#  Elementary groups in ${\rm PSL}(3,\mathbb{C})$

W. Barrera  Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo
Periférico Norte Tablaje Cat 13615, Mérida, Yucatán, México<EMAIL_ADDRESS>,  A. Cano  Instituto de Matemáticas UNAM , Av. Universidad S/N, Col. Lomas de
Chamilpa, C.P. 62210, Cuernavaca, Morelos, México<EMAIL_ADDRESS>,
J.P. Navarrete  Facultad de Matemáticas, Universidad Autónoma de Yucatán,
Anillo Periférico Norte Tablaje Cat 13615, Mérida, Yucatán, México.
<EMAIL_ADDRESS>and  J. Seade  Instituto de Matemáticas UNAM , Av.
Universidad S/N, Col. Lomas de Chamilpa, C.P. 62210, Cuernavaca, Morelos,
México<EMAIL_ADDRESS>
###### Abstract.

In this paper, we give a classification of the subgroups of
$\textrm{PSL}(3,\mathbb{C})$ that act on $\mathbb{P}_{\mathbb{C}}^{2}$ in such
a way that their Kulkarni limit set has finitely many lines in general
position lines. These are the elementary groups.

$\qquad;\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ To Ravi Kulkarni, with
great admiration

###### Key words and phrases:

Kleinian groups, projective complex plane, discrete groups, limit set

###### 2010 Mathematics Subject Classification:

Primary: 37F30, 32Q45; Secondary 37F45, 22E40

## Introduction

Kleinian groups are discrete subgroups of $\textrm{PSL}(2,\mathbb{C})$ acting
on $\mathbb{S}^{2}\cong\mathbb{P}_{\mathbb{C}}^{1}$ in such a way that their
limit set is not all of $\mathbb{S}^{2}$. These are classified into elementary
and non-elementary groups. The elementary groups are those whose limit set
consists of $0,1$ or $2$ points, and they are classified, see for instance
(see [20]).

In this work we look at complex Kleinian groups, that is, discrete subgroups
of $\textrm{PSL}(3,\mathbb{C})$ acting properly and discontinuously on some
open subset of $\mathbb{P}_{\mathbb{C}}^{2}$. In [11] it is proved that the
Kulkarni limit set $\Lambda_{\rm Kul}$ (see Definition 1.1) of every infinite
complex Kleinian group contains at least one complex projective line. In fact
we known from [1] that under some generic hypothesis on the group, the
Kulkarni limit set is a union of complex projective lines in general position,
i.e. no three of them intersect. By definition, the exception are the
elementary groups. These have finitely many lines in general position in their
limit set, and there are two types ef such groups: those of the first kind
have a finite number of lines in their limit set, and the groups of the second
kind have infinitely many lines in their limit set, but only finitely many in
general position.

As an example of elementary groups of the first kind, take the cyclic group
generated by a $3\times 3$ diagonal matrix with non-zero eigenvalues of
different norms; this has two projective lines as limit set in
${\mathbb{P}}_{\mathbb{C}}^{2}$. On the other hand (see [9]), take a
$\mathbb{C}$-Fuchsian group in ${\rm PSL}(3,\mathbb{C})$. Then its limit set
in ${\mathbb{P}}_{\mathbb{C}}^{2}$ consists of a cone of projective lines with
base a circle (see [9] ; it has infinitely many lines, but all of them pass
through the vertex of the cone, so there are only two in general position. We
remark that if we take an $\mathbb{R}$-Fuchsian group in ${\rm
PSL}(3,\mathbb{C})$, then it is non-elementary and its limit set actually has
infinitely many lines in general position, see [9]

In this work we study and describe all the elementary groups in
$\textrm{PSL}(3,\mathbb{C})$.

An interesting class of elementary groups are the discrete purely parabolic
groups. One finds that the limit set of such a group consists of either one
line, or a cone of lines over the circle, or the whole of
${\mathbb{P}}_{\mathbb{C}}^{2}$. We describe these in Section 2.

We know (see Theorem 1.5 below) that if $\Gamma\subset{\rm PSL}(3,\mathbb{C})$
is an infinite discrete subgroup, then the number of complex projective lines
in its limit set $\Lambda_{{\rm Kul}}(\Gamma)$ is equal to $1,2,3$ or
$\infty$, and the number of lines $\Lambda_{{\rm Kul}}(\Gamma)$ in general
position can be $1,2,3,4$ or $\infty$. Also, if the number of lines in
$\Lambda_{{\rm Kul}}(\Gamma)$ is exactly 3, then these lines are in general
position.

It follows from Theorem 1.5 and the results in this note, that if
$\Lambda_{{\rm Kul}}(\Gamma)$ has more than one line, then it is a union of
lines. The only case where $\Lambda_{{\rm Kul}}(\Gamma)$ can have isolated
points is when this set actually consists of one line and one point.

Section 1 contains the basic definitions and preliminary results that we use
in the sequel. Sections 4 and 7 are expository and we cite in each the
appropriate bibliography. The results in these sections are taken from [3] and
[2] and we include them here for completeness. In [3] the authors discuss the
groups with limit set a line, and in [2] those with four lines in general
position.

Section 5 focuses on the case of two lines. These can be of the first kind,
with exactly 2 lines in $\Lambda_{\rm Kul}$, or of the second kind, with limit
set a cone of lines over a circle, but only 2 in general position.

In section 6 we describe the groups with exactly three lines in their limit
set, and section 8 discusses the only case where the limit set is not a union
of lines. Then $\Lambda_{\rm Kul}$ consists of a line and a point.

##  1\. Preliminaries

###  1.1. Pseudo-projective transformations

We let
$\mathbb{P}^{2}_{\mathbb{C}}:=(\mathbb{C}^{3}\setminus\\{0\\})/\mathbb{C}^{*},$
be the complex projective plane. This is a compact connected complex
$2$-dimensional manifold. Let $[\mbox{
}]:\mathbb{C}^{3}\setminus\\{0\\}\rightarrow\mathbb{P}^{2}_{\mathbb{C}}$ be
the quotient map. If $\beta=\\{e_{1},e_{2},e_{3}\\}$ is the standard basis of
$\mathbb{C}^{3}$, we write $[e_{j}]=e_{j}$ and if
$w=(w_{1},w_{2},w_{3})\in\mathbb{C}^{3}\setminus\\{0\\}$ then we write
$[w]=[w_{1}:w_{2}:w_{3}]$. Also, $\ell\subset\mathbb{P}^{2}_{\mathbb{C}}$ is
said to be a complex line if $[\ell]^{-1}\cup\\{0\\}$ is a complex linear
subspace of dimension $2$. Given $p,q\in\mathbb{P}^{2}_{\mathbb{C}}$ distinct
points, there is a unique complex projective line passing through $p$ and $q$,
such complex projective line is called a _line_ , for short, and it is denoted
by $\overleftrightarrow{p,q}$.

Consider the action of $\mathbb{Z}_{3}$ (viewed as the cubic roots of the
unity) on $\textrm{SL}(3,\mathbb{C})$ given by the usual scalar
multiplication, then the Lie group

$\textrm{PSL}(3,\mathbb{C})=\textrm{SL}(3,\mathbb{C})/\mathbb{Z}_{3}$

is the group of automorphisms of ${\mathbb{P}}_{\mathbb{C}}^{2}$. Let
$[[\mbox{ }]]:\textrm{SL}(3,\mathbb{C})\rightarrow\textrm{PSL}(3,\mathbb{C})$
be the quotient map, $\gamma\in\textrm{PSL}(3,\mathbb{C})$ and
$\widetilde{\gamma}\in\textrm{GL}(3,\mathbb{C})$, we say that $\tilde{\gamma}$
is a lift of $\gamma$ if there is a cubic root $\tau$ of $Det(\gamma)$ such
that $[[\tau\widetilde{\gamma}]]=\gamma$. Also, we use the notation
$(\gamma_{ij})$ to denote elements in $\textrm{SL}(3,\mathbb{C})$.

We denote by $M_{3\times 3}(\mathbb{C})$ the space of all $3\times 3$ matrices
with entries in $\mathbb{C}$ equipped with the standard topology. The quotient
space

${\rm SP}(3,\mathbb{C}):=(M_{3\times
3}(\mathbb{C})\setminus\\{\mathbf{0}\\})/\mathbb{C}^{*}$

is called the space of _pseudo-projective maps of_
$\mathbb{P}_{\mathbb{C}}^{2}$ and it is naturally identified with the
projective space $\mathbb{P}_{\mathbb{C}}^{8}$. Since
$\textrm{GL}(3,\mathbb{C})$ is an open, dense, $\mathbb{C}^{*}$-invariant set
of $M_{3\times 3}(\mathbb{C})\setminus\\{\mathbf{0}\\}$, we obtain that the
space of pseudo-projective maps of $\mathbb{P}_{\mathbb{C}}^{2}$ is a
compactification of $\textrm{PSL}(3,\mathbb{C})$. As in the case of projective
maps, if $\mathbf{s}\in M_{3\times 3}(\mathbb{C})\setminus\\{\mathbf{0}\\}$,
then $[\mathbf{s}]$ denotes the equivalence class of the matrix $\mathbf{s}$
in the space of pseudo-projective maps of $\mathbb{P}_{\mathbb{C}}^{2}$. Also,
we say that $\mathbf{s}\in M_{3\times 3}(\mathbb{C})\setminus\\{\mathbf{0}\\}$
is a lift of the pseudoprojective map $S$, whenever $[\mathbf{s}]=S$.

Let $S$ be an element in $(M_{3\times
3}(\mathbb{C})\setminus\\{\mathbf{0}\\})/\mathbb{C}^{*}$ and $\mathbf{s}$ a
lift to $M_{3\times 3}(\mathbb{C})\setminus\\{\mathbf{0}\\}$ of $S$. The
matrix $\mathbf{s}$ induces a non-zero linear transformation
$s:\mathbb{C}^{3}\rightarrow\mathbb{C}^{3}$, which is not necessarily
invertible. Let $Ker(s)\subsetneq\mathbb{C}^{3}$ be its kernel and let
$Ker(S)$ denote its projectivization to $\mathbb{P}_{\mathbb{C}}^{2}$, taking
into account that $Ker(S):=\varnothing$ whenever $Ker(s)=\\{(0,0,0)\\}$. We
refer to [10] for more details about this subject.

###  1.2. The limit set

There are two types of limit sets relevant for this work. These are the
Kulkarni limit set $\Lambda_{{\rm Kul}}$ and the Myrberg limit set
$\Lambda_{{\rm Myr}}$. Let us define these. Let
$\Gamma\subset\textrm{PSL}(n+1,\mathbb{C})$ be a discrete subgroup.

######  Definition 1.1.

The Kulkarni limit set of $\Gamma$ is:

$\Lambda(\Gamma)=L_{0}(\Gamma)\cup L_{1}(\Gamma)\cup L_{2}(\Gamma),$

where:

  * •

$L_{0}(\Gamma)$ is the closure of the points in $\mathbb{P}^{2}_{\mathbb{C}}$
with infinite isotropy group.

  * •

$L_{1}(\Gamma)$ is the closure of the set of accumulation points of the
$\Gamma$-orbits of points in $\mathbb{P}^{2}_{\mathbb{C}}\setminus
L_{0}(\Gamma)$.

  * •

$L_{2}(\Gamma)$ is the closure of the union of accumulation points of
$\\{\gamma(K):\gamma\in\Gamma\\}$, where $K$ runs over all the compact sets in
$\mathbb{P}^{2}_{\mathbb{C}}\setminus(L_{0}(\Gamma)\cup L_{1}(\Gamma))$.

This is a closed $\Gamma$-invariant set. Its complement

$\Omega_{{\rm
Kul}}(\Gamma)=\mathbb{P}^{n}_{\mathbb{C}}\setminus\Lambda(\Gamma),$

is the Kulkarni discontinuity region of $\Gamma$. We know from [18] that the
$\Gamma$-action on $\Omega_{{\rm Kul}}(\Gamma)$ is properly discontinuous, and
we further know from [1] that $\Omega_{{\rm Kul}}(\Gamma)$ contains the
equicontinuity region ${\rm Eq}(\Gamma)$. However, as we will see later,
$\Omega_{{\rm Kul}}(\Gamma)$ generically is the largest open subset of
${\mathbb{P}}_{\mathbb{C}}^{2}$ where the group acts properly and
discontinuously, but this is not always so.

Recall that ${\rm Eq}(\Gamma)$ is the set of points in
${\mathbb{P}}_{\mathbb{C}}^{2}$ for which there is an open neighborhood where
the family of transformations defined by $\Gamma$ is a normal family.

Now, given $\Gamma\subset\textrm{PSL}(n+1,\mathbb{C})$, a discrete subgroup,
we let $\Gamma^{\prime}$ be the set of pseudo-projective maps of
${\mathbb{P}}_{\mathbb{C}}^{2}$ which are limits of sequences in $\Gamma$.
That is:

$\Gamma^{\prime}:=\large\\{S\in{\rm SP}(3,\mathbb{C})\,|\,S\;\hbox{is an
accumulation point of}\;\Gamma\large\\}\;.$

The following notion is due to Myrberg [21] (cf. [1, Definition 3.3]):

######  Definition 1.2.

The Myrberg limit set of $\Gamma$ is:

${\Lambda_{{\rm Myr}}}(\Gamma):={\cup_{S\in\Gamma^{\prime}}Ker(S)}\;,$

where $Ker(S)$ is the kernel of the pseudoprojective transformation $S$
defined above.

Notice that if a pseudoprojective map $S$ is not in ${\rm PSL}(3,\mathbb{C})$,
then its kernel is either a line or a point. The following useful notion is
introduced in [4]:

######  Definition 1.3.

We say that a line $\ell\subset{\mathbb{P}}_{\mathbb{C}}^{2}$ is an effective
line for $\Gamma$ if there exists a pseudoprojective transformation
$S\in\Gamma^{\prime}$ with kernel $\ell$.

The Myrberg limit set of $\Gamma$ contains all the effective lines, and it is
not hard to see that, by [22], it contains the Kulkarni limit set. It is
immediate from [21] that $\Gamma$ acts properly and discontinuously on the
complement of ${\Lambda_{{\rm Myr}}}(\Gamma)$. Furthermore, one has the
following theorem.

######  Theorem 1.4.

The Myrberg limit set is the complement of the equicontinuity region: $\;{\rm
Eq}(\Gamma)={\mathbb{P}}_{\mathbb{C}}^{2}\setminus{\Lambda_{{\rm
Myr}}}(\Gamma)\,.$ Also, the set ${\Lambda_{{\rm Myr}}}(\Gamma)$ equals the
union of all effective lines of $\Gamma$ except when it is the disjoint union
of one line and one point. Moreover, $\Lambda_{{\rm
Kul}}(\Gamma)\subset{\Lambda_{{\rm Myr}}}(\Gamma)$.

The first statement is a special case of [10, Lemma 4.1] (see also [4, Theorem
3.4]). The second statement is Corollary 4.5 in [4].

Throughout this work, when we say limit set we refer to the Kulkarni limit
set, otherwise we will say it explicitly.

###  1.3. Classification of the elements in ${\rm PSL}(3,\mathbb{C})$

Recall that the elements of ${\rm PSL}(2,\mathbb{C})$ are classified as
elliptic, parabolic or loxodromic: $g$ is elliptic if regarded as a Möbius
transformation in $S^{2}\cong{\mathbb{P}}_{\mathbb{C}}^{1}$, up to conjugation
it is a rotation; parabolic elements are translations up to conjugation and
loxodromic elements are multiplication by a complex number of norm $\neq 1$.
Equivalently, given $g\in{\rm PSL}(2,\mathbb{C})$ and a lift $\tilde{g}$ to
${\rm SL}(2,\mathbb{C})$, $g$ is elliptic if $\tilde{g}$ is diagonalizable
with unitary eigenvalues, it is parabolic if $\tilde{g}$ is not diagonalizable
and it has unitary eigenvalues, and $g$ is loxodromic otherwise. The
classification given in these terms extends to all dimensions (see [22, 8]).

The _elliptic_ elements in ${\rm PSL}(3,\mathbb{C})$ are those elements
$\gamma$ that have a lift to ${\rm SL}(3,\mathbb{C})$ whose Jordan canonical
form is

$\left(\begin{array}[]{ccc}e^{i\theta_{1}}&0&0\\\ 0&e^{i\theta_{2}}&0\\\
0&0&e^{i\theta_{3}}\end{array}\right).$

The limit set for (the cyclic group generated by) $\gamma$ elliptic is either
empty or all of $\mathbb{P}_{\mathbb{C}}^{2}$, according to whether the order
of $\gamma$ is finite or infinite. The subgroups of ${\rm PSL}(3,\mathbb{C})$
containing an elliptic element of infinite order cannot be discrete.

The _parabolic_ elements in ${\rm PSL}(3,\mathbb{C})$ are the elements
$\gamma$ with limit set $\Lambda_{{\rm Kul}}(\gamma)$ equal to one single
complex line. If $\gamma$ is parabolic then it has a lift to ${\rm
SL}(3,\mathbb{C})$ with Jordan canonical form one of the following matrices:

$\left(\begin{array}[]{ccc}1&1&0\\\ 0&1&0\\\
0&0&1\end{array}\right),\left(\begin{array}[]{ccc}1&1&0\\\ 0&1&1\\\
0&0&1\end{array}\right),\left(\begin{array}[]{ccc}e^{2\pi it}&1&0\\\ 0&e^{2\pi
it}&0\\\ 0&0&e^{-4\pi it}\end{array}\right)\,,\,e^{2\pi it}\neq 1\;.$

In the first case $\Lambda_{{\rm Kul}}(\gamma)$ is the complex line consisting
of all the fixed points of $\gamma$, in the second case $\Lambda_{{\rm
Kul}}(\gamma)$ is the unique $\gamma$-invariant complex line. In the last case
$\Lambda_{{\rm Kul}}(\gamma)$ is the complex line determined by the two fixed
points of $\gamma$.

There are four kinds of _loxodromic_ elements in ${\rm PSL}(3,\mathbb{C})$:

  * •

The _complex homotheties_ are the elements that have a lift to ${\rm
SL}(3,\mathbb{C})$ with Jordan canonical form:

$\left(\begin{array}[]{ccc}\lambda&0&0\\\ 0&\lambda&0\\\
0&0&\lambda^{-2}\end{array}\right),\quad|\lambda|\neq 1.$

The limit set $\Lambda_{{\rm Kul}}(\gamma)$ is the set of fixed points of
$\gamma$, consisting of a complex line and a point.

  * •

The _screws_ are those elements $\gamma\in{\rm PSL}(3,\mathbb{C})$ that have a
lift to ${\rm SL}(3,\mathbb{C})$ whose Jordan canonical form is

$\left(\begin{array}[]{ccc}\lambda&0&0\\\ 0&\mu&0\\\
0&0&(\lambda\mu)^{-1}\end{array}\right),\quad\lambda\neq\mu,\,|\lambda|=|\mu|\neq
1.$

The limit set consists of a complex line $l$, on which $\gamma$ acts as an
elliptic transformation of ${\rm PSL}(2,\mathbb{C})$, and the fixed point of
$\gamma$ not lying in $l$.

  * •

The _loxoparabolic_ elements $\gamma\in{\rm PSL}(3,\mathbb{C})$ have a lift to
${\rm SL}(3,\mathbb{C})$ whose Jordan canonical form is

$\left(\begin{array}[]{ccc}\lambda&1&0\\\ 0&\lambda&0\\\
0&0&\lambda^{-2}\end{array}\right),\quad|\lambda|\neq 1.$

The limit set $\Lambda_{{\rm Kul}}(\gamma)$ consists of two $\gamma$-invariant
complex lines. The element $\gamma$ acts on one of these complex lines as a
parabolic element of ${\rm PSL}(2,\mathbb{C})$ and on the other as a
loxodromic element of ${\rm PSL}(2,\mathbb{C})$.

  * •

The _strongly loxodromic_ elements $\gamma\in{\rm PSL}(3,\mathbb{C})$ have a
lift to ${\rm SL}(3,\mathbb{C})$ whose Jordan canonical form is

$\left(\begin{array}[]{ccc}\lambda_{1}&0&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{3}\end{array}\right),\quad|\lambda_{1}|<|\lambda_{2}|<|\lambda_{3}|.$

This kind of transformation has three fixed points, one of them is attracting,
another is repelling and the other point is a saddle. The limit set is the
union of the complex line determined by the attracting and saddle points and
the complex line determined by the saddle and repelling points.

###  1.4. Elementary groups in ${\rm PSL}(3,\mathbb{C})$

In the previous section we have examples of elements in ${\rm
PSL}(3,\mathbb{C})$ where the limit set consists of one line, or one line and
one point, or two lines. We have the following theorem from [4].

######  Theorem 1.5.

Let $\Gamma\subset{\rm PSL}(3,\mathbb{C})$ be an infinite discrete subgroup.
Then:

  1. (i)

The number of complex projective lines in $\Lambda_{{\rm Kul}}(\Gamma)$ is
equal to $1,2,3$ or $\infty$.

  2. (ii)

The number of lines $\Lambda_{{\rm Kul}}(\Gamma)$ in general position can be
$1,2,3,4$ or $\infty$.

  3. (iii)

If the number of lines in $\Lambda_{{\rm Kul}}(\Gamma)$ is exactly 3, then
these lines are in general position.

  4. (iv)

If there are infinitely many lines in $\Lambda_{{\rm Kul}}(\Gamma)$, then the
effective lines form a perfect set in $\Lambda_{{\rm Kul}}(\Gamma)$.

  5. (v)

There can be at most one isolated point in $\Lambda_{{\rm Kul}}(\Gamma)$, and
in that case this limit set is the disjoint union of that point and one line.

Statements (i), (ii) and (iv) are theorems 1.1 and 1.2 in [4]. Statement (iii)
is a corollary of [4, Proposition 5.4]. Statement (v) is not in the literature
so we now give a short proof of it.

Recall that a discrete subgroup of ${\rm PSL}(2,\mathbb{C})$ is elementary if
its limit set has finite cardinality. We remark that in the case of ${\rm
PSL}(3,\mathbb{C})$ there are groups with infinitely many lines in their limit
set $\Lambda_{{\rm Kul}}$, but only two of them in general position. In view
of the previous theorem, this leads to the following definition:

######  Definition 1.6.

A discrete subgroup of ${\rm PSL}(3,\mathbb{C})$ is elementary of the first
kind if its limit set $\Lambda_{{\rm Kul}}$ has finitely many lines. The group
is elementary of the second kind if $\Lambda_{{\rm Kul}}$ has finitely many
lines in general position.

So every elementary group of the first kind also is of the second kind, but
not conversely. In the following sections we describe the classification of
elementary groups.

###  1.5. The control group

We refer to [8] for a discussion about this section. Consider
$\Gamma\subset{\rm PSL}(3,\mathbb{C})$ a (discrete or not) subgroup which acts
on ${\mathbb{P}}_{\mathbb{C}}^{2}$ with a point $p$ which is fixed by all of
$\Gamma$. Choose an arbitrary line $\ell$ in
${\mathbb{P}}_{\mathbb{C}}^{2}\setminus\\{p\\}$, and notice we have a
canonical projection:

$\pi=\pi_{p,\ell}:\mathbb{P}^{2}_{\mathbb{C}}\setminus\\{p\\}\longrightarrow\ell\,,$

given by $\pi(x)=\overleftrightarrow{x,p}\cap\ell$. It is clear that this map
is holomorphic and it allows us to define a group homomorphism:

$\Pi=\Pi_{p,\ell}:\Gamma\longrightarrow Bihol(\ell)\cong{\rm
PSL}(2,\mathbb{C})\,,$

by $\Pi(g)(x)=\pi(g(x))$. If we choose another line, say $\ell^{\prime}$, one
gets similarly a projection
$\pi^{\prime}=\pi_{p,\ell^{\prime}}:\mathbb{P}^{2}_{\mathbb{C}}\setminus\\{p\\}\to\ell^{\prime}\,,$
and a group homomorphism $\Pi^{\prime}=\Pi_{p,\ell^{\prime}}:\Gamma\to{\rm
PSL}(2,\mathbb{C})$. It is an exercise to see that $\Pi$ and $\Pi^{\prime}$
are equivalent in the sense that there is a biholomorphism
$h:\ell\to\ell^{\prime}$ inducing an automorphism $H$ of ${\rm
PSL}(2,\mathbb{C})$ such that $H\circ\Pi=\Pi^{\prime}$. As before, the line
$\ell$ is called the horizon.

This leads to the following definition:

######  Definition 1.7.

Let $\Gamma\subset{\rm PSL}(3,\mathbb{C})$ be a discrete group as above. We
call $\Pi=\Pi_{p,\ell}$ the control morphism (or map) and its image
$\Pi(\Gamma)\subset{\rm PSL}(2,\mathbb{C})$, is the control group. These are
well-defined and independent of $\ell$ up to an automorphism of ${\rm
PSL}(2,\mathbb{C})$.

The control map and the control group allow us to get information about the
dynamics of $\Gamma$ by looking at a subgroup of ${\rm PSL}(2,\mathbb{C})$,
which is far easier to handle. The prize we pay is that the control group in
${\rm PSL}(2,\mathbb{C})$ may not be discrete.

##  2\. Purely parabolic groups

We now follow [5] and look at the discrete subgroups in ${\rm
PSL}(3,\mathbb{C})$ that, besides the identity, have only parabolic elements.
These are called purely parabolic and there are five families of such groups;
three of them split into various subfamilies according to their limit set (and
their control group, see [8]). All of these are elementary.

The simplest purely parabolic groups are cyclic, generated by a parabolic
element. As described above, there are three types of such elements in ${\rm
PSL}(3,\mathbb{C})$, described by the Jordan normal form of their lifts to
${\rm SL}(3,\mathbb{C})$. Each of these belongs to a different type of the
families we describe below. The first type generates torus groups (see
definitions below), the second generates Abelian Kodaira groups and the
ellipto-parabolic elements generate elliptic groups.

  1. (i)

Elliptic groups. These are the only purely parabolic groups that are not
conjugate to subgroups of the Heisenberg group ${\rm Heis}(3,\mathbb{C})$ and
they are subgroups of fundamental groups of elliptic surfaces. These have
limit set a single line. Up to conjugation these groups are of the form:

${{\rm Ell}}(W,\mu)=\left\\{\left[\begin{array}[]{lll}\mu(w)&\mu(w)w&0\\\
0&\mu(w)&0\\\ 0&0&\mu(w)^{-2}\\\ \end{array}\right]:w\in W\right\\},$

where $W\subset\mathbb{C}$ is an additive discrete subgroup and
$\mu:W\rightarrow\mathbb{S}^{1}$ is a group morphism.

  2. (ii)

Torus groups. These are subgroups of fundamental groups of complex tori. These
are of the form:

$\mathcal{T}(\mathfrak{L})=\left\\{\left[\begin{array}[]{lll}1&0&a\\\ 0&1&b\\\
0&0&1\\\ \end{array}\right]:(a,b)\in\mathfrak{L}\right\\}\;,$

where $\mathfrak{L}$ is an additive discrete subgroup of $\mathbb{C}^{2}$.
These groups also have a single line as limit set, so they are elementary of
the first kind.

  3. (iii)

Dual torus groups,

$\mathcal{T}^{*}(\mathfrak{L})=\left\\{\left[\begin{array}[]{lll}1&a&b\\\
0&1&0\\\ 0&0&1\\\ \end{array}\right]:(a,b)\in\mathfrak{L}\right\\}\;.$

These split into three types: the first have Kulkarni limit set a complex
projective line, so these are elementary of the first kind. The second type
have limit set a cone of projective lines over a circle, so these are of the
second kind. The third type have all ${\mathbb{P}}_{\mathbb{C}}^{2}$ as limit
set and they are non-elementary.

  4. (iv)

Inoue groups and their extensions. Inoue groups are proper subgroups of
fundamental groups of Inoue surfaces. To define these, let
$\mathfrak{L}\subset\mathbb{C}^{2}$ be an additive discrete subgroup and
consider a dual torus group.

${\mathcal{I}}={\mathcal{I}}(u,v):=\left\langle\begin{bmatrix}1&u&v\\\
0&1&0\\\ 0&0&1\\\ \end{bmatrix}\;,\;(u,v)\in\mathfrak{L}\right\rangle\;.$

Inoue groups are obtained by taking a generator ${\mathcal{I}}$ as above and
adding to it a generator of the form

$\gamma_{1}=\gamma_{1}(x,y,z):=\begin{bmatrix}1&x+z&y\\\ 0&1&z\\\
0&0&1\end{bmatrix}\;x,y,z\in\mathbb{C}\;.$

The limit set is a cone of lines over a circle, so these are elementary of the
second kind.

Then one has the extended Inoue group, which are purely parabolic as well,
with limit set all of ${\mathbb{P}}_{\mathbb{C}}^{2}$, so they are non-
elementary and do not fall within the scope of this article. We refer to [5]
for details.

  5. (v)

Finally one has the Kodaira groups ${\rm Kod}_{0}$, which are Abelian, and
their extensions. A Kodaira group is a discrete group in ${\rm
PSL}(3,\mathbb{C})$ such that each element $h$ in the group can be written as:

$\left[\begin{array}[]{lll}1&a&b\\\ 0&1&a\\\ 0&0&1\\\ \end{array}\right]\,.$

One can show that these are extensions of dual torus groups. Their limit set
is a line, so they are elementary of the first kind.

There are five types of extensions $\widetilde{\mathcal{K}_{i}}$,
$i=1,\cdots,5$, which are purely parabolic and discrete. These are all non-
Abelian and they split into five types according to their limit set and the
control group. The first type $\widetilde{\mathcal{K}_{1}}$ have limit set a
projective line, so they are elementary of the first kind; the second type
$\widetilde{\mathcal{K}_{2}}$ have limit set a cone of projective lines over
the circle, so they are of the second kind, while the remaining three types
have limit set all of ${\mathbb{P}}_{\mathbb{C}}^{2}$ and they are non-
elementary. We refer to [5] for details.

##  3\. Solvable groups

An essential step towards understanding elementary groups in ${\rm
PSL}(3,\mathbb{C})$ is studying the dynamics of solvable groups in ${\rm
PSL}(3,\mathbb{C})$. Unlike the classical case of ${\rm PSL}(2,\mathbb{C})$,
now there is great richness.

######  Definition 3.1.

A group $H$ is called virtually solvable if it contains a finite index
subgroups $G$ and subgroups $G_{i}$, $i=0,...,k$, such that:

  * •

$e=G_{0}\subset G_{1}\subset\ldots\subset G_{k}=G$, and

  * •

$G_{j-1}$ is normal in $G_{j}$, and $G_{j}/G_{j-1}$ is an abelian group, for
$j=1,\ldots,k$.

For instance, for a subgroup of ${\rm PSL}(2,\mathbb{C})$, virtually solvable
is equivalent to saying that the group is elementary, i.e. its limits set has
finite cardinality, see[11] and [20], or in an equivalent way (via Tit’s
dichotomy) a Möbius subgroup is non-virtually solvable if it contains either a
non-Ciclyc Schottky group or a purely elliptic free group whose rank is at
least two, see [7, 26] for a general discussion or [16, 20] for standard
arguments in the one-dimensional case.

One has the following theorem from [27]:

######  Theorem 3.2.

For subgroups of ${\rm PSL}(3,\mathbb{C})$, solvable is equivalent to having a
finite index subgroup which is conjugate to group of upper triangulable
matrices.

So the following are examples of solvable groups:

  1. (i)

All purely parabolic groups.

  2. (ii)

Hyperbolic Toral groups (we discuss these later, in section 7).

  3. (iii)

Fundamental groups of Inoue surfaces, see [2, 5, 11].

So, after our previous section 2, to have the complete picture of solvable
groups we must look at solvable groups with loxodromic elements. This is done
in [27, 28]. In those articles the author determined the corresponding limit
set and the representations of solvable groups in ${\rm PSL}(3,\mathbb{C})$
containing loxodromic element. The main results can be summarized as follows.
These summarize Theorems 1.1 to 1.4 in [28] and theorems 1 and 2 in [27]:

######  Theorem 3.3.

The solvable groups in ${\rm PSL}(3,\mathbb{C})$ can be of three main types:

  1. (i)

Commutative.

  2. (ii)

Cyclic extensions of torus groups by a strongly loxodromic element.

  3. (iii)

Dual torus groups extensions.

######  Theorem 3.4.

If a solvable group $\Gamma$ is commutative, then its limit set consists of
either two lines, three lines in general position, or it is a line and a
point, and $\Gamma$ can be either triangular or a fake Hopf group:

  * •

Diagonal groups:

$\left\\{diag(\alpha^{n},\alpha^{m},1):n,m\in\mathbb{Z}\right\\}$

here $\alpha,\beta\in\mathbb{C}^{*}$, $|\alpha|\neq 1$ or $|\beta|\neq 1$.

  * •

Fake Hopf:

$\left\\{\begin{bmatrix}\mu(w)&w\mu(w)&0\\\ 0&\mu(w)&0\\\
0&0&\mu(w)^{-2}\end{bmatrix}\right\\}$

where $W\subset\mathbb{C}$ is a discrete group and
$\mu:W\rightarrow\mathbb{C}^{*}$ is a group morphism, satisfying some
technical conditions, see [28].

######  Theorem 3.5.

If the group is an extension of either a torus group or a dual torus group,
then its limit set can be a single projective line, or two lines, or else it
has infinitely many lines but either it is a cone of lines over a circle or it
has four lines in general position. And one has:

  1. (i)

If the group $\Gamma$ is a cyclic extension of a torus group, then it is a
semi-direct product of a torus group and the cyclic group generated by a
strongly loxodromic element.

  2. (ii)

If the group is an extension of a dual torus groups, then it can be:

     * •

An extension by a loxodromic element (of any type); or

     * •

An extension by two loxo-parabolic elements, and in this case the group is a
semi-direct product of the dual torus group and the group generated by the two
loxodromic elements.

In the sequel we describe all groups with limit set as stated in this theorem.

##  4\. Groups with limit set exactly one line

In the previous section we described several purely parabolic groups with
limit set a single line. Here we follow [3] and describe all the discrete
groups in ${\rm PSL}(3,\mathbb{C})$ with limit set a line. In particular,
every such group is virtually nilpotent.

Let $\Gamma$ be complex Kleinian with limit set a line $\ell$. Then $\Gamma$
acts properly and discontinuously on the complement of $\ell$. From the above
described classification of the elements in ${\rm PSL}(3,\mathbb{C})$ we known
that the elements in $\Gamma$ are all either elliptic, parabolic or
loxoparabolic. If the group contains a loxoparabolic element, then one finds
that $\Gamma$ is conjugate to a group such that every element can be
represented by a matrix of the form:

$\left(\begin{array}[]{lll}a&0&v\\\ 0&d&0\\\ 0&0&1\\\ \end{array}\right)\,,$

where $\ell$ becomes the line $\overleftrightarrow{e_{1},e_{2}}$. Moreover,
$\Gamma$ acts as an Euclidean group on the line
$\overleftrightarrow{e_{1},e_{3}}$. By well-known facts on Euclidean groups
(see [20]), $a$ is a root of unity of order $1,2,3,4$ or $6$, and the rank of
$\Gamma$ considered as acting on $\overleftrightarrow{e_{1},e_{3}}$ is equal
to $1$ or $2$. One can show that $a$ cannot be a root of unity of order
$2,3,4$ nor $6$, otherwise $\Gamma$ would contain a complex homothety.
Finally, the rank of $\Gamma$ considered as acting on
$\overleftrightarrow{e_{1},e_{3}}$ is not equal to $1$, otherwise, the
Kulkarni limit set of $\Gamma$ would be equal to two lines.

On the other hand, if the group does not contain any loxoparabolic elements,
then there are two cases. If it acts on $\ell$ without parabolic elements then
$\Gamma$ can be considered as a discrete group of Euclidean isometries of
$\mathbb{R}^{4}$. If some element acts on $\ell$ as parabolic element then the
group can be identified with a group of triangular matrices. Then one can rule
out the existence of irrational ellipto-parabolic elements and show that there
exists a unipotent subgroup of finite index.

In this way we arrive to the following [3, Theorem 1.1]:

######  Theorem 4.1.

Let $\Gamma$ be a subgroup of ${\rm PSL}(3,\mathbb{C}$) such that its Kulkarni
limit consists of precisely one complex projective line $\ell$. Then:

  1. (i)

If $\Gamma$ does not contain loxoparabolic elements nor an element which acts
as a parabolic element on the line $\ell$, then $\Gamma$ is a group of
isometries of $\mathbb{C}^{2}$ and it contains a free abelian normal subgroup
of finite index and rank $\leq 4$.

  2. (ii)

If $\Gamma$ does not contain loxoparabolic elements but it does contain an
element which acts as a parabolic element on $\ell$, then $\Gamma$ does not
contain any irrational ellipto-parabolic elements and it is a finite extension
of a unipotent subgroup that consists of unipotent parabolic maps. Hence it is
a finite extension of a group of the form
$\mathbb{Z},\mathbb{Z}^{2},\mathbb{Z}^{3},\mathbb{Z}^{4}$, $\Delta_{k}$ or
$G_{k}$, where

$\Delta_{k}=\large\langle A,B,C,D:\,C,D\;\hbox{are central
and}\;\,[A,B]=C^{k}\large\rangle\,,k\in\mathbb{N}\;,$ $G_{k}=\large\langle
A,B,C:\,C\;\hbox{is central
and}\;\,[A,B]=C^{k}\large\rangle\,,k\in\mathbb{N}\;,$

  3. (iii)

If $\Gamma$ does contain a loxoparabolic element, then $\Gamma$ is isomorphic
to the group $\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}_{n_{0}}$, where
$n_{0}\in\mathbb{N}$ is arbitrary. The $\mathbb{Z}_{n_{0}}$ summand is a group
of complex reflections, while $\mathbb{Z}\oplus\mathbb{Z}$ is generated by a
loxoparabolic element and another element which can be loxoparabolic or
parabolic.

##  5\. Groups with limit set two lines

We must distinguish two cases:

a) Groups that have exactly two lines in their limit set. These are elementary
of the first kind; and

b) Groups with more than two lines in the limit set, but only two in general
position. In this case, we know from Theorem 1.5 that the total number of
lines actually is infinite. These are elementary of the second kind.

For instance, cyclic groups generated by a strongly loxodromic element have
exactly two lines in their limit set. On the other hand, for instance, the
dual torus groups of the second type, described before, have limit set a cone
of projective lines over a circle. In this case, there are infinitely many
lines in the limit set, but only two in general position.

Let us describe these elementary groups. As in [4], for every open
$\Gamma$-invariant set $U$ we denote by $\lambda(U)$ the maximum number of
complex projective lines contained in $\mathbb{P}^{2}_{\mathbb{C}}\setminus
U$, and by $\mu(U)$ the maximum number of such lines in general position. If
$U$ is the Kulkarni discontinuity region $\Omega_{{\rm Kul}}(\Gamma)$, then
its complement $\mathbb{P}^{2}_{\mathbb{C}}\setminus U$ is the limit set
$\Lambda_{{\rm Kul}}=\Lambda_{{\rm Kul}}(\Gamma)$.

######  Theorem 5.1.

Let $\Gamma\subset{\rm PSL}(3,\mathbb{C})$ be a group satisfying
$\mu(\Omega_{{\rm Kul}}(\Gamma))=2$, then either the limit set is the union of
two lines or it is a pencil of lines over a nowhere dense perfect set.
Moreover, one of the following facts occurs:

  1. (i)

if $\lambda(\Omega_{{\rm Kul}}(\Gamma))=2$, then $\Gamma$ is virtually
solvable and contains a loxodromic element, see [27, 28] for a precise
description.

  2. (ii)

if $\lambda(\Omega_{{\rm Kul}}(\Gamma))=\infty$, then $\Gamma$ either is
virtually purely parabolic or contains a loxodromic element, and:

    1. (a)

If the group is virtually purely parabolic then it contains a finite index
subgroup conjugate to one of the following groups: a dual torus group of type
II, a Kleinian Inoue group, or an extended Kodaira group $\mathcal{K}_{2}$.

    2. (b)

If the group contains a loxodromic element, then either $\Gamma$ is virtually
conjugate to a solvable group (described in [28]), or it is a weakly
controllable group whose control group is non-elementary.

###### Proof.

Since $\mu(\Omega_{{\rm Kul}}(\Gamma))=2$, we have that $\Gamma$ has a global
fixed point $p$, the point where those two lines intersect, and
$\lambda(\Omega_{{\rm Kul}}(\Gamma))$ is either 2 or $\infty$ by Theorem 1.1
[4]. In fact, will see in the next section that if $\lambda(\Omega_{{\rm
Kul}}(\Gamma))=3$, then also $\mu(\Omega_{{\rm Kul}}(\Gamma))=3$. And we know
from Theorem 1.5 that if $\lambda(\Omega_{{\rm Kul}}(\Gamma))>3$, then this
number actually is $\infty$.

Also, if $\lambda(\Omega_{{\rm Kul}}(\Gamma))=2$, then, we claim, that
$\Gamma$ is virtually solvable. To see this notice that the group is weakly
semi-controllable. Choose a line $\ell$ as the ”horizon”. Then $\Lambda_{{\rm
Kul}}$ meets $\ell$ in two points which are invariant under the control group
$\Pi(\Gamma)$. Hence $\Pi(\Gamma)$ is virtually diagonalizable and therefore
$\Gamma$ itself is virtually diagonalizable. Hence it is virtually solvable.
Then, by Theorem 0.1 in [5], we deduce that $\Gamma$ contains a loxodromic
element. Then, by [27, 28], its limit set consists of exactly two lines.

In case $\lambda(\Omega_{{\rm Kul}}(\Gamma))=\infty$, we have that $\Gamma$ is
either virtually purely parabolic or it contains a loxodromic element. In the
first case by part (b) item (2) of Theorem 0.1 in [5] we have that $\Gamma$
contains a finite index subgroup conjugate to one of the following groups: a
dual torus group of type II, a Kleinian Inoue group, or an extended Kodaira
group $\mathcal{K}_{2}$, thus, in this case, the limit set is a cone of lines
over a Euclidean circle, see section 2 of [5]. Now the proof splits into the
following cases:

Case 1. The group $\Gamma$ is virtually solvable. In this case by Theorems
1.1, 1.2, 1.3, 1.4 in [28], the limit set of $\Gamma$ is either two lines or a
cone of lines over an Euclidean circle.

Case 2. The group $\Gamma$ is non-virtually solvable. In this case it is clear
that the control group $\Pi_{p,\ell}\Gamma$ is non-elementary, here $\ell$ is
any line not containing $p$, and

$\Lambda_{{\rm
Kul}}(\Gamma)=\bigcup_{z\in\Lambda(\Pi(\Gamma))}\overleftrightarrow{p,z}.$

That $\Pi_{p,\ell}\Gamma$ is non-elementary follows from the fact that,
essentially by Borel’s fixed point theory, a subgroup of ${\rm
PSL}(2,\mathbb{C})$ is elementary if and only if it is virtually solvable (see
[11]). ∎

##  6\. Groups with limit set three lines

We now describe the subgroups of ${\rm PSL}(3,\mathbb{C})$ with three lines in
their limit set. As in the previous section, we denote by
$\lambda=\lambda(\Omega_{{\rm Kul}}(\Gamma))$ the maximum number of complex
projective lines contained in $\Lambda_{{\rm Kul}}$, and by
$\mu=\mu(\Omega_{{\rm Kul}}(\Gamma))$ the maximum number of such lines in
general position. Then there are groups with $\mu=\lambda=3$, and there are
groups with $\lambda=\infty$ but $\mu=3$. In the case $\lambda=\infty$ the
groups in question are “suspensions” of non-elementary Kleinian groups in
${\rm PSL}(2,\mathbb{C})$, see Chapter 5 in [8]:

######  Theorem 6.1.

Let $\Gamma\subset{\rm PSL}(3,\mathbb{C})$ be a group for which $\mu=3$. Then
$\lambda$ can be either $3$ or $\infty$ and:

  1. (i)

If $\lambda=3$, then these lines are in general position and $\Gamma$ contains
a finite index subgroup conjugate to:

$G_{0}=\left\\{diag[\alpha^{n},\beta^{m},1]:n,m\in\mathbb{Z}\right\\}\;,$

where $\alpha,\beta\in\mathbb{C}^{*}$ are non-unitary complex numbers.

  2. (ii)

If $\lambda=\infty$, then the limit set is a cone of lines plus a non-
concurrent line and $\Gamma$ contains a finite index subgroup conjugate to:

$H_{\Sigma,\rho,\alpha}=\left\\{\begin{bmatrix}\alpha^{n}\rho([g])&0\\\ 0&g\\\
\end{bmatrix}:n\in\mathbb{Z},g\in{\rm
SL}(2,\mathbb{C}),\,\,[g]\in\Sigma\right\\}\;,$

where $\alpha\in\mathbb{C}^{*}$ is a non-unitary complex number, $\Sigma$ is a
non-elementary discrete group in ${\rm PSL}(2,\mathbb{C})$ with non-empty
discontinuity region, and $\rho$ is a function
$\Sigma\rightarrow\mathbb{C}^{*}$. In this case

$\Lambda_{{\rm
Kul}}(H_{\Sigma,\rho,\alpha})=\overleftrightarrow{[e_{2}],[e_{3}]}\cup\bigcup_{p\in\Lambda(\Sigma)}\overleftrightarrow{[e_{1}],p}\;.$

###### Proof.

We know from statement (iii) in Theorem 1.5 that if $\lambda=3$ then $\mu=3$.
That is, if the group has exactly three lines in its limit set, then these
lines are in general position.

Now we assume $\mu=3$ and we choose three distinct lines in general position
contained in $\Lambda_{{\rm Kul}}(\Gamma)$, say $\ell_{1},\ell_{2},\ell_{3}$.
We define

$P=\\{p_{ij}=\ell_{i}\cap\ell_{j}:1\leq i<j\leq 3\\}\,,$

then any line contained in $\Lambda_{{\rm Kul}}(\Gamma)$ must intersect $P$.

We will say $p\in P$ is a vertex whenever there are infinite lines in
$\Lambda_{{\rm Kul}}(\Gamma)$ passing through it. If the set of vertices is
empty, then by Proposition 5.6 in [4] the set $P$ is a $\Gamma$-invariant set.
Then $H=Isot(P,\Gamma)$ is a finite index subgroup of $\Gamma$. Moreover,
$\Lambda_{{\rm Kul}}(H)=\Lambda_{{\rm Kul}}(\Gamma)$ and $H$ is conjugate to a
group $G_{0}$ where every element has a diagonal lift. So by Theorem 1.1 in
[28] we conclude $G_{0}=\\{diag[\alpha^{n},\beta^{m},1]:n,m\in\mathbb{Z}\\}$
for some $\alpha,\beta\in\mathbb{C}^{*}$ non-unitary complex numbers.

If the set of vertices is non-empty then by the proof of Proposition 6.5 in
[4] we must have that this set is $\Gamma$-invariant. Observe that in case the
number of vertices is at least two, we must have four lines in general
position (compare with Proposition 6.5 in [4]) which is not possible,
therefore we have exactly one vertex, say $v$, which is $\Gamma$-invariant.
Let $\ell\in\\{\ell_{1},\ell_{2},\ell_{3}\\}$ be a line not containing $v$.
Since $\mu(\Omega_{{\rm Kul}}(\Gamma))=3$, we conclude that $\ell$ is
$\Gamma$-invariant. After conjugating with a projective transformation, if
necessary, let us assume that $v=[e_{1}]$ and
$\ell=\overleftrightarrow{[e_{2}],[e_{3}]}$, as in Section 1.5 take
$\pi=\pi_{v,\ell}$ and $\Pi=\Pi_{v,\ell}$, now we claim:

Claim 1. If $\Pi(\Gamma)$ is non-virtually solvable, then $\Pi(\Gamma)$ is
discrete. On the contrary let us assume that $\Pi(\Gamma)$ is non-discrete,
thus by Theorem 1 and Proposition 12 in [16] the principal component of
$\overline{\Pi(\Gamma)}$ is ${\rm PSL}(2,\mathbb{C})$ or conjugate to ${\rm
SO}(3)$ or ${\rm PSL}(2,\mathbb{R})$. In the first two cases, the orbit of any
line passing through $v$ is dense in $\ell$, and this is not possible. In the
latter case, let $(g_{n})\subset\Pi(G)$ be a sequence such that
$\lim_{n\to\infty}g_{n}=Id$ and $h_{n}=[g_{1},g_{n}]\neq Id$ for $n\geq 2$. If
$G_{m}\in\Gamma$ satisfies $\Pi(G_{m})=g_{m}$, then a straightforward
computation shows $\lim_{n\to\infty}[G_{1},G_{n}]=Id$, which is a
contradiction.

Claim 2. The group $\Gamma$ cannot be virtually solvable. On the contrary, let
us assume that $\Gamma$ is solvable, then by Theorem 0.1 in [5] the group
$\Gamma$ contains a loxodromic element. By Theorems 1.1, 1.2, 1.3 and 1.4 in
[28], there is $W\subset\mathbb{C}$ a discrete additive subgroup with
$rank(W)\leq 2$ and $\alpha,\beta\in\mathbb{C}^{*}$ such that $\alpha\neq 1$
and $\alpha\beta^{2}\notin\mathbb{R}$ and $\Gamma$ is conjugate to:

$\left\\{\begin{bmatrix}1&0&0\\\ 0&1&w\\\ 0&0&1\end{bmatrix}:w\in
W\right\\}\rtimes\left\\{\begin{bmatrix}\alpha^{2n}\beta^{n}&0&0\\\
0&\alpha^{n}\beta^{2n}&0\\\ 0&0&1\end{bmatrix}:n\in\mathbb{Z}\right\\}.$

Now a simple computation shows $W$ is non-discrete, which is a contradiction.

By the previous claims, we conclude that $\Pi(\Gamma)$ is a discrete non-
elementary with a non-empty ordinary region. Now the rest of the proof is
trivial. ∎

##  7\. Groups with limit set four lines in general position lines

In this section we provide an algebraic characterization of the subgroups of
${\rm PSL}(3,\mathbb{C})$ with exactly four lines lines in general position in
their limit set $\Lambda_{{\rm Kul}}$ . Of course that by Theorem 1.5 this
implies that $\Lambda_{{\rm Kul}}$ actually has infinitely many lines, but
only four in general position. The basic reference is [2].

We recall first that an element $A\in{\rm SL}(2,\mathbb{Z})$ is a hyperbolic
toral automorphism if none of its eigenvalues has norm 1 (see [17]).

######  Definition 7.1.

A subgroup $\Gamma$ of ${\rm PSL}(3,\mathbb{C})$ is a hyperbolic toral group
if it is conjugate to a group of the form:

$\Gamma_{A}\,=\left\\{\,\left(\begin{array}[]{lll}A^{k}&b\\\ 0&1\\\
\end{array}\right)\,\;\Big{|}\;b=\left(\begin{array}[]{lll}b_{1}\\\ b_{2}\\\
\end{array}\right)\,,\;b_{1},b_{2},k\in\mathbb{Z}\,\right\\}$

where $A\in{\rm SL}(2,\mathbb{Z})$ is a hyperbolic toral automorphism.

One has:

######  Theorem 7.2.

Let $\Gamma\subset{\rm PSL}(3,\mathbb{C})$ be a discrete group. The maximum
number of complex lines in general position contained in its Kulkarni limit
set is equal to four if, and only if, $\Gamma$ contains a hyperbolic toral
group whose index is at most 8. [In this case the limit set of the subgroup
coincides with that of $\Gamma$, and the discontinuity region has four
components each one diffeomorphic to $\mathbb{H}\times\mathbb{H}$, where
$\mathbb{H}$ denotes the half open plane in $\mathbb{C}$.]

To prove the theorem above, the first step is showing that a toral group
$\Gamma_{A}$ as above is the subgroup of ${\rm PSL}(3,\mathbb{Z})$ generated
by the unipotent parabolic elements $P_{1},P_{2}$ and the strongly loxodromic
element $L$ given by:

$P_{1}=\left(\begin{array}[]{lll}1&0&0\\\ 0&1&1\\\ 0&0&1\\\
\end{array}\right)\;,\;P_{2}=\left(\begin{array}[]{lll}1&0&1\\\ 0&1&0\\\
0&0&1\\\ \end{array}\right)\;,\;L=\left(\begin{array}[]{lll}A&0\\\ 0&1\\\
\end{array}\right)\;.\;$

Therefore $\Gamma_{A}$ is discrete. The dynamical properties of the hyperbolic
toral group are used to show that $\Lambda_{\textrm{{\rm Kul}}}(L)$ is
contained in Kulkarni limit set of $\Gamma_{A}$. Also, a straightforward
computation shows that $\Lambda_{\textrm{{\rm Kul}}}(P_{1})$ is contained in
$\Lambda_{\textrm{{\rm Kul}}}(\Gamma_{A})$. Thus,

$\Lambda_{\textrm{{\rm
Kul}}}(\Gamma_{A})=\overline{\bigcup_{\gamma\in\Gamma_{A}}\Lambda_{\textrm{{\rm
Kul}}}(\gamma)},$

because both sets in this equation contain three lines in general position
(see Theorem 1.2 in [1]).

Now, it is possible to compute the sets $\Lambda_{{{\rm Kul}}}(\gamma)$ for
every $\gamma\in\Gamma_{A}$. One finds that
$\Lambda_{\textrm{Kul}}(\Gamma_{A})$ can be described as a union of two
pencils of lines over two distinct circles, with vertices two distinct points,
which share the line determined by the two vertices. Therefore, the maximum
number of lines in general position contained in
$\Lambda_{\textrm{Kul}}(\Gamma_{A})$ is equal to four. Moreover, this
description shows that the Kulkarni discontinuity region of $\Gamma_{A}$ is a
disjoint union of four copies of $\mathbb{H}\times\mathbb{H}$, where
$\mathbb{H}$ denotes the upper half plane in $\mathbb{C}$.

Conversely, if we assume that the maximum number of lines in general position
contained in the Kulkarni limit set of a group $\Gamma$ is equal to four, then
one shows that there are two distinguished points (called vertices) contained
in infinitely many lines in this limit set. The subgroup which stabilizes
these two vertices is denoted by $\Gamma_{0}$ and it has index at most two in
$\Gamma$. Furthermore, it can be proved that
$\Omega_{\textrm{Kul}}(\Gamma_{0})$ has four components, each one
diffeomorphic to a copy of $\mathbb{H}\times\mathbb{H}$. Finally, the subgroup
of $\Gamma_{0}$ stabilizing each of these components has finite index (at most
four) in $\Gamma_{0}$, and it can be shown that it is a hyperbolic toral
group.

A description of the quotient spaces
$\Omega_{\textrm{Kul}}(\Gamma_{A})/\Gamma_{A}$ is given by the following
theorem (see [6].)

######  Theorem 7.3.

If $\Gamma_{A}$ is a hyperbolic toral group then

  1. (i)

The group $\Gamma_{A}$ is isomorphic to a lattice of the group Sol (see
Definition 7.4).

  2. (ii)

The quotient space $\Omega_{\textrm{Kul}}(\Gamma_{A})/\Gamma_{A}$ is a
disjoint union of four copies of

$\left(\textrm{Sol}/\Gamma_{A}\right)\times\mathbb{R}.$

######  Definition 7.4.

The group Sol is $\mathbb{R}^{3}$ equipped with the operation

$\left(\left(\begin{array}[]{c}x_{1}\\\
y_{1}\end{array}\right),t_{1}\right)\left(\left(\begin{array}[]{c}x_{2}\\\
y_{2}\end{array}\right),t_{2}\right)=\left(\left(\begin{array}[]{c}x_{1}+e^{t_{1}}x_{2}\\\
y_{1}+e^{-t_{1}}y_{2}\end{array}\right),t_{1}+t_{2}\right)$

The proof of 7.3 is done by giving an explicit smooth foliation of
$\mathbb{H}\times\mathbb{H}$ where the leaves are diffeomorphic to Sol, and by
showing that $\mathbb{H}\times\mathbb{H}$ is diffeomorphic to
$\textrm{Sol}\times\mathbb{R}$ by a $\Gamma_{A}$-equivariant diffeomorphism.

It can be shown that $\Gamma_{A}$ is isomorphic to
$\mathbb{Z}^{2}\rtimes_{A}\mathbb{Z}$, and this group is determined by the
conjugacy class in $\textrm{GL}(2,\mathbb{Z})$ of $A$ (see [23] pages 96,97).
It follows from Theorem 7.2 that there is a countable number of nonisomorphic
Complex Kleinian groups such that the maximum number of lines in general
position contained in its Kulkarni limit set is equal to four.

##  8\. Groups with limit set having an isolated point

Let us suppose now that $\Gamma\subset{\rm PSL}(3,\mathbb{C})$ is a group with
at least an isolated point in its limit set. Then the results explained in
this article show that $\Gamma$ must have only one line in its limit set, for
otherwise $\Lambda_{{\rm Kul}}$ is a union of lines.

Let $\ell$ be the unique line in $\Lambda_{{\rm Kul}}(\Gamma)$ and let
$p\in\Lambda_{{\rm Kul}}(\Gamma)$ be an isolated point. Then $\ell$ is an
invariant set and we look at the action of $\Gamma$ on $\ell$ to deduce from
it consequences on the global dynamics of $\Gamma$.

We note that this is somehow similar to the controllable groups, where we have
a global fix point and a line $L$ that we call the horizon, which is not
invariant but we get an action on it. Then we know a lot about the action of
the group on ${\mathbb{P}}_{\mathbb{C}}^{2}$ by looking at the induced action
on $L$. In the case we now envisage, we have the invariant line $\ell$ and the
action on it says a lot about the dynamics of the group on
${\mathbb{P}}_{\mathbb{C}}^{2}$. There are two possibilities as the action of
$\Gamma$ on $\ell$ can be either:

  1. a.

Virtually Solvable.

  2. b.

Non- Virtually solvable.

The idea of the proof consists in showing first that the non-virtually
solvable case cannot happen, so the action must be virtually solvable. This
uses the Tits Alternative.

In the solvable case, the dynamics implies that on the invariant line $\ell$
we must have either one or two fixed points. Then one shows that in the first
case, when there is only one fixed point, one actually has an invariant flag
and up to conjugation the elements are all of the form

$\begin{bmatrix}\alpha&0&0\\\ 0&\beta&\delta\\\ 0&0&\gamma\end{bmatrix}\;.$

In the second case we find that, up to conjugation, all the elements are
diagonal, and the group consists of lodrodromic elements that are screws (see
Section 1.3), so the limit set of each element consists of a point and a line.
Then we show that the limit set of the whole group is a point and a line.

The above conisderations lead to the following:

######  Theorem 8.1.

Let $\Gamma\subset{\rm PSL}(3,\mathbb{C})$ be a group with at least an
isolated point in its limit set. Then the limit set consists of exactly one
line and one point, and $\Gamma$ is conjugate to either:

  1. (i)

$G_{0}=\left\\{{\rm diag}\,[\alpha^{n},e^{2\pi i\theta},e^{-2\pi
i\theta}]:n,m\in\mathbb{Z}\right\\}$

where $\theta\in\mathbb{R}$ and $\alpha\in\mathbb{C}^{*}$ is non-unitary
complex number.

  2. (ii)

or

$H_{0}=\left\\{\begin{bmatrix}\mu(w)^{-2}&0&0\\\ 0&\mu(w)&w\mu(w)\\\
0&0&\mu(w)\\\ \end{bmatrix}:w\in W\right\\}$

where $W\subset\mathbb{C}$ is a discrete additive group and
$\mu:(\mathbb{C},+)\rightarrow(\mathbb{C}^{*},\cdot)$ is a group morphism
satisfying that for every sequence of distinct elements $(w_{n})\subset W$,
the limit $\lim_{n\to\infty}\mu(w_{n})$ is either $0$ or $\infty$.

###### Proof.

By Theorem 1.2 in [2] we have $\mu(\Omega_{{\rm Kul}}(\Gamma))\leq 2$, and
from Theorem 5.1 we conclude $\mu(\Omega_{{\rm Kul}}(\Gamma))=1$. On the other
hand, by Theorem 0.1 in [5], the group $\Gamma$ contains a Loxodromic element.
Let $\ell$ be the unique line in $\Lambda_{{\rm Kul}}(\Gamma)$,
$p\in\Lambda_{{\rm Kul}}(\Gamma)$ be an isolated point and $\gamma\in\Gamma$ a
loxodromic element. We must consider the following cases:

Case 1. The action of $\Gamma$ restricted to $\ell$ is given by a virtually
solvable group. In this case the group $\Gamma$ is virtually solvable and the
theorem follows from Claim 2 and Theorems 1,1, 1.2, 1.3 and 1.4 in [28], see
also section 1 above .

Case 2. The action of $\Gamma$ restricted to $\ell$ is given by a non-
virtually solvable group. Under this assumption we claim:

Claim. There are $g_{1},g_{2}\in\Gamma$ such that the group generated by
$\\{\gamma_{1}|_{\ell},\gamma_{2}|_{\ell}\\}$ is a rank two free group which
is either purely loxodromic or purely elliptic. If the action of $\Gamma$ on
$\ell$ is given by a discrete group $H$, then it is well know that $H$
contains a rank two Shottky group, see Chapter X in [20], these will do the
job. If the action on $\ell$ is given by a non discrete group $G$, thus by
Theorem 1 and Proposition 12 in [16] the principal component of $\overline{G}$
is ${\rm PSL}(2,\mathbb{C})$, or conjugate to ${\rm SO}(3)$ or ${\rm
PSL}(2,\mathbb{R})$. In any such case we deduce that $G$ contains a non-
virtualy solvable finitelly generated subgroup, as a consequence of Tit’s
alternative we deduce that $G$ contains a rank two free subgroup which is
either purely loxodromic or purely elliptic. Which proves the claim.

Now let $g_{1},g_{2}$ be the elements given by the previous claim. After
conjugating with a projective transformation we can assume that
$\ell=\overleftrightarrow{[e_{2}],[e_{3}]}$ so $h=[g_{1},g_{2}]$ is a
loxodromic element and

$h=\begin{bmatrix}1&0&0\\\ c&d&e\\\ f&g&h\end{bmatrix}$

where $dh-eg=1$ and $(d+h)^{2}\in\mathbb{C}-\\{4\\}$. Thus $h$ is either an
elliptic element of infinite order, which cannot happen by the discreteness of
$\Gamma$, or a strongly loxodromic element whose attracting and repelling
points are contained in $\ell$. We conclude that the attracting and repelling
lines of $h$ are distinct from $\ell$. On the other hand, from the
$\lambda$-Lemma (see Lemma 7.3.6 in [8]) we know that the attracting and
repelling line of $h$ are both contained in $\Lambda_{{\rm Kul}}(\Gamma)$,
which is a contradiction. ∎

## References

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Miguel Neves, Laura Duarte, and Pedro Neto 11institutetext: University of
Coimbra, CEMMPRE, ARISE, Department of Mechanical Engineering, 3030-788,
Coimbra, Portugal
11email<EMAIL_ADDRESS>
# A Collaborative Robot-Assisted Manufacturing Assembly Process

Miguel Neves 11 Laura Duarte 11 Pedro Neto 11

###### Abstract

An effective human-robot collaborative process results in the reduction of the
operator’s workload, promoting a more efficient, productive, safer and less
error-prone working environment. However, the implementation of collaborative
robots in industry is still challenging. In this work, we compare manual and
robot-assisted assembly processes to evaluate the effectiveness of
collaborative robots while featuring different modes of operation
(coexistence, cooperation and collaboration). Results indicate an improvement
in ergonomic conditions and ease of execution without substantially
compromising assembly time. Furthermore, the robot is intuitive to use and
guides the user on the proper sequencing of the process.

###### keywords:

Human-robot Interaction, Collaborative robotics, Manufacturing, Assembly

##  1 Introduction

Manufacturing assembly often requires the human operator to conduct complex
tasks in non-ergonomic conditions. An effective human-robot interactive and
collaborative process stands out among the possible solutions to reduce human
workload and improve comfort, by having a robot to assist/answer the human
needs in the manufacturing context [1]. This approach has the main advantage
of combining the flexibility and dexterity of human operators with the
consistency and precision of collaborative robots.

The human-robot interactive process can be subdivided into three main types,
human-robot coexistence, human-robot cooperation and human-robot collaboration
[2, 3]. Human-robot coexistence, as the name suggests, occurs when the human
operator and the robot share a physical space but do not share a workspace.
However, since there is no physical separation between them, the robot must be
equipped with safety systems that stop the robot when a human operator enters
its workspace. Human-robot cooperation refers to the scenario where the human
and robot share a workspace but their tasks are organized in a sequential way
and the physical contact between the two is not required. Lastly, human-robot
collaboration corresponds to the scenario where the human operator and the
robot share a workspace, work simultaneously on the same task and physical
contact often occurs [4]. The ISO/TS 15066 standard extends the generic
guidelines of the ISO 10218 standard of safety requirements for industrial
robots by defining passive and active safeguards that robot designers,
integrators and users must acknowledge when deploying collaborative robot
applications. [5]

Human-robot collaborative scenarios are complex to implement due to the
requirement of being adaptable and safe for the human operator [6]. Due to
these difficulties, frameworks have been proposed to aid the implementation
processes [7] and the related task allocation processes [2]. Human-robot
interaction and collaboration must be dynamic to accommodate the current
workflow of the human operator, reducing the workload felt by the human
operator [8]. In fact, the human operator’s state and mental workload are some
of the most important aspects to be considered in human-robot interactive and
collaborative scenarios. As a result, various studies have focused on
capturing this information through the usage of wearable sensors coupled with
machine-learning solutions [9, 10]. Alternatively, virtual environments offer
a safe space to verify the plausibility of the real-world implementation,
facilitating the realisation of an integrated, flexible and collaborative
manufacturing environment [6, 11]. These environments can also be used to add
useful and relevant information to the real-world environment through
techniques such as augmented reality (AR) [12].

In this work, inspired by a real industrial application, we propose to use a
collaborative robot as a third hand to assist/help a human operator in a
manufacturing assembly process. The proposed methodology aims to identify
opportunities for the collaborative robot to assist the operator in the
various assembly tasks. Finally, this human-robot collaborative process is
compared with the existing manual process, highlighting and discussing the
pros and cons of each modality.

Figure 1: Components from the roller conveyor section, including the contents
of the boxes transported by the robot to the operator.

##  2 Methodology

This study considers the real industrial use case of the assembly of a roller
conveyor section, Fig 1. It has a total of 10 parts (P1 to P10) plus the
components delivered in boxes (Box 1 to Box 6), and the assembly is subdivided
into 6 steps. The process is studied considering two different scenarios,
namely the manual assembly and the human-robot collaborative assembly to
identify whether the addition of a collaborative robot can positively impact
the assembly task. The metrics considered are the ergonomic conditions, the
correctness of the assembly sequence and the execution time. In the first
scenario, manual assembly, all bolts and components are within reach of the
human operator. In the second scenario, the collaborative robot is introduced
with two major functions: bringing boxes with small components and bolts to
the workspace and holding components during the assembly, operating as a third
hand to the human operator. In this scenario, bigger components remain within
reach of the human operator, Fig 2.

Figure 2: Sequence of actions done by the human, the robot or in collaboration
in the collaborative scenario.

The collaborative robot is a KUKA lbr iiwa 7 r800 [13] equipped with a SCHUNK
Co-act EGP-C 64 gripper [14]. The human operator has an input button, which
prompts the pre-programmed sequence processor to send the next sequence of
commands to the collaborative robot through the iiwaPy3 library [15]. Certain
sequences require the human operator to guide the collaborative robot into
position through an intuitive process of hand-guiding at the end-effector
level [16]. For each assembly step, we defined the tasks being performed
according to the current process of manual assembly in the company, and the
associated collaborative mode when the robot is used. The coexistence refers
to physical space sharing, but not workspace sharing, while the human is
performing assembly actions. Cooperation refers to when the robot brings the
boxes of components to a specific area in a given sequence, while the human is
performing assembly actions within the workpiece. The collaboration
corresponds to the hand-guiding of the robot to specific areas to hold
components while the operator is performing assembly actions such as screwing.
In this case, the robot acts as a third hand to the operator. Ergonomy levels
can be differentiated into three categories: Undesirable if the user
experiences physical discomfort due to unnatural body positioning or handling
heavy loads; average if the user experiences some discomfort but it does not
impact their performance; and good if the user experiences minimal effort.

##  3 Results and discussion

Results indicate that for this specific part, the measured manual assembly
time was about 6 minutes, while the collaborative assembly took approximately
7 minutes. This equates to a $15\%$ increase in task time duration when using
the human-robot collaborative approach. However, the benefits gained from
using the collaborative robot outweigh the savings in manual assembly time.
The human operator struggles in the manual assembly tasks which require both
hands, particularly steps 1 and 2, Fig. 3. Notably, step 2 demands the
operator to hold two components while simultaneously placing a screw to hold
them together. The robot acting as a third hand can be used to hold one of the
components, facilitating the assembly process. As a consequence, the
introduction of the collaborative robot simplifies the execution of assembly
steps 1 and 2. Most assembly tasks benefit from collaborative assistance, even
when it is not essential to the assembly process. By assigning the robot the
task of holding certain components in place, the operator can be in a more
ergonomic position and, thus, perform the assembly task with more ease. The
human operator also benefits from a more intuitive workflow. The robot
interaction is triggered by a single button, which is selected depending on
the current step of the assembly process.

Owing to the diversity of the components used in the assembly process, a
variety of screw sizes are needed to join them. Therefore, the robot is able
to deliver the correct screws for each step of the assembly process, reducing
the chance of errors by the human operator and clearing up the workspace. Due
to the low number of trials performed, it was impossible to make a thorough
quantitative analysis of the error reduction in this regard, but it was
possible to conclude that the mental workload was lower in the collaborative
scenario.

Figure 3: Ergonomics comparison between manual and collaborative assemblies
for each different assembly step.

##  4 Conclusions

This study demonstrates the usage of a robot as a collaborative assistant to a
human operator in a manufacturing assembly process. Results demonstrated that
collaborative robots can be introduced in an intuitive way in the assembly
process, as the human operator only has the additional basic and intuitive
tasks of positioning the robot through hand-guiding and prompting the robot to
continue the sequence using a single button.

By using a collaborative robot, the human operator benefits from improved
ergonomic conditions, a well-organized workspace, and less mistakes in the
assembly sequencing. The use of the robot as a third hand to hold components
facilitates the most strenuous assembly tasks. However, the total assembly
time increases $15\%$ when compared to the manual assembly.

While the proposed assembly process covers some of the main primitive tasks in
assembly future work will be dedicated to testing with other components,
namely from electronics and textile industries.

### Acknowledgements

This research is sponsored by national funds through FCT – Fundação para a
Ciência e a Tecnologia, under the project UIDB/00285/2020 and LA/P/0112/2020,
and the grants 2021.06508.BD and 2021.08012.BD.

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