|\>[x_{1},x_{2}]=z_{1}^{\alpha}z_{2}^{\beta},[x_{1},x_{3}]=z_{2}^{\gamma}\rangle$

with $\alpha,\gamma\in{\wiskunde Z}\setminus\\{0\\}$ and $\beta\in{\wiskunde
Z}$.

###### Proof.

Since $G\in\mathcal{I}(3,2)$ it has a presentation

$G=\langle
x_{1},x_{2},x_{3},z_{1},z_{2}\>|\>[x_{1},x_{2}]=z_{1}^{s_{12}}z_{2}^{t_{12}},[x_{1},x_{3}]=z_{1}^{s_{13}}z_{2}^{t_{13}},[x_{2},x_{3}]=z_{1}^{s_{23}}z_{2}^{t_{23}}\rangle$

where $s_{ij},t_{ij}\in{\wiskunde Z}$. Hence, $G/\langle
z_{2}\rangle\in\mathcal{I}(3,1)$ and by Proposition 2.1 we may assume that
$G/\langle z_{2}\rangle$ has the following presentation

$G/\langle
z_{2}\rangle=\langle\overline{x_{1}},\overline{x_{2}},\overline{x_{3}},\overline{z_{1}}\>|\>[\overline{x_{1}},\overline{x_{2}}]=\overline{z_{1}}^{\alpha}\rangle$

for some $\alpha\neq 0$. Thus, we obtain that $G$ has a presentation (with
$\beta=t_{12}$) of the form

$G=\langle
x_{1},x_{2},x_{3},z_{1},z_{2}\>|\>[x_{1},x_{2}]=z_{1}^{\alpha}z_{2}^{\beta},[x_{1},x_{3}]=z_{2}^{t_{13}},[x_{2},x_{3}]=z_{2}^{t_{23}}\rangle.$

Now we can assume that $t_{13}$ and $t_{23}$ are both non-zero. Indeed, if
only one of both is zero, the result trivially holds (by swapping the roles of
$x_{1}$ and $x_{2}$ if necessary). If $t_{13}=t_{23}=0$, then one can prove
that $G\not\in\mathcal{I}(3,2)$ (by for example using Proposition 6.2.3 in
[Dek96]). Moreover, we can assume that $t_{13}>0$ by inverting the generator
$z_{2}$ if necessary. For two integers $a,b\in{\wiskunde Z}$ with $b>0$, we
denote with $a\text{ mod }b$ the remainder of the division of $a$ by $b$, i.e.
$0\leq a\text{ mod }b<b$ and there exists a unique $q\in{\wiskunde Z}$ such
that $a=qb+(a\text{ mod }b)$. In particular, there exists a unique
$k\in{\wiskunde Z}$ such that $t_{23}+kt_{13}=(t_{23}\text{ mod }t_{13})$. We
fix a new set of generators of $G$

$x_{1}^{\prime}:=x_{1},\>x_{2}^{\prime}:=x_{1}^{k}x_{2},\>x_{3}^{\prime}:=x_{3},\>z_{1}^{\prime}:=z_{1},\>z_{2}^{\prime}:=z_{2}.$

Using the bilinearity of the commutator (see Lemma 1.2) one can easily check
that the new presentation (where we omit the accents) of $G$ becomes

$G=\langle
x_{1},x_{2},x_{3},z_{1},z_{2}\>|\>[x_{1},x_{2}]=z_{1}^{\alpha}z_{2}^{\beta},[x_{1},x_{3}]=z_{2}^{t_{13}},[x_{2},x_{3}]=z_{2}^{t_{23}\text{
mod }t_{13}}\rangle.$

If $t_{23}\text{ mod }t_{13}=0$ then we are done. Otherwise, we take
$l\in{\wiskunde Z}$ such that $t_{13}+l(t_{23}\text{ mod
}t_{13})=(t_{13}\text{ mod }(t_{23}\text{ mod }t_{13}))$ and define the new
set of generators of $G$ by

$x_{1}^{\prime}:=x_{2}^{l}x_{1},\>x_{2}^{\prime}:=x_{2},\>x_{3}^{\prime}:=x_{3},\>z_{1}^{\prime}:=z_{1},\>z_{2}^{\prime}:=z_{2}.$

It is easy to verify that now $G$ has the following presentation (where we
omit the accents)

$G=\langle
x_{1},x_{2},x_{3},z_{1},z_{2}\>|\>[x_{1},x_{2}]=z_{1}^{\alpha}z_{2}^{\beta},[x_{1},x_{3}]=z_{2}^{t_{13}\text{
mod }(t_{23}\text{ mod }t_{13})},[x_{2},x_{3}]=z_{2}^{t_{23}\text{ mod
}t_{13}}\rangle.$

We can keep on reducing the powers of $z_{2}$ in the presentation using these
two tricks. Eventually, this procedure will end (since it is nothing else than
Euclid’s algorithm on the two occurring powers of $z_{2}$). Hence, $G$ will
have one of the following presentations

$\displaystyle G$ $\displaystyle=\langle
x_{1},x_{2},x_{3},z_{1},z_{2}\>|\>[x_{1},x_{2}]=z_{1}^{\alpha}z_{2}^{\beta},[x_{1},x_{3}]=z_{2}^{\gcd(t_{13},t_{23})}\rangle\text{
or }$ $\displaystyle G$ $\displaystyle=\langle
x_{1},x_{2},x_{3},z_{1},z_{2}\>|\>[x_{1},x_{2}]=z_{1}^{\alpha}z_{2}^{\beta},[x_{2},x_{3}]=z_{2}^{\gcd(t_{13},t_{23})}\rangle$

Define $\gamma:=\gcd(t_{13},t_{23})$ and remark that $\gamma\neq 0$ (since
$t_{13}$ and $t_{23}$ are both non-zero). Thus, in the first case we are done.
In the second case, by taking the following new generators for $G$

$x_{1}^{\prime}:=x_{2},\>x_{2}^{\prime}:=x_{1}^{-1},\>x_{3}^{\prime}:=x_{3},\>z_{1}^{\prime}:=z_{1},\>z_{2}^{\prime}:=z_{2}$

we obtain the desired presentation. ∎

The description in Lemma 3.1 allows us to describe for each group in
$\mathcal{I}(3,2)$ an automorphism with finite Reidemeister number.

######  Theorem 3.2.

Let $G\in\mathcal{I}(3,2)$, then $G$ does not have the $R_{\infty}$–property.

###### Proof.

By Lemma 3.1 we know that

$G=\langle
x_{1},x_{2},x_{3},z_{1},z_{2}\>|\>[x_{1},x_{2}]=z_{1}^{\alpha}z_{2}^{\beta},[x_{1},x_{3}]=z_{2}^{\gamma}\rangle$

with $\alpha,\gamma\in{\wiskunde Z}\setminus\\{0\\}$ and $\beta\in{\wiskunde
Z}$. Take $k,l\in{\wiskunde Z}\setminus\\{0\\}$ such that
$4+\alpha\gamma^{2}kl\neq 0$ and define the map $\varphi$ on $G$ that is
induced by

$\varphi(x_{1}):=x_{1}^{-1},\quad\varphi(x_{2}):=x_{2}^{-\beta\gamma
l-\alpha\gamma^{2}kl-1}x_{3}^{\beta^{2}l+\alpha\beta\gamma kl-\alpha
k},\quad\varphi(x_{3}):=x_{2}^{-\gamma^{2}l}x_{3}^{\beta\gamma l-1},$
$\varphi(z_{1}):=z_{1}^{\alpha\gamma^{2}kl+1}z_{2}^{\gamma
k},\quad\varphi(z_{2}):=z_{1}^{\alpha\gamma l}z_{2}.$

Using the bilinearity of the commutator (see Lemma 1.2) one can check that the
relations are preserved by $\varphi$. Hence, $\varphi$ induces a morphism on
$G$. Moreover, the matrix of the induced morphism $\varphi_{1}\in{\rm
End}(G/\sqrt{\gamma_{2}(G)})$ with respect to the ${\wiskunde Z}$-basis
$\\{x_{1}\sqrt{\gamma_{2}(G)},x_{2}\sqrt{\gamma_{2}(G)},x_{3}\sqrt{\gamma_{2}(G)}\\}$
equals

$\begin{pmatrix}-1&0&0\\\ 0&-\beta\gamma
l-\alpha\gamma^{2}kl-1&-\gamma^{2}l\\\ 0&\beta^{2}l+\alpha\beta\gamma
kl-\alpha k&\beta\gamma l-1\end{pmatrix}$

and is an invertible matrix over ${\wiskunde Z}$. Hence, Proposition 1.7 tells
us that $\varphi$ is an automorphism of $G$. Using Lemma 1.6, one can now
calculate that

$R(\varphi)=R(\varphi_{1})R(\varphi_{2})=(2|4+\alpha\gamma^{2}kl|_{\infty})(|\alpha\gamma^{2}kl|_{\infty})=2|\alpha\gamma^{2}kl(4+\alpha\gamma^{2}kl)|_{\infty}<\infty$

where we used the assumption that $4+\alpha\gamma^{2}kl\neq 0$ with
$k,l\in{\wiskunde Z}\setminus\\{0\\}$. ∎

##  4 Hirsch length 6

By Theorem 2.3 we know that all groups in $\mathcal{I}(5,1)$ have Reidemeister
spectrum equal to $2{\wiskunde N}_{>0}\cup\\{\infty\\}$ or $4{\wiskunde
N}_{>0}\cup\\{\infty\\}$. The only other classes to consider are
$\mathcal{I}(3,3)$ and $\mathcal{I}(4,2)$. Malfait argued in [Mal00, Theorem
3.8] that all groups in $\mathcal{I}(3,3)$ admit a hyperbolic automorphism.

######  Definition 4.1.

Let $G$ be a finitely generated torsion-free c-step nilpotent group, then
$\varphi\in{\rm Aut}(G)$ is called a hyperbolic automorphism if it has no
eigenvalues of absolute value one.

In particular, Theorem 1.4 tells us that a hyperbolic automorphism has a
finite Reidemeister number.

######  Theorem 4.2 ([Mal00, Theorem 3.8]).

Any group $G\in\mathcal{I}(3,3)$ admits a hyperbolic automorphism. In
particular, no group in $\mathcal{I}(3,3)$ has the $R_{\infty}$–property.

######  Remark 4.3.

In the article by Malfait, they also prove (see [Mal00, Proposition 3.1]) that
each automorphism of a group in $\mathcal{I}(3,2)$ has an eigenvalue of
absolute value one (and thus the group does not admit a hyperbolic
automorphism). Nevertheless, in Theorem 3.2 we constructed for each group in
$\mathcal{I}(3,2)$ an automorphism that does not have one as an eigenvalue (or
equivalently has finite Reidemeister number).

What rests is to consider the groups in $\mathcal{I}(4,2)$. We start by
describing the classification of all the groups in $\mathcal{I}(4,2)$. More
details about this classification can be found in [GSS82]. Let
$G\in\mathcal{I}(4,2)$. Note that $\sqrt{\gamma_{2}(G)}/\gamma_{2}(G)$ is a
finitely generated abelian torsion group. Hence, it follows that

$\frac{\sqrt{\gamma_{2}(G)}}{\gamma_{2}(G)}\cong\frac{{\wiskunde
Z}}{\delta{\wiskunde Z}}\times\frac{{\wiskunde Z}}{\lambda\delta{\wiskunde
Z}}$

where $\delta,\lambda\in{\wiskunde N}_{>0}$. We define $\delta(G):=\delta$ and
$\lambda(G):=\lambda$ to be these unique integers. Besides the invariants
$\delta(G)$ and $\lambda(G)$, we also need the notion of $\lambda$-equivalent
binary quadratic forms for the classification of the groups in
$\mathcal{I}(4,2)$.

######  Definition 4.4.

Let $\Phi$ and $\Psi$ be two binary quadratic forms over ${\wiskunde Z}$ and
$\lambda\in{\wiskunde Z}\setminus\\{0\\}$ some non-zero integer. Then $\Phi$
and $\Psi$ are called $\lambda$-equivalent, denoted by
$\Phi\overset{\lambda}{\sim}\Psi$, if there exists an invertible integer
matrix

$\begin{pmatrix}a&b\\\ \lambda c&d\end{pmatrix}\in{\rm GL}_{2}({\wiskunde
Z}),\text{ with }c\in{\wiskunde Z}$

such that

$\Psi(X,Y)=\pm\Phi(aX+bY,\lambda cX+dY).$

One can easily verify that the relation “$\lambda$-equivalence” is indeed an
equivalence relation on the set of binary quadratic forms over ${\wiskunde
Z}$.

Let $\Phi(X,Y)=aX^{2}+bXY+cY^{2}$ be a binary quadratic form over ${\wiskunde
Z}$ and $\delta,\lambda\in{\wiskunde N}_{>0}$. We define the group
$G(\delta,\lambda,\Phi)$ by

$G(\delta,\lambda,\Phi):=\Bigg{\langle}x_{1},x_{2},x_{3},x_{4},z_{1},z_{2}\>\bBigg@{3}|\>\begin{array}[]{l}[x_{1},x_{3}]=z_{2}^{\delta\lambda},\>[x_{1},x_{4}]=z_{1}^{\delta}\\\
[x_{2},x_{3}]=z_{1}^{a\delta}z_{2}^{b\delta\lambda},\>[x_{2},x_{4}]=z_{2}^{-c\delta\lambda}\\\
[x_{1},x_{2}]=[x_{3},x_{4}]=1\text{ and }z_{1},\>z_{2}\text{
central}\end{array}\Bigg{\rangle}.$

Note that $G(\delta,\lambda,\Phi)\in\mathcal{I}(4,2)$,
$\lambda(G(\delta,\lambda,\Phi))=\lambda$ and
$\delta(G(\delta,\lambda,\Phi))=\delta$. Grunewald, Segal and Sterling gave a
classification of the groups in $\mathcal{I}(4,2)$ using $\lambda$-equivalent
binary quadratic forms and the groups $G(\delta,\lambda,\Phi)$ defined above.

######  Theorem 4.5 ([GSS82, Theorem 1]).

Let $\delta,\lambda\in{\wiskunde N}_{>0}$. The assignment

$\Phi\mapsto G(\delta,\lambda,\Phi)$

induces a bijective correspondence between the set of $\lambda$-equivalence
classes of binary quadratic forms over ${\wiskunde Z}$ and the set of
isomorphism classes of groups $G\in\mathcal{I}(4,2)$ with $\lambda(G)=\lambda$
and $\delta(G)=\delta$.
In other words, any group $G\in\mathcal{I}(4,2)$ is isomorphic to some
$G(\delta(G),\lambda(G),\Phi)$ and two such groups $G(\delta,\lambda,\Phi)$
and $G(\delta^{\prime},\lambda^{\prime},\Psi)$ are isomorphic if and only if
$\delta=\delta^{\prime}$, $\lambda=\lambda^{\prime}$ and
$\Phi\overset{\lambda}{\sim}\Psi$.

Using the classification in Theorem 4.5, we are now able to describe for each
group in $\mathcal{I}(4,2)$ an automorphism with a finite Reidemeister number.

######  Theorem 4.6.

Let $G\in\mathcal{I}(4,2)$, then $G$ does not have the $R_{\infty}$–property.

###### Proof.

By Theorem 4.5 we can assume without loss of generality that
$G=G(\delta,\lambda,\Phi)$ with $\delta,\lambda\in{\wiskunde N}_{>0}$ and
$\Phi(X,Y)=aX^{2}+bXY+cY^{2}$ a binary quadratic form over ${\wiskunde Z}$.
Let us first assume that $\Phi(0,1)=c\neq 0$. Consider the map $\varphi$ on
$G$ that is induced by

$\varphi(x_{1}):=x_{1}^{-1-4c}x_{4}^{2},\quad\varphi(x_{2}):=x_{2}^{-1-4c}x_{3}^{-2c},\quad\varphi(x_{3}):=x_{2}^{2}x_{3},\quad\varphi(x_{4}):=x_{1}^{-2c}x_{4},$
$\varphi(z_{1}):=z_{1}^{-1},\quad\varphi(z_{2}):=z_{2}^{-1}.$

One can check using the bilinearity of the commutator (see Lemma 1.2) that all
relations are preserved by $\varphi$ and thus it induces a morphism on $G$.
Moreover, the matrix of $\varphi_{1}\in{\rm End}(G/\sqrt{\gamma_{2}(G)})$ with
respect to the ${\wiskunde Z}$-basis
$\\{x_{1}\sqrt{\gamma_{2}(G)},x_{2}\sqrt{\gamma_{2}(G)},x_{3}\sqrt{\gamma_{2}(G)},x_{4}\sqrt{\gamma_{2}(G)}\\}$
equals

$\begin{pmatrix}-1-4c&0&0&-2c\\\ 0&-1-4c&2&0\\\ 0&-2c&1&0\\\
2&0&0&1\end{pmatrix}$

and is an invertible matrix over ${\wiskunde Z}$. Proposition 1.7 now tells us
that $\varphi$ is an automorphism of $G$. Using Lemma 1.6 we obtain that

$R(\varphi)=R(\varphi_{1})R(\varphi_{2})=(16|c^{2}|_{\infty})4=64|c^{2}|_{\infty}<\infty$

where we used that $c\neq 0$.

Assume that $c=0$. Note that if also $a=b=0$, then

$\displaystyle G(\delta,\lambda,0)$
$\displaystyle=\Bigg{\langle}x_{1},x_{2},x_{3},x_{4},z_{1},z_{2}\>\bBigg@{3}|\>\begin{array}[]{l}[x_{1},x_{3}]=z_{2}^{\delta\lambda},\>[x_{1},x_{4}]=z_{1}^{\delta}\\\
[x_{3},x_{4}]=1\text{ and }x_{2},\>z_{1},\>z_{2}\text{
central}\end{array}\Bigg{\rangle}$
$\displaystyle=\Bigg{\langle}x_{1},x_{3},x_{4},z_{1},z_{2}\>\bBigg@{3}|\>\begin{array}[]{l}[x_{1},x_{3}]=z_{2}^{\delta\lambda},\>[x_{1},x_{4}]=z_{1}^{\delta}\\\
[x_{3},x_{4}]=1\text{ and }z_{1},\>z_{2}\text{
central}\end{array}\Bigg{\rangle}\times{\wiskunde Z}$
$\displaystyle=:G^{\prime}\times{\wiskunde Z}$

where $G^{\prime}\in\mathcal{I}(3,2)$. By Theorem 3.2 we can take some
$\psi^{\prime}\in{\rm Aut}(G^{\prime})$ with $R(\psi^{\prime})<\infty$.
Consider now the automorphism $\varphi:=\psi^{\prime}\times-{\rm
Id}_{{\wiskunde Z}}\in{\rm Aut}(G)$. Using Proposition 2.4 in [Sen21], it it
follows that $R(\varphi)=R(\psi^{\prime})R(-{\rm Id}_{{\wiskunde
Z}})=2R(\psi^{\prime})<\infty$.
So we can assume without loss of generality that $a$ and $b$ are not both zero
(but $c=0$). We now argue that in this case, $G$ is isomorphic to some
$G(\delta,\lambda,\Psi)$ with $\Psi(0,1)\neq 0$. For this, define
$\Psi_{k}(X,Y):=\Phi((k\lambda+1)X+kY,\lambda X+Y)$ for any $k\in{\wiskunde
Z}$. Note that the matrix

$\begin{pmatrix}k\lambda+1&k\\\ \lambda&1\end{pmatrix}\in{\rm
GL}_{2}({\wiskunde Z})$

and thus $\Phi\overset{\lambda}{\sim}\Psi_{k}$ for all $k\in{\wiskunde Z}$.
Since we know that $\Phi(X,Y)=aX^{2}+bXY$, some calculation yields that

$\Psi_{k}(X,Y)=\left(a(k\lambda+1)^{2}+b\lambda(k\lambda+1)\right)X^{2}+\left(2ak(k\lambda+1)+b(2k\lambda+1)\right)XY+k\left(ak+b\right)Y^{2}.$

Since not both $a$ and $b$ are zero, we can take some $k_{0}\in{\wiskunde Z}$
such that $k_{0}(ak_{0}+b)\neq 0$. Hence, $G\cong
G(\delta,\lambda,\Psi_{k_{0}})$ and $\Psi_{k_{0}}(0,1)=k_{0}(ak_{0}+b)\neq 0$.
Thus we reduced this case to the first one and we get an automorphism with a
finite Reidemeister number. ∎

##  5 Conclusion and minimal example

In the previous sections, we argued that all finitely generated torsion-free
2-step nilpotent groups of Hirsch length at most 6 do not have the
$R_{\infty}$–property. In Remark 7.8 in [DL23] we gave an example of a
finitely generated torsion-free 2-step nilpotent group of Hirsch length 7 that
has the $R_{\infty}$–property.

######  Theorem 5.1.

All finitely generated torsion-free 2-step nilpotent groups of Hirsch length
at most 6 do not have the $R_{\infty}$–property. This upper bound is sharp,
i.e. there exists a finitely generated torsion-free 2-step nilpotent group of
Hirsch length 7 with the $R_{\infty}$–property.

###### Proof.

Combining Theorems 2.2, 2.3, 3.2, 4.2 and 4.6 yields that there do not exist
finitely generated torsion-free 2-step nilpotent groups of Hirsch length at
most 6 with the $R_{\infty}$–property.
We now give an example of such a group of Hirsch length 7 with the
$R_{\infty}$–property. Consider the group $G$ defined by

$G:=\Bigg{\langle}x_{1},x_{2},x_{3},x_{4},z_{1},z_{2},z_{3}\>\bBigg@{3}|\>\begin{array}[]{l}[x_{1},x_{2}]=z_{1},\>[x_{2},x_{3}]=z_{2},\>[x_{3},x_{4}]=z_{3},\\\
[x_{1},x_{3}]=[x_{1},x_{4}]=[x_{2},x_{4}]=1\text{ and
}z_{1},\>z_{2},\>z_{3}\text{ central}\end{array}\Bigg{\rangle}.$

Note that $G\in\mathcal{I}(4,3)$ is indeed a finitely generated torsion-free
2-step nilpotent group with $h(G)=7$. In [DL23, Remark 7.8] we argue that $G$
has the $R_{\infty}$–property by using that this group can be associated to
the path graph on 4 vertices. In [GW09, Example 4.1], it is proven directly
that $G$ has the $R_{\infty}$–property. ∎

######  Remark 5.2.

For higher nilpotency classes the analogous problem is quite trivial. If $G$
is a finitely generated torsion-free nilpotent group of class $c$, then the
Hirsch length of $G$ is at least $c+1$. It can be easily shown that for any
$c\geq 3$ there exists a finitely generated torsion-free $c$-step nilpotent
group of Hirsch length $c+1$ with the $R_{\infty}$–property.

In fact, the case $c=3$ can be proved in a similar way as Example 5.2 of
[GW09], while the claim for such groups $G$ of higher nilpotency classes then
reduces to the case $c=3$ by considering $G/\gamma_{4}(G)$.

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