---
abstract: |
    We study some basic properties of the variety of characters in ${\mathrm{PSL}_2 (\C)}$ of a finitely generated group. In particular we give an interpretation of its points as characters of representations. We construct 3-manifolds whose variety of characters has arbitrarily many components that do not lift to ${\mathrm{SL}_2 (\C)}$. We also study the singular locus of the variety of characters of a free group.\
    [*MSC*: 57M50, 57M05, 20C15\
    *Keywords*: Representation spaces; variety of characters; ${\mathrm{PSL}_2 (\C)}$]{}
author:
- Michael Heusener and Joan Porti
title: 'The variety of characters in ${\mathrm{PSL}_2 (\C)}$'
---

Introduction
============

The varieties of representations and characters have many applications in 3-dimensional topology and geometry. The variety of ${\mathrm{SL}_2 (\C)}$-characters has been intensively studied since the seminal paper of Culler and Shalen [@CS], but for many applications it is more convenient to work with ${\mathrm{PSL}_2 (\C)}$ instead of ${\mathrm{SL}_2 (\C)}$ (see [@BZ] and [@BMP] for instance). The purpose of this note is to study some basic properties of the variety of characters in ${\mathrm{PSL}_2 (\C)}$. Most of the results of invariant theory that we use can be found in any standard reference (e.g. [@KSS], [@Kraft], [@PV]).

Throughout this paper, $\Gamma$ will denote a finitely generated group.

The set of all representations of $\Gamma$ in ${\mathrm{PSL}_2 (\C)}$ is denoted by $
       R(\Gamma)
$ and it is called the *variety of representations of $\Gamma$ in ${\mathrm{PSL}_2 (\C)}$*.

The variety of representations $R(\Gamma)$ has a natural structure as an affine algebraic set over the complex numbers given as follows: the group ${\mathrm{PSL}_2 (\C)}$ is algebraic (see Section \[sec:invariants\]). Given a presentation $\Gamma=\langle
\gamma_1,\ldots,\gamma_s\mid (r_i)_{i\in I} \rangle$ we have a natural embedding: $$\begin{array}{rcl}
      R(\Gamma)&\to &{\mathrm{PSL}_2 (\C)}\times\cdots\times {\mathrm{PSL}_2 (\C)}\\
      \rho & \mapsto & (\rho(\gamma_1),\ldots,\rho(\gamma_s))
\end{array}$$ and the defining equations are induced by the relations. This structure can be easily seen to be independent of the presentation. In fact using the isomorphism ${\mathrm{PSL}_2 (\C)}\cong {\mathrm{SO}_{3} (\C)}$, $R(\Gamma)$ has a structure of an affine set (see Lemma \[lem:SO3\]).

The action of ${\mathrm{PSL}_2 (\C)}$ on $R(\Gamma)$ by conjugation is algebraic. The quotient $R(\Gamma)/{\mathrm{PSL}_2 (\C)}$ may be not Hausdorff and it is more convenient to consider the algebraic quotient of invariant theory, because ${\mathrm{PSL}_2 (\C)}$ is reductive.

The *variety of ${\mathrm{PSL}_2 (\C)}$-characters* $X(\Gamma)$ is the quotient $R(\Gamma)/\!/{\mathrm{PSL}_2 (\C)}$ of invariant theory.

This definition means that $X(\Gamma)$ is an affine algebraic set together with a regular map $t\co R(\Gamma) \to X(\Gamma)$ which induces an isomorphism $$t^{*}\co \C[X(\Gamma)]\to\C[R(\Gamma)]^{{\mathrm{PSL}_2 (\C)}}$$ (i.e.the regular functions on $X(\Gamma)$ are precisely the regular functions on $R(\Gamma)$ invariant by conjugation). We will use the notation $R(M)=R(\pi_1M)$ and $X(M)=X(\pi_1M)$ if $M$ is a path-connected topological space.

In this paper we study the basic properties of $X(\Gamma)$.

First we explain the name “variety of characters": given a representation $\rho\co\Gamma\to {\mathrm{PSL}_2 (\C)}$, its character is the map $$\begin{array}{rcl}
      \chi_{\rho}\co\Gamma&\to&\C\\
          \gamma&\mapsto &{\operatorname{tr}}^2(\rho(\gamma))
\end{array}$$

\[thm:PSL2characters\] There is a natural bijection between $X(\Gamma)$ and the set of characters of representations $\rho\in
R(\Gamma)$. This bijection maps every $t(\rho)\in X(\Gamma)$ to the character $\chi_\rho$.

In many cases the representations of $R(\Gamma)$ lift to ${\mathrm{SL}_2 (\C)}$, for instance if $\Gamma$ is a free group. In such a case, $X(\Gamma)$ is just a quotient of the usual variety of characters in ${\mathrm{SL}_2 (\C)}$ (See Proposition \[prop:naturaliso\]). This quotient is the definition already used in [@Bur90], [@HLM1],[@HLM2] and [@Ril84] for 2-bridge knot exteriors. The explicit computation for the figure eight knot exterior is found in [@GM].

There are cases where representations do not lift to ${\mathrm{SL}_2 (\C)}$, for instance the holonomy representation of an orientable hyperbolic 3-orbifold with 2 torsion. The next result proves that there are manifolds with arbitrarily many components of characters that do not lift.

\[thm:nolifts\] For every $n$, there exist a compact irreducible 3-manifold $M$ with $\partial M$ a 2-torus such that $X(M)$ has at least $n$ irreducible one dimensional components whose characters do not lift to ${\mathrm{SL}_2 (\C)}$.

In Section \[sec:invariants\] we prove Theorem \[thm:PSL2characters\]. In Section \[sec:irreducibility\] we study the fiber of the projection $t\co R(\Gamma)\to X(\Gamma)$, introducing the different notions of irreducibility. Section \[sec:lifts\] is devoted to the study of lifts of representations and the proof of Theorem \[thm:nolifts\]. In the last section we determine the singular set of $X(\Gamma)$ when $\Gamma\cong F_n$ is the free group of rank $n\geq 3$.

Invariants of ${\mathrm{PSL}_2 (\C)}$ {#sec:invariants}
=====================================

Before proving Theorem \[thm:PSL2characters\] we quickly review some basic notions of algebraic geometry and invariant theory (that the reader may prefer to skip and go directly to the proof in Subsection \[ss:ProofTheorem\]). For details see [@KSS], [@Kraft] or [@PV].

Basic notions of invariant theory
---------------------------------

A closed algebraic subset $Z\subset \C^{N}$ is called *affine*. We denote by $\C[Z]$ the ring of regular functions on $Z$. An algebraic group $G$ that acts algebraically on $Z$ acts naturally on $\C[Z]$ via $g f (z) := f(g^{-1} z) $. We denote by $\C[Z]^{G} $ the ring of invariant functions, i.e. functions $f\in\C[Z]$ for which $g
f = f$ for all $g\in G$.

The group $G$ is called *reductive* if it has the following property: for each finite dimensional rational representation $\rho\co G\to \mathrm{GL}(V)$ and every $G$-invariant subspace $W\subset V$ there exist a complementary $G$-invariant subspace $W'\subset V$, i.e. $V= W'\oplus W$.

If $Z$ is affine and $G$ is reductive, then the ring $\C[Z]^{G} $ is finitely generated. The affine set $Y$ such that $\C[Y]\cong
\C[Z]^{G} $ is called the *algebraic quotient* and it is denoted by $Z /\!/\/ G$.

We shall use the following properties of reductive groups:

-   By Maschke’s theorem, finite groups are reductive.

-   More generally, let $G\subset \mathrm{GL}_{n}(\C)$ be a linear algebraic group. The group $G$ is reductive if there is a Zariski-dense subgroup $K\subset G$ which is compact in the classical topology. It follows that $\mathrm{GL}_{n}(\C)$, $\mathrm{SL}_{n}(\C)$, $\textrm{O}_{n}(\C)$, $\textrm{SO}_{n}(\C)$ and $\textrm{Sp}_{n}(\C)$ are reductive.

-   Let $G$ be a reductive linear algebraic group. Let $Y$ and $Z$ be varieties on which $G$ acts and let $f\co X\to Y$ be a $G$-invariant regular map. If $f^{*}\co \C[Y]\to\C[X]$ is surjective then $f^{*}(\C[Y]^{G}) = \C[X]^{G}$ holds.

Algebraic structure of ${\mathrm{PSL}_2 (\C)}$
----------------------------------------------

The group ${\mathrm{PSL}_2 (\C)}$ is algebraic, it is the quotient of ${\mathrm{SL}_2 (\C)}$ by the finite group $\{\pm {\mathrm{Id} }\}$.

It is useful to recall the isomorphism with ${\mathrm{SO}_{3} (\C)}$, that we construct next. We denote by $${\operatorname{Ad}}\co{\mathrm{PSL}_2 (\C)}\to {{\operatorname{Aut}}({\mathfrak{sl}_2 (\C)})}$$ the adjoint action of ${\mathrm{PSL}_2 (\C)}$ on its Lie algebra ${\mathfrak{sl}_2 (\C)}$. The Killing form on ${\mathfrak{sl}_2 (\C)}$ is a non degenerate symmetric bilinear form over $\C$. For each $A\in {\mathrm{PSL}_2 (\C)}$, ${\operatorname{Ad}}(A)$ preserves the Killing form and $\det ({\operatorname{Ad}}(A)) = 1$, hence ${\operatorname{Ad}}({\mathrm{PSL}_2 (\C)}) \subseteq
{\mathrm{SO}_{3} (\C)}$. The following lemma is well known from representation theory (see for instance [@FultonHarris]):

\[lem:SO3\] The action of ${\mathrm{PSL}_2 (\C)}$ on the Lie algebra induces an isomorphism ${\operatorname{Ad}}\co{\mathrm{PSL}_2 (\C)}\to {\mathrm{SO}_{3} (\C)} $.

In this paper the trace will be abbreviated by ${\operatorname{tr}}$, and ${\operatorname{tr}}^{2}(A)$ stands for $({\operatorname{tr}}(A))^{2}$. By direct computation we obtain the equality $$\label{eqn:traceAd}
      {\operatorname{tr}}({\operatorname{Ad}}(A))={\operatorname{tr}}^{2}(A)-1={\operatorname{tr}}(A^2)+1
      \quad\text{ for all } A\in {\mathrm{PSL}_2 (\C)}$$ that will be used later.

Given $\gamma\in\Gamma$, we have a well defined function $$\begin{array}{rcl}
      \tau_{\gamma}\co R(\Gamma)&\to&\C\\
                                       \rho&\mapsto &{\operatorname{tr}}^{2}(\rho(\gamma))
\end{array}$$ Since it is invariant by conjugation, it induces a function $$J_{\gamma}\co X(\Gamma) \to\C.$$

Proof of Theorem \[thm:PSL2characters\] {#ss:ProofTheorem}
---------------------------------------

Theorem \[thm:PSL2characters\] is a consequence of:

\[prop:generators\] The ring of invariant functions $\C[R(\Gamma)]^{{\mathrm{PSL}_2 (\C)}}$ is generated by the functions $\tau_{\gamma}$, with $\gamma\in\Gamma$.

There is a surjection $\psi\co F_{n}\to \Gamma$ where $F_n$ is a free group of rank $n\in\N$. We obtain an inclusion $\psi^{*}\co R(\Gamma) \subset R(F_{n})$. This inclusion induces a surjection $\psi_{*} \co\C[R(F_{n})]\to \C[R(\Gamma)]$. Now, ${\mathrm{PSL}_2 (\C)}$ is reductive and acts regularly by conjugation on the representation varieties. Hence we obtain a surjection $$\psi_{*}\co \C[R(F_{n})]^{{\mathrm{PSL}_2 (\C)}}\to \C[R(\Gamma)]^{{\mathrm{PSL}_2 (\C)}}$$ and it is sufficient to prove the proposition for $\Gamma =
F_{n}$ since $\psi_{*} (\tau_{\gamma}) = \tau_{\psi(\gamma)}$.

Using Lemma \[lem:SO3\] and (\[eqn:traceAd\]), we have to prove that $\C[R(F_n)]^{{\mathrm{SO}_{3} (\C)}}$ is generated by the trace functions on elements of $F_n$. Equivalently, we claim that $$\C[{\mathrm{SO}_{3} (\C)}\times\cdots\times{\mathrm{SO}_{3} (\C)}]^{{\mathrm{SO}_{3} (\C)}}$$ is generated by traces of products of matrices and their transposes.

Let ${\mathrm{M}_3(\C)}$ denote the algebra of $3\times 3$ matrices with complex coefficients. The group ${\mathrm{PSL}_2 (\C)}\cong {\mathrm{SO}_{3} (\C)}$ acts on the product ${\mathrm{M}_3(\C)}\times \cdots \times{\mathrm{M}_3(\C)}$ diagonally by conjugation. A theorem of Aslaksen, Tan and Zhu (see [@ATZ]) states that the algebra of invariant functions $$\C[{\mathrm{M}_3(\C)}\times \cdots \times {\mathrm{M}_3(\C)}]^{{\mathrm{SO}_{3} (\C)}}$$ is generated by the traces of products of matrices and their transposes. Thus the proof of the proposition reduces to show that we have a natural surjection $$\C[{\mathrm{M}_3(\C)}\times \cdots \times {\mathrm{M}_3(\C)}]^{{\mathrm{SO}_{3} (\C)}} \to
\C[{\mathrm{SO}_{3} (\C)}\times \cdots \times {\mathrm{SO}_{3} (\C)}]^{{\mathrm{SO}_{3} (\C)}}\,.$$ Since ${\mathrm{SO}_{3} (\C)}\times \cdots \times {\mathrm{SO}_{3} (\C)}\subset {\mathrm{M}_3(\C)}\times \cdots \times
{\mathrm{M}_3(\C)}$ is a closed subvariety we obtain a natural surjection $$\C[{\mathrm{M}_3(\C)}\times \cdots \times {\mathrm{M}_3(\C)}]\to \C[{\mathrm{SO}_{3} (\C)}\times \cdots\times {\mathrm{SO}_{3} (\C)}]$$ which is of course ${\mathrm{SO}_{3} (\C)}$-invariant. Using the fact that ${\mathrm{SO}_{3} (\C)}$ is reductive gives the surjection $\C[{\mathrm{M}_3(\C)}\times \cdots \times {\mathrm{M}_3(\C)}]^{{\mathrm{SO}_{3} (\C)}} \to
\C[{\mathrm{SO}_{3} (\C)}\times \cdots \times {\mathrm{SO}_{3} (\C)}]^{{\mathrm{SO}_{3} (\C)}}$.

Since $\C[X(\Gamma)]=\C[R(\Gamma)]^{{\mathrm{SO}_{3} (\C)}}$ is finitely generated, we also obtain:

\[cor:embedding\] There are finitely many elements $\gamma_{1},\ldots,\gamma_{N}$ in $\Gamma$ such that $J_{\gamma_1}\times\cdots\times J_{\gamma_N}\co X(M)\to\C^N$ is an embedding and its image is a closed algebraic set.

Other invariant functions {#ssec:others}
-------------------------

There are other natural functions to consider. Let $\Gamma^{2}$ be the subgroup of $\Gamma$ generated by the squares $\gamma^{2}$ of all elements $\gamma$ of $\Gamma$. It is well known that we have an exact sequence: $$1\to\Gamma^2\to\Gamma\to H_1(\Gamma,{C_2})\to 1,$$ where $C_2=\{\pm 1\}$ is the group with $2$ elements. For instance, if $\Gamma$ is a finite group of odd order, then $\Gamma^{2}=\Gamma$. In general, if $ \gamma,\mu \in\Gamma$ the commutator $[\gamma,\mu]=\gamma\mu\gamma^{-1}\mu^{-1}=
(\gamma\mu)^{2}(\mu^{-1}\gamma^{-1}\mu)^{2}\mu^{-2}$ is in $\Gamma^{2}$ and hence $\Gamma^{2}$ contains the commutator group $\Gamma' = [\Gamma,\Gamma]$. Notice that $$\Gamma^2=\bigcap_{\epsilon\in H^1(\Gamma,C_2)} {\operatorname{Ker}}(\epsilon)$$ where $H^1(\Gamma,C_2)={{\operatorname{Hom}}((,\Gamma)},C_{2})$. Let $R(\Gamma,{\mathrm{SL}_2 (\C)})$ denote the variety of representations of $\Gamma$ in ${\mathrm{SL}_2 (\C)}$. The cohomology group $H^1(\Gamma,{C_2})$ acts on this variety of representations as follows: an homomorphism $\epsilon\co\Gamma\to{C_2}=\{\pm 1\}$ maps the representation $\rho\in R(\Gamma,{\mathrm{SL}_2 (\C)})$ to the product of representations $\epsilon\cdot \rho$ (which maps $\gamma\in\Gamma$ to $
{\epsilon(\gamma)}\cdot \rho(\gamma)$).

### Invariant functions for the free group

Let $F$ be a finitely generated free group. For $\gamma\in F^{2}$ and $\rho\in R(F)$, ${\operatorname{tr}}(\rho(\gamma))$ is well defined since the representation $\rho\co F \to {\mathrm{PSL}_2 (\C)}$ lifts to $\tilde{\rho}\co F \to {\mathrm{SL}_2 (\C)}$ and for $\gamma\in F^{2}$ the trace ${\operatorname{tr}}(\tilde{\rho}(\gamma))$ depends only on $\gamma$. Note that two lifts $\tilde{\rho}_{1}$ and $\tilde{\rho}_{2}$ of $\rho$ differ by a homomorphism $\epsilon\in H^1(F,C_2) $ and that $F^{2}\subset {\operatorname{Ker}}(\epsilon)$ for each $\epsilon\in H^{1}(F,C_{2})$.

\[prop:sigmafunction\] Let $F$ be a free group. For every $k$-tuple $\gamma_1,\ldots,\gamma_k\in F$ such that the product $\gamma_1\cdots\gamma_k\in F^2$, the function $$\begin{array}{rcl}
     \sigma_{\gamma_1,\ldots,\gamma_k}\co R(F)&\to&\C \\
     \rho&\mapsto&{\operatorname{tr}}(\tilde\rho(\gamma_1))\cdots
     {\operatorname{tr}}(\tilde\rho(\gamma_k))
\end{array}$$ is regular (i.e.$\sigma_{\gamma_1,\ldots,\gamma_k}\in\C[R(F)]$). Here, $\tilde\rho\co F \to{\mathrm{SL}_2 (\C)}$ denotes a lift of $\rho$.

In order to prove this proposition we shall use the following:

\[lem:RFn\] Let $F_n$ be the free group of rank $n$. We have a natural isomorphism $$R(F_n,{\mathrm{SL}_2 (\C)})/\! /H^1(F_n,{C_2})\cong R(F_n)\,.$$

Since $R(F_n,{\mathrm{SL}_2 (\C)})\cong {\mathrm{SL}_2 (\C)}^n$, $R(F_n)\cong {\mathrm{PSL}_2 (\C)}^n$ and ${\mathrm{SL}_2 (\C)}/{C_2}\cong {\mathrm{PSL}_2 (\C)} $, we have the lemma.

For a free group $F$ and $\gamma_1,\ldots,\gamma_k\in F$, the function $\tilde \sigma \co R(F,{\mathrm{SL}_2 (\C)})\to \C$ given by $\tilde\sigma(\rho)={\operatorname{tr}}(\rho(\gamma_1))\cdots {\operatorname{tr}}(\rho(\gamma_k))$ is regular. Moreover, we have $\tilde\sigma(\epsilon\cdot\rho) =
\epsilon(\gamma_1\cdots\gamma_k) \tilde\sigma(\rho)$. Since the product $\gamma_1\cdots\gamma_k\in F^{2}$ we get that $\tilde\sigma\in \C[R(F_n,{\mathrm{SL}_2 (\C)})]^{H^1(F_n,{C_2})}$ is an invariant regular function on the ${\mathrm{SL}_2 (\C)}$ representation variety. By Lemma \[lem:RFn\], this function factors through $R(F)$ and gives the regular function $\sigma_{\gamma_1,\ldots,\gamma_k}\in\C[R(F)] $.

\[ex:sigmagammaeta\] Given $\gamma,\eta\in F$, by Proposition \[prop:sigmafunction\], $\sigma_{\gamma,\eta,\gamma\eta}\in \C[R(F)]$, thus by Proposition \[prop:generators\], $\sigma_{\gamma,\eta,\gamma\eta}$ is a polynomial on the functions $\tau$.

To compute explicitly the polynomial of Example \[ex:sigmagammaeta\], we recall some identities of traces in ${\mathrm{SL}_2 (\C)}$: $${\operatorname{tr}}(AB)={\operatorname{tr}}(BA)\quad\textrm{and}\quad {\operatorname{tr}}(A)={\operatorname{tr}}(A^{-1})\qquad
\forall A,B\in {\mathrm{SL}_2 (\C)}\,.$$ In addition, we have the fundamental identity: $$\label{eqn:sl2fundamental}
      {\operatorname{tr}}(AB)+{\operatorname{tr}}(A^{-1}B)={\operatorname{tr}}(A)\,{\operatorname{tr}}(B)
      \qquad \forall A,B\in
{\mathrm{SL}_2 (\C)} .$$ This identity can be deduced from $A^2-({\operatorname{tr}}A)A+{\mathrm{Id} }=0$ multiplying by $A^{-1}B$ and taking traces. Taking the square of ${\operatorname{tr}}(AB^{-1})={\operatorname{tr}}(A)\,{\operatorname{tr}}(B)-{\operatorname{tr}}(AB)$ we deduce: $$2 {\operatorname{tr}}(A){\operatorname{tr}}(B){\operatorname{tr}}(AB) = {\operatorname{tr}}^{2}(A) \, {\operatorname{tr}}^{2}(B)+{\operatorname{tr}}^{2}(AB) -
{\operatorname{tr}}^{2}(AB^{-1})\,.$$ Thus $$\label{eqn:sigmaABAB}
\sigma_{\gamma,\eta,\gamma\eta}=
\frac12(\tau_{\gamma}\tau_{\eta}+\tau_{\gamma\eta}-\tau_{\gamma\eta^{-1}}).$$

For every $\gamma,\mu\in F$, the commutator $[\gamma,\mu]=\gamma\mu\gamma^{-1}\mu^{-1}$ belongs to $F^2$ and therefore $\sigma_{[\gamma,\mu]}\in\C[R(F)]$. Using the the same method as for Equation (\[eqn:sigmaABAB\]) one can find: $$\label{ex:traces}
\sigma_{[\gamma,\eta]}= \tau_{\gamma} + \tau_{\eta} + \frac 1 2
\tau_{\gamma\eta} + \frac 1 2 \tau_{\gamma\eta^{-1}} - \frac 1 2
\tau_{\gamma}\tau_{\eta} -2 \,.$$

### Invariant functions for other groups

Let $\Gamma$ be a finitely generated group, $F$ a free group and $\psi\co F\to\Gamma$ a surjection. It induces another surjection $\psi_{*}\co \C[ R(F)]\to \C[R(\Gamma)]$, $\psi_{*}f
(\rho) = f (\rho\circ \psi)$. Hence we obtain for all $\eta_{1},\ldots,\eta_{k}\in F$ such that the product $\eta_{1}\cdots\eta_{k}\in F^{2}$ a regular function $\psi_{*}\sigma_{\eta_{1},\ldots,\eta_{k}}\in \C[R(\Gamma)]$. Note that the functions $\psi_{*}\sigma_{\eta_1}$ and $\psi_{*}\sigma_{\eta_2}$ might be different even if $\psi(\eta_1)=\psi(\eta_2)$ in $\Gamma$. This reflects the fact that in general not every representation $\rho\co\Gamma\to{\mathrm{PSL}_2 (\C)}$ lifts to ${\mathrm{SL}_2 (\C)}$.

Let $\psi\co F\to\Gamma$ be the canonical projection where $F=\langle x,y \mid - \rangle$ and $\Gamma = \langle x,y \mid [x,y] =1 \rangle$. We consider the representation $\rho\co\Gamma\to{\mathrm{PSL}_2 (\C)}$ given by $ \rho (x) = \pm A_{x}$ and $\rho (y) = \pm A_{y}$ where $$A_{x} =\begin{pmatrix} i & 0 \\ 0& -i \end{pmatrix} \text{ and }
        A_{y} = \begin{pmatrix} 0 & 1 \\ -1& 0 \end{pmatrix}\,.$$ We obtain ${\operatorname{tr}}([A_{x},A_{y}]) = -2$ and hence $\psi_{*}\sigma_{[x,y]}(\rho) = -2$. On the other hand we have $[x,y] =1$ in $\Gamma$ and $\psi_{*}\sigma_{1}= 2$ is a constant function.

If the representation $\rho\in R(\Gamma)$ admits a lift $\tilde\rho\co\Gamma\to{\mathrm{SL}_2 (\C)}$ then $$\label{equ:sigma}
\psi_{*}\sigma_{\eta_{1},\ldots,\eta_{k}}(\rho)=
{\operatorname{tr}}(\tilde\rho(\psi(\eta_{1})))\cdots
{\operatorname{tr}}(\tilde\rho(\psi(\eta_{k})))$$ only depends on the elements $\psi(\eta_1),\ldots,
\psi(\eta_k)\in\Gamma$.

Irreducibility {#sec:irreducibility}
==============

To study the fiber of the map $t\co R(\Gamma)\to X(\Gamma)$ we shall consider two different notions of irreducibility for $\rho\in
R(\Gamma)$, the usual one as a representation in ${\mathrm{PSL}_2 (\C)}$ and the so called ${\operatorname{Ad}}$-irreducibility, for the three dimensional representation ${\operatorname{Ad}}\circ\rho\co \Gamma\to SO_3(\C)$.

Irreducible representations
---------------------------

A representation $\rho\in R(\Gamma)$ is called *reducible* if $\rho(\Gamma)$ preserves a point of $P^1(\C)$. Otherwise it is called irreducible. A character $\chi\co\Gamma\to\C$ is called *reducible* if it is the character of a reducible representation.

\[rem:reducible\] Up to conjugation, the image of a reducible representation is contained in the set of upper-triangular matrices $\left(\begin{smallmatrix}
* & *\\ 0 & *\end{smallmatrix} \right)$.

We shall require the following well known lemma (see [@Beardon § 4.3]).

\[lem:fixedpoint\] Two non-trivial elements $g,h\in{\mathrm{PSL}_2 (\C)}$ have a common fixed point in $P^{1}(\C)$ if and only if ${\operatorname{tr}}([g,h])=2$. In addition, this fixed point is unique if $[g,h]$ is not the identity.

Irreducibility is a property that can be detected from characters:

\[lem:tracered\] A representation $\rho\in R(\Gamma)$ is reducible iff ${\operatorname{tr}}([\rho(\gamma),\rho(\eta)])=2$ for all elements $\gamma,\eta$ in $\Gamma$.

If $\rho$ is reducible then all the $\rho(\gamma)$ have a common fixed point and Lemma \[lem:fixedpoint\] gives the result.

Assume now that ${\operatorname{tr}}([\rho(\gamma),\rho(\eta)])=2$ for all elements $\gamma,\eta$ in $\Gamma$.

*Case 1:* There are two elements $\gamma$ and $\eta$ in $\Gamma$ such that $[\rho(\gamma),\rho(\eta)]$ is not the identity. Then $A=[\rho(\gamma),\rho(\eta)]$ is a non-trivial parabolic element in the image of $\Gamma$. For any $\mu\in\Gamma$, either $\rho(\mu)$ commutes with $A$ or $[\rho(\mu),A]$ is non-trivial. The former possibility implies that $\rho(\mu)$ fixes the unique fixed point of $A$, the latter too by Lemma \[lem:fixedpoint\].

*Case 2:* The image of $\rho$ is an abelian group. Abelian subgroups of ${\mathrm{PSL}_2 (\C)}$ are well-known: either they have a global fixed point in $P^{1}(\C)$ or they are conjugated to the group with four elements generated by $\pm\left(\begin{smallmatrix} 0 & 1\\ -1 &
0\end{smallmatrix} \right)$ and $\pm \left(\begin{smallmatrix} i &
0\\ 0 & -i\end{smallmatrix} \right)$. Since the commutator of these two generators is $\left(\begin{smallmatrix} -1 & 0\\ 0 &
-1\end{smallmatrix} \right)$, this possibility does not occur.

\[def:Klein\] A non-cyclic abelian subgroup of ${\mathrm{PSL}_2 (\C)}$ with four elements is called *Klein’s 4-group*. Such a group is realized by rotations about three orthogonal geodesics and it is conjugated to the one generated by $\pm\left(\begin{smallmatrix} 0 & 1\\ -1 & 0\end{smallmatrix} \right)$ and $\pm \left(\begin{smallmatrix} i & 0\\ 0 & -i\end{smallmatrix} \right)$.

Let $R^{red}(\Gamma)$ denote the set of reducible representations and $X^{red}(\Gamma)=t(R^{red}(\Gamma))$. Let $F$ be a free group and let $\psi\co F\to\Gamma$ be surjective. Lemma \[lem:tracered\] implies that $$R^{red}(\Gamma) = \{\rho\in R(\Gamma)\mid
\psi_{*}\sigma_{[\gamma,\eta]}(\rho)=2 \quad \forall \gamma,\eta\in F \}$$ is a Zariski closed subset invariant by conjugation. Thus, by invariant theory we have:

\[cor:red\] The set $X^{red}(\Gamma)$ is Zariski closed and $R^{red}(\Gamma)=t^{-1}(X^{red}(\Gamma))$.

\[rem:rhodiagonal\] Every reducible character $\chi$ is the character of a diagonal representation, because if $\rho(\gamma)=
        \pm\left(\!\begin{smallmatrix}a_{\gamma}&b_{\gamma}\\
                                      0&
c_{\gamma}\end{smallmatrix}\!\right)$ is a representation, then $\rho'(\gamma)=\pm\left(\!\begin{smallmatrix}a_{\gamma}&0\\ 0&
c_{\gamma}\end{smallmatrix}\!\right)$ is also a representation with $\chi_{\rho}=\chi_{\rho'}$.

${\operatorname{Ad}}$-irreducibility
------------------------------------

A representation $\rho\in R(\Gamma)$ is *${\operatorname{Ad}}$-reducible* if ${\mathfrak{sl}_2 (\C)}$ has a proper invariant subspace by the action of ${\operatorname{Ad}}\circ\rho$. Otherwise it is ${\operatorname{Ad}}$-irreducible.

Let $\H^3$ denote the three-dimensional hyperbolic space and $\partial_{\infty}\H^3$ its ideal boundary. We use the isomorphism $\operatorname{Isom}^+(\H^3)\cong {\mathrm{PSL}_2 (\C)}$ and the natural identification $\partial_{\infty}\H^3\cong  P^1(\C)$.

A representation $\rho\co\Gamma\to{\mathrm{PSL}_2 (\C)}$ is ${\operatorname{Ad}}$-reducible if and only if $\rho(\Gamma)$ preserves either a point in $\partial_{\infty}\H^3$ or a geodesic in $\H^{3}$.

Let $V$ be a proper subspace of ${\mathfrak{sl}_2 (\C)}$ invariant by ${\operatorname{Ad}}\circ\rho(\Gamma)$. Up to taking $V^{\bot}$, we may assume $\dim V=1$, because the Killing form is not degenerate. We have then two possibilities: either the Killing form restricted to $V$ vanishes or not. In the first case $V$ consists of parabolic Killing fields, in particular the 1-parameter group $\{\exp(v)\mid
v\in V\}\cong \C$ is parabolic and fixes a unique point at infinity, that has to be fixed also by $\rho$. In the second case, when the Killing form restricted to $V$ does not vanish, the 1-parameter group $\{\exp(v)\mid v\in V\}\cong \C^{*}$ is a subgroup of index two in the group of isometries which preserve a geodesic in $\H^3$. This geodesic has to be preserved by the representation. Conversely, if a representation preserves a point in $\partial_{\infty}\H^3$ or a geodesic, the previous argument shows how to construct an invariant subspace of ${\mathfrak{sl}_2 (\C)}$.

Reducible representations are also ${\operatorname{Ad}}$-reducible.

A representation ${\operatorname{Ad}}$-reducible but not reducible is a ${C_2}$-extension of an abelian one that fixes an oriented geodesic. Thus it preserves an unoriented geodesic

We call a representation $\rho\in R(\Gamma)$ *abelian* respectively *metabelian* if its image is an abelian respectively metabelian subgroup of ${\mathrm{PSL}_2 (\C)}$

\[lem:Ad-red\] A representation $\rho\in R(\Gamma)$ is ${\operatorname{Ad}}$-reducible iff it is metabelian.

If $\rho$ is ${\operatorname{Ad}}$-reducible then its image is contained in the stabilizer of either a point in $P^1(\C)$ or a geodesic in $\mathbb
   H^3$. Those stabilizers are metabelian, since they are respectively the group of affine transformations of $\C$ and the semidirect product $\C^{*}\rtimes C_{2}$.

Now assume that $\rho(\Gamma)\subset{\mathrm{PSL}_2 (\C)}$ is a metabelian subgroup. We use the fact that an abelian subgroup of ${\mathrm{PSL}_2 (\C)}$ preserves a unique point of $P^1(\C) $, a unique geodesic or it is Klein’s 4-group (Def. \[def:Klein\]). If $\rho([\Gamma,\Gamma])$ is trivial then $\rho$ is ${\operatorname{Ad}}$-reducible by this fact. If $\rho([\Gamma,\Gamma])$is not trivial, then we look at those *unique* invariant objects: the *unique* point in $P^1(\C) $, the *unique* geodesic, or the *unique* three geodesics if it is Klein’s 4-group. Since $[\Gamma,\Gamma]$ is normal in $\Gamma$, uniqueness implies that $\rho(\Gamma)$ preserves the same objects, hence $\rho$ is ${\operatorname{Ad}}$-reducible.

The set of characters of ${\operatorname{Ad}}$-reducible representations is Zariski closed.

Lemma \[lem:Ad-red\] gives that the set of ${\operatorname{Ad}}$-reducible representations is $$R^{{\operatorname{Ad}}-red}=\{ \rho\in R(\Gamma)\mid \rho(c) =
\pm{\mathrm{Id} }\quad \forall c \in \Gamma''\}$$ where $\Gamma''$ denotes the second commutator group of $\Gamma$. This is a closed subset of $R(\Gamma)$ invariant under conjugation. Hence we have $X^{{\operatorname{Ad}}-red}(\Gamma)=t(R^{{\operatorname{Ad}}-red})$ is a closed subset of $X(\Gamma)$.

The image of an ${\operatorname{Ad}}$-reducible representation is elementary, but elementary groups also include groups that fix a point in $\mathbb H^3$.

The fibers of $t\co R(\Gamma)\to X(\Gamma)$
-------------------------------------------

\[lem:fiberirred\] The fiber of an irreducible character consists of a single closed orbit (i.e. two irreducible representations have the same character iff they are conjugate).

Let $\rho_1,\rho_2\in R(\Gamma)$ be two irreducible representations with $\chi_{\rho_1}=\chi_{\rho_2}$.

We assume first that each $\rho_i$ is irreducible but ${\operatorname{Ad}}$-reducible. Thus each $\rho_i$ preserves a geodesic $l$, that we may assume to be the same after conjugation. The action of $\rho_i(\gamma)$ on $l$ is determined by the value of $\chi_{\rho_i}(\gamma)$, except in the case $\chi_{\rho_i}(\gamma)=0$, which means that $\rho_i(\gamma)$ is a rotation through angle $\pi$, but it can be either about $\gamma$ or about an axis perpendicular to $\gamma$. Thus if there exists an element $\gamma_0\in \Gamma$ with $\chi_{\rho_i}(\gamma_0)\neq 4,0$ (i.e. $\rho_i(\gamma_0)$ is either a loxodromic element or a rotation of angle $\neq\pi$) then $\forall\gamma\in\Gamma$ the action of $\rho_i(\gamma)$ on the geodesic $l$ is determined by $ \chi_{\rho_i}(\gamma)$ and $\chi_{\rho_i}(\gamma\gamma_0)$. In particular $\rho_i$ is unique up to conjugation. The exceptional case occurs when $\chi_{\rho_i}(\gamma)=0$ or $4$ for every $\gamma\in\Gamma$. In this special case, $\rho_i$ is necessarily a representation into Klein’s 4-group. The lemma is also clear in this case.

When $\rho_i$ are ${\operatorname{Ad}}$-irreducible, we can assume that $\Gamma$ is a free group. Thus we can lift $\rho_i$ to $
\tilde\rho_i\co\Gamma\to{\mathrm{SL}_2 (\C)} $. By Example \[ex:sigmagammaeta\], for every pair $\gamma,\gamma'\in \Gamma$ we obtain a regular function $\sigma_{\gamma,\gamma',\gamma\gamma'}\co X(\Gamma)\to\C$, given by $$\sigma_{\gamma,\gamma',\gamma\gamma'}(\chi_{\rho}) =
   {\operatorname{tr}}\tilde\rho(\gamma\gamma')\,{\operatorname{tr}}\tilde\rho(\gamma)\,
    {\operatorname{tr}}\tilde\rho(\gamma')$$ where $\tilde\rho\co\Gamma\to{\mathrm{SL}_2 (\C)}$ is any lift of $\rho$. Thus: $$\label{eqn:traceslifts}
{\operatorname{tr}}\tilde\rho_1(\gamma\gamma')\,
{\operatorname{tr}}\tilde\rho_1(\gamma)\,
{\operatorname{tr}}\tilde\rho_1(\gamma') =
{\operatorname{tr}}\tilde\rho_2(\gamma\gamma')\,
{\operatorname{tr}}\tilde\rho_2(\gamma)\,
{\operatorname{tr}}\tilde\rho_2(\gamma')\, .$$

We define $\epsilon\co\Gamma\to{C_2}=\{\pm 1\}$ by the formula: $$\phantom{aa}\qquad {\operatorname{tr}}\tilde\rho_1(\gamma)= {\epsilon(\gamma)}
{\operatorname{tr}}\tilde\rho_2(\gamma), \qquad \forall\gamma\in\Gamma\textrm{
such that } \chi_{\rho_1}(\gamma)\neq 0.$$ When $\chi_{\rho_1}(\gamma)=0$, since we assume that $\rho_i$ is ${\operatorname{Ad}}$-irreducible, we can find $\gamma_0\in \Gamma$ with $\chi_{\rho_i}(\gamma_0)\neq 0$ and $\chi_{\rho_i}(\gamma\gamma_0)\neq 0$. In this case we define $\epsilon (\gamma)=\epsilon(\gamma_0)\cdot \epsilon(\gamma
\gamma_0)$.

By (\[eqn:traceslifts\]), $\epsilon$ is a morphism. Hence $\tilde\rho_1$ and $ {\epsilon}\cdot \tilde\rho_2$ are irreducible representations in ${\mathrm{SL}_2 (\C)}$ with the same character. By [@CS] they are conjugate.

\[lem:stabilizers\]

-   A character $\chi$ is irreducible iff ${\mathrm{PSL}_2 (\C)}$ acts transitively on the fiber and with finite stabilizer.

-   A character is ${\operatorname{Ad}}$-irreducible iff ${\mathrm{PSL}_2 (\C)}$ acts faithfully on the fiber.

\(i) By Lemma \[lem:fiberirred\], if $\chi$ is irreducible then ${\mathrm{PSL}_2 (\C)}$ acts transitively on $t^{-1}(\chi)$. Assume now that the stabilizer is infinite: i.e. there exists nontrivial $A\in {\mathrm{PSL}_2 (\C)}$ of order $\geq 3$ (possibly infinite) and $\rho$ in the fiber such that $A$ commutes with $\rho$. If $A$ is parabolic, then it has a fixed point in $P^1(\C)$ and therefore $\rho$ fixes this point. Otherwise $A$ has an invariant geodesic; since $A$ has order $\geq 3$, $\rho$ preserves the oriented geodesic, and therefore $\rho$ is also reducible.

Assume the character is reducible, then it has a diagonal representation $\rho$ on the fiber (Rem. \[rem:rhodiagonal\]), and therefore the group of diagonal matrices stabilizes it. Thus the stabilizer is infinite.

\(ii) Assume ${\mathrm{PSL}_2 (\C)}$ does not act faithfully on the fiber, i.e. there exists nontrivial $A\in {\mathrm{PSL}_2 (\C)}$ and $\rho$ in the fiber such that $A$ commutes with $\rho$. If $A$ is parabolic, then $\rho$ fixes a point in $P^1(\C)$ by the previous argument. Otherwise $A$ has an invariant geodesic, and by commutativity, $\rho$ must preserve this geodesic. In both cases, $\rho$ is ${\operatorname{Ad}}$-reducible.

If the character is irreducible but ${\operatorname{Ad}}$-reducible, then it preserves a geodesic, and the rotation through angle $\pi$ about this geodesic commutes with $\rho$. Hence the stabilizer is nontrivial.

The projection $t\co R(\Gamma)\to X(\Gamma)$ induces a bijection between irreducible components.

A priori $R(\Gamma)$ could have more components than $X(\Gamma)$, but the number of components is the same, because ${\mathrm{PSL}_2 (\C)}$ is irreducible.

From Corollary \[cor:red\] and Proposition \[lem:stabilizers\] we deduce:

Let $\rho\in R(\Gamma)$ be an irreducible representation. Let $R_0$ denote an irreducible component of $R(\Gamma)$ that contains $\rho$ and let $X_0$ denote the corresponding irreducible component of $X(\Gamma)$. Then $$\dim R_0=\dim X_0+3.$$

Lifts of representations to ${\mathrm{SL}_2 (\C)}$ {#sec:lifts}
==================================================

Let $\overline R(\Gamma)\subset R(\Gamma)$ denote the set of representations $\rho\in R(\Gamma)$ that lift to ${\mathrm{SL}_2 (\C)}$. According to [@Culler Thm. 4.1] $\overline R(\Gamma)$ is a union of connected components of $R(\Gamma)$. In particular $\overline R
(\Gamma) $ is a Zariski-closed algebraic subset of $R(\Gamma)$, since irreducible complex varieties are connected in the $\C$-topology [@Sha VII, §2]. Moreover, $\overline R (\Gamma)$ is invariant under conjugation and hence the algebraic quotient $$\overline X(\Gamma)=\overline
R(\Gamma)/\! /{\mathrm{PSL}_2 (\C)}$$ is a well defined closed subset of $X(\Gamma)$.

In many cases, $\overline X(\Gamma)=X(\Gamma)$. For instance this is clear when $\Gamma$ is a free group. It is also true if $H^2(\Gamma,{C_2})=0$ by the following remark (see [@GM] or [@Culler]).

Let $\rho\co\Gamma\to{\mathrm{PSL}_2 (\C)}$ be a representation. There is a second Stiefel-Whitney class $w_{2}(\rho)\in H^{2}(\Gamma,{C_2})$ which is exactly the obstruction for the existence of a lift $\overline \rho\co\Gamma\to{\mathrm{SL}_2 (\C)}$.

Properties of $\overline X(\Gamma)$
-----------------------------------

Let $R(\Gamma,{\mathrm{SL}_2 (\C)})$ and $X(\Gamma,{\mathrm{SL}_2 (\C)})$ denote the variety of representations and characters in ${\mathrm{SL}_2 (\C)}$. The ring $\C[R(\Gamma,{\mathrm{SL}_2 (\C)})]^{{\mathrm{SL}_2 (\C)}}$ is generated by the trace functions $\tilde\tau_{\gamma}\co R(\Gamma,{\mathrm{SL}_2 (\C)})\to\C$, $\tilde\tau_{\gamma}(\rho)={\operatorname{tr}}(\rho(\gamma))$. The function induced by $\tilde\tau_{\gamma}$ is denoted by $I_{\gamma}\co
X(\Gamma)\to\C$, therefore $\C[X(\Gamma)]$ is finitely generated by the functions $I_{\gamma}$, $\gamma\in\Gamma$ [@CS].

Elements of the cohomology group $  H^1( \Gamma,{C_2})$ are homomorphisms $\theta\co\Gamma\to{C_2}=\{\pm 1\}$ that act on representations by multiplication. The action of $\epsilon \in H^1( \Gamma,C_2)$ on $I_{\gamma}$ is given by: $ \epsilon \cdot I_{\gamma} = \epsilon
(\gamma) I_{\gamma}$. Since $ H^1( \Gamma,{C_2})$ is finite, it is reductive and we may take the quotient of invariant theory.

Let $F$ be a finitely generated free group and $\psi\co F\to\Gamma$ be a surjection. We fix a $k$-tuple $\gamma_{1},\ldots,\gamma_{k}\in
\Gamma$ such that the product $\gamma_{1}\cdots\gamma_{k}\in\Gamma^{2}$. Moreover, we choose $\eta_{i}\in F$ such that $\psi(\eta_{i})=\gamma_{i}$ and such that the product $\eta_{1}\cdots\eta_{k}\in F^{2}$. The function $\psi^{*}\sigma_{\eta_1,\ldots,\eta_k}\in \C[\overline R(\Gamma)]$ is invariant under conjugation and gives us a function $\psi^{*}\sigma_{\eta_1,\ldots,\eta_k}\in\C[\overline X(\Gamma)]$. By Equation (\[equ:sigma\]) we have $\psi^{*}\sigma_{\eta_1,\ldots,\eta_k} (\chi) =
\tilde\chi(\gamma_{1})\cdots\tilde\chi(\gamma_{k})$ where $\tilde\chi\in X(\Gamma,{\mathrm{SL}_2 (\C)})$ is a character such that $\pi(\tilde\chi)= \chi$. Note that $\pi\co X(\Gamma,{\mathrm{SL}_2 (\C)})\to
\overline X(\Gamma)$ is surjective. The function $$\label{eqn:Sigma}
 \Sigma_{\gamma_1,\ldots,\gamma_k} :=
\phi^{*}\sigma_{\eta_1,\ldots,\eta_k} \in\C[\overline X(\Gamma)]$$ depends only on the elements $\gamma_{i}\in\Gamma$.

\[prop:naturaliso\] There is a natural isomorphism: $$X( \Gamma,{\mathrm{SL}_2 (\C)})/\! /H^1( \Gamma,C_2)\cong \overline X(\Gamma).$$

Composition with the projection ${\mathrm{SL}_2 (\C)}\to{\mathrm{PSL}_2 (\C)}$ induces a surjection $$\pi\co X(\Gamma,{\mathrm{SL}_2 (\C)}) \to \overline X(\Gamma),$$ which is easily seen to be algebraic and is given by $\pi(\chi) = \chi^{2}$. At the level of function rings it induces an injection $$\pi^*\co  \C[\overline X(\Gamma)]\hookrightarrow \C[X(\Gamma,{\mathrm{SL}_2 (\C)})].$$ We have $\pi^{*}f (\chi)= f(\chi^{2})$ for $f\in\C[\overline X(\Gamma)]$ and $\chi\in X(\Gamma,{\mathrm{SL}_2 (\C)})$. The image of $\pi^*$ is contained in the set of invariant functions: $${\operatorname{Im}}\pi^{*} \subseteq
     \C[X(\Gamma,{\mathrm{SL}_2 (\C)})]^{H^1(\Gamma,C_2)}.$$ More precisely, we have $\pi^{*}f (\epsilon\chi)=
     f(\epsilon^{2}\chi^{2}) = \pi^{*}f (\chi)$ for all $\epsilon\in H^{1}(\Gamma,C_{2})$. It remains to prove that this inclusion is an equality.

Since $ \C[X(\Gamma,{\mathrm{SL}_2 (\C)})]$ is generated as $\C$-algebra by the functions $I_{\gamma}$ with $\gamma\in\Gamma$, the monomials $$I_{\gamma_1} I_{\gamma_2}\cdots I_{\gamma_k}$$ generate $\C[X(\Gamma,{\mathrm{SL}_2 (\C)})]$ as a $\C$-vector space. Taking the average of the action of $H^1(\Gamma,C_2)$, we deduce that the subspace of invariant functions $\C[X(\Gamma,{\mathrm{SL}_2 (\C)})]^{H^1(\Gamma,C_2)}$ is generated by $$\frac 1 {2^r} \sum_{\epsilon\in H^1(\Gamma,C_2)}
            \epsilon \cdot I_{\gamma_{1}}\cdots
             I_{\gamma_{k}} =
             \big(\frac 1 {2^r}\sum_{\epsilon\in H^1(\Gamma,C_2)}
             \epsilon (\gamma_{1}\cdots\gamma_{k}) \big)
              I_{\gamma_{1}}\cdots
             I_{\gamma_{k}}$$ where $r$ is the rank of $H^{1}(\Gamma,C_2)$ (see [@Kraft II.3.6] for instance). Using the fact that $$\frac 1 {2^r}\sum_{\epsilon\in H^1(\Gamma,C_2)}
         \epsilon(\gamma) =
         \begin{cases}
             1 & \text{ if } \gamma\in\Gamma^{2}\\
             0         & \text{ otherwise }
         \end{cases}$$ we deduce that $\C[X(\Gamma,{\mathrm{SL}_2 (\C)})]^{H^1(\Gamma,C_2)}$ is generated by the monomials $
     I_{\gamma_1}\cdots I_{\gamma_k}
     $ such that the product $\gamma_1\ldots\gamma_k\in\Gamma^2$.

On the other hand we have for $\chi\in X(\Gamma,{\mathrm{SL}_2 (\C)})$: $$\pi^{*}\Sigma_{\gamma_{1},\ldots,\gamma_{k}} (\chi)
= \Sigma_{\gamma_{1},\ldots,\gamma_{k}}(\chi^{2}) =
\chi(\gamma_{1})\cdots\chi(\gamma_{k}) = I_{\gamma_{1}}\cdots
I_{\gamma_{k}} (\chi)\, ,$$ where $\Sigma_{\gamma_{1},\ldots,\gamma_{k}}$ is the function defined in (\[eqn:Sigma\]). This gives that the monomials $I_{\gamma_1}\cdots I_{\gamma_k}$ such that the product $\gamma_1\ldots\gamma_k\in\Gamma^2$ is in the image of $\pi^{*}$ and therefore $\C[X(\Gamma,{\mathrm{SL}_2 (\C)})]^{H^1(\Gamma,C_2)}={\operatorname{Im}}\pi^{*}$.

Let $p\co X(\Gamma,{\mathrm{SL}_2 (\C)})\to\overline X(\Gamma)$ denote the projection. If $\chi\in \overline X(\Gamma)$ is ${\operatorname{Ad}}$-irreducible, then $p^{-1}(\chi)$ has $2^{r}$ points where $r$ is the rank of $H^{1}(\Gamma,C_{2})$. If $\chi$ is ${\operatorname{Ad}}$-reducible then the cardinality of $p^{-1}(\chi)$ is strictly less than $2^{r}$. Thus $p$ is a branched covering with branching locus the set of ${\operatorname{Ad}}$-reducible characters.

\[ex:F2\] Let $F_2$ be the free group of rank 2, with generators $\alpha$ and $\beta$. There is an isomorphism: $$(I_{\alpha},I_{\beta},I_{\alpha\beta})\co X(F_2,{\mathrm{SL}_2 (\C)})\to\C^3$$ where $I_{\gamma}$ denotes the regular function induced by $\tilde\tau_{\gamma}$. In particular $X(F_2,{\mathrm{SL}_2 (\C)})$ is smooth.

Since every representation in $R(F_2)$ lifts to ${\mathrm{SL}_2 (\C)}$, we deduce $$X(F_2)=X(F_2,{\mathrm{SL}_2 (\C)})/\!/{H^{1}(F_2,C_{2})}.$$ The group $H^{1}(F_2,C_2)\cong ({C_2})^2$ has four elements, and its action on $X(F_2,{\mathrm{SL}_2 (\C)})$ is generated by the involutions $$\begin{array}{rcl}
(I_{\alpha},I_{\beta},I_{{\alpha}{\beta}})&\mapsto&(-I_{\alpha},I_{\beta},-I_{{\alpha}{\beta}})
\\
(I_{\alpha},I_{\beta},I_{{\alpha}{\beta}})&\mapsto&(I_{\alpha},-I_{\beta},-I_{{\alpha}{\beta}}).
\end{array}$$ Thus $\C[X(F_2),{\mathrm{SL}_2 (\C)}]^{H^{1}(F_2,C_{2})}$ is generated by $X=I_{\alpha}^2$, $Y=I_{\beta}^2$, $Z=I_{{\alpha}{\beta}}^2$ and $W=I_{\alpha} I_{\beta}
I_{{\alpha}{\beta}}$. Hence $$\label{eqn:XF2}
   X(F_2)\cong\{(X,Y,Z,W)\in\C^4\mid W^2=XYZ\}$$ The relationship with Corollary \[cor:embedding\] is given by the change of coordinates (cf. Equality (\[eqn:sigmaABAB\])) $$\left\{
\begin{array}{l}
      J_{\alpha}=X\\
      J_{\alpha}=Y\\
      J_{{\alpha}{\beta}}=Z\\
      J_{{\alpha}{\beta}^{-1}}=XY+Z-2W.
\end{array}
\right.$$

From Equality (\[eqn:XF2\]) we remark that the singular set of $X(F_2 )$ consists of those points such that two of $\{ X,Y,Z\}$ vanish. This is the same as the set of characters of representations generated by two rotations of angle $\pi$. This is also the set of ${\operatorname{Ad}}$-reducible but non-reducible representations.

If $M$ is a knot exterior in $S^3$, then $H_2(\pi_1M)\cong H_2(M)\cong
0$ and therefore $X(M)=\overline X(M)$. When in addition $M$ is a 2-bridge knot exterior, explicit methods of how to compute $X(M)$ are given in [@HLM1] and [@HLM2], where $X(M)$ for this particular case was already defined as $X(M,{\mathrm{SL}_2 (\C)})/\!/C_2$. The explicit computation for the figure eight knot exterior is found in [@GM], for instance.

Representations that do not lift
--------------------------------

The manifold $M$ is a bundle over $S^1$ with fiber $\dot{T}^2$ a torus minus a disk. Up to homeomorphism, $M$ is described by the action of the monodromy on $H_1(\dot{T}^2,\mathbb Z)$, which is given by the matrix $$\left(\begin{matrix} 1 & m_2 \\ m_1 & 1+m_1 m_2
\end{matrix}
\right)$$ with $m_i\in 2\mathbb Z$, $m_i>0$. We shall show that $X(M)-\overline X(M)$ has arbitrarily many components by choosing $m_i$ sufficiently large.

To have a presentation of $\pi_1M$, we use an automorphism $f$ of $\pi_1\dot T^2$ induced by the monodromy. Since $\pi_1\dot T^2$ is the free group of rank $2$ generated by $\alpha$ and $\beta$, $$\pi_1M=\langle \alpha,\beta,\mu\mid \mu\alpha\mu^{-1}=f(\alpha),
\mu\beta\mu^{-1}=f(\beta)\rangle$$ We choose $f$ such that: $$\left\{
\begin{array}{rcl}
      \mu\alpha\mu^{-1}&=&\alpha\beta^{m_2}\\
      \mu\beta\mu^{-1}&=&\beta(\alpha\beta^{m_2})^{m_1}
\end{array}
\right.$$ We choose odd numbers $p_1,p_2\in  2\mathbb Z+1$, with $1\leq
p_i\leq m_i/2$ and an arbitrary complex number $z\in \C$. By Example \[ex:F2\], there exist matrices $A_{z},B_{z}\in {\mathrm{SL}_2 (\C)}$ with $${\operatorname{tr}}(A_{z})=2\cos\frac{\pi p_1}{m_1},\
{\operatorname{tr}}(B_{z})=2\cos\frac{\pi p_2}{m_2} \text{ and }
{\operatorname{tr}}(A_{z} B_{z})=z\,.$$ Those trace equalities imply that $A_{z}^{m_1}=B_{z}^{m_2}=-{\mathrm{Id} }$. In particular $$\begin{array}{rcl}
      A_{z} B_{z}^{m_2} &=& -A_{z},\\
      B_{z} (A_{z} B_{z}^{m_2})^{m_1}&=&-B_{z}.
\end{array}$$ Let $\rho_{z}\in R(\Gamma)$ be the representation that $\rho_{z}(\alpha)=\pm A_{z} $, $\rho_z(\beta)=\pm B_{z}$ and $\rho_z(\mu)=\pm {\mathrm{Id} }$. Since $m_1$ and $m_2$ are even, this representation does not lift to ${\mathrm{SL}_2 (\C)}$. In addition, for each value of $p_1$ and $p_2$ we have defined a one parameter family of characters, with parameter $z={\operatorname{tr}}(A_{z}B_{z})\in\C$. By [@CCGLS Proposition 2.4] the dimension of each component of $X(M)$ is at most one, hence different values of $p_1$ and $p_2$ give different components.

The singular set of $X(F_n)$
============================

In this section we compute the singular set of $X(F_n)$, but before we need two preliminary subsections: in Subsection \[ss:ZariskiTS\] we recall some basic facts about the Zariski tangent space and Luna’s slice theorem, and in Subsection \[ss:cohomology\] we compute the cohomology of free groups with twisted coefficients.

The Zariski tangent space {#ss:ZariskiTS}
-------------------------

Given a representation $\rho\in R(\Gamma)$, we define the space of cocycles $$Z^1(\Gamma,{\operatorname{Ad}}\circ\rho) = \left\{
      \theta\co\Gamma\to {\mathfrak{sl}_2 (\C)}\ \left\vert
      \begin{array}{c}
      \theta(\gamma_1\gamma_2)=\theta(\gamma_1)+{\operatorname{Ad}}_{\rho(\gamma_1)}
(\theta(\gamma_2)),
      \\
      \forall \gamma_1,\gamma_2\in\Gamma
      \end{array}
      \right.
      \right\}\, .$$ Given a smooth path of representations $\rho_t$, with $t$ in a neighborhood of the origin, one can construct a cocycle as follows: $$\begin{array}{rcl}
          \Gamma &\to &{\mathfrak{sl}_2 (\C)}\\
          \gamma&\mapsto&
          \frac{d\phantom{t}}{dt}\rho_t(\gamma)\rho_0(\gamma)^
{-1}\vert_{t=0}
\end{array} \,.$$ This construction defines an isomorphism, due to Weil [@Weil]:

\[thm:Weil\] The previous construction defines an isomorphism $${T^{\mathrm{Zar}}_{\rho}(R(\Gamma))}\cong Z^1(\Gamma,{\operatorname{Ad}}\circ\rho).$$

Here ${T^{\mathrm{Zar}}_{\rho}(R(\Gamma))}$ denotes the Zariski tangent space in the scheme sense (i.e. the defining ideals are not necessary reduced).

We also consider the space of coboundaries $$B^1(\Gamma,{\operatorname{Ad}}\circ\rho) = \left\{
      \theta\co\Gamma\to{\ensuremath{R} }^2\ \left\vert
      \begin{array}{c}
      \textrm{ there exists } a\in {\mathfrak{sl}_2 (\C)}\textrm{ such that }\\
      \theta(\gamma)={\operatorname{Ad}}_{\rho(\gamma)} (a)- a,\,
         \forall \gamma\in\Gamma
      \end{array}
      \right.
      \right\}\,.$$

The isomorphism of Theorem \[thm:Weil\] identifies the subspace of the Zariski tangent space corresponding to the orbits by conjugation with $
B^1(\Gamma,{\operatorname{Ad}}\circ\rho)$. So it seems natural that in some cases ${T^{\mathrm{Zar}}_{\chi}(X(\Gamma))}$ is isomorphic to the cohomology group $$H^1(\Gamma,{\operatorname{Ad}}\circ\rho)=
Z^1(\Gamma,{\operatorname{Ad}}\circ\rho)/B^1(\Gamma,{\operatorname{Ad}}\circ\rho)$$ as we will show next.

The stabilizer of a representation $\rho\in R(\Gamma)$ is denoted by $$Stab_{\rho}=\{A\in {\mathrm{PSL}_2 (\C)}\mid A\rho A^{-1}=\rho\}\,.$$ In particular, for and ${\operatorname{Ad}}$-irreducible representation $Stab_{\rho}$ is trivial.

\[prop:TZ\] If $\rho$ is a smooth point of $R(\Gamma)$ with closed orbit, then $${T^{\mathrm{Zar}}_{\chi_{\rho}}(X(\Gamma))}\cong
{T^{\mathrm{Zar}}_{0}(H^1(\Gamma,{\operatorname{Ad}}\circ\rho)/\!/ Stab_{\rho})}\,.$$

We use the slice theorem of Luna: there exists an algebraic subvariety $S\subset R(\Gamma)$ that contains $\rho$ and that is $  Stab_{\rho}$-invariant, such that $$\label{eqn:complement}
Z^1(\Gamma,{\operatorname{Ad}}\circ\rho)=B^1(\Gamma,{\operatorname{Ad}}\circ\rho)\oplus
{T^{\mathrm{Zar}}_{\rho}(S)}$$ and the map induced by the projection $$S/\!/Stab_{\rho}\to X(\Gamma)$$ is an étale isomorphism (in particular their tangent spaces are isomorphic). Since we assume that $\rho$ is a smooth point, Luna’s theorem shows that $S/\!/Stab_{\rho}$ and ${T^{\mathrm{Zar}}_{\rho}(S)}/\!/Stab_{\rho}$ are étale equivalent (see [@KSS p. 97 ]). Since ${T^{\mathrm{Zar}}_{\rho}(S)}$ and $H^1(\Gamma,{\operatorname{Ad}}\circ\rho)$ are isomorphic as $Stab_{\rho}$-modules (by Equation (\[eqn:complement\])), the proposition follows.

Cohomology of Free groups {#ss:cohomology}
-------------------------

We start with irreducible characters:

\[lem:cohomologyirreducible\] Let $\chi_{\rho}\in X(F_n)$ be an irreducible character. Then $$\dim H^1(F_n,{\operatorname{Ad}}\circ\rho)=3n-3.$$

Notice first that $Z^1(F_{n},{\operatorname{Ad}}\circ\rho)\cong
{\mathfrak{sl}_2 (\C)}^n\cong \C^{3n}$. Irreducibility implies that $\dim
B^1(F_{n},{\operatorname{Ad}}\circ\rho)=3$, which is maximal (even if ${\operatorname{Ad}}$-reducible representations have invariant subspaces, irreducibility implies that the eigenvalues are different from $1$).

We are interested in computing $H^1(F_n,{\operatorname{Ad}}\circ\rho)$ as a $Stab_{\rho}$-module. If $ \rho$ is ${\operatorname{Ad}}$-irreducible, then $Stab_{\rho}$ is trivial, and therefore $H^1(F_n,{\operatorname{Ad}}\circ\rho)$ is the trivial module $\C^{3n-3}$. In the reducible and ${\operatorname{Ad}}$-reducible cases we need further computations.

. Let $\chi\in X(F_n)$ be a non trivial reducible character. There exists a representation $\rho\in R(F_n)$ with character $\chi$ such that $\rho$ consists of diagonal matrices, constructed in Remark \[rem:rhodiagonal\].

We decompose the Lie algebra ${\mathfrak{sl}_2 (\C)}=h_0\oplus h_-\oplus h_+$, where $h_0$, $h_+$ and $h_-$ are the one dimensional $\C$-vector spaces generated respectively by $ \left(\!\begin{smallmatrix} 1&0\\
0&-1
   \end{smallmatrix}\!\right)$, $ \left(\!\begin{smallmatrix} 0&1\\
   0&0
   \end{smallmatrix}\!\right)$ and $ \left(\!\begin{smallmatrix} 0&0\\
   1&0
   \end{smallmatrix}\!\right)$.

\[lem:splitdiagonal\] If $\rho$ is diagonal then ${\operatorname{Ad}}\circ\rho$ preserves the splitting ${\mathfrak{sl}_2 (\C)}=h_0\oplus h_-\oplus
h_+$. If in addition $\rho$ is non-trivial, then $Stab_{\rho}$ preserves the splitting ${\mathfrak{sl}_2 (\C)}=h_0\oplus (h_-\oplus h_+)$ (some elements may permute $h_+$ and $h_-$).

The first assertion is clear, because diagonal matrices preserve each factor $h_0$ and $h_{\pm}$.

When the image of $\rho$ has order $\geq 3$, the group $Stab_{\rho}$ is precisely the set of diagonal matrices. When the image has order precisely $2$, then $Stab_{\rho}$ is the group of diagonal and anti-diagonal ones $\left(\!\begin{smallmatrix} 0&*\\ *&0
\end{smallmatrix}\!
\right)$. Antidiagonal matrices preserve $h_0$ and permute $h_-$ with $h_+$, hence the second assertion is proved.

\[lem:cohmologydiagonal\] Let $\rho\in R(F_n)$ be a non-trivial diagonal representation, then $H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong h_0^n\oplus (h_+\oplus h_-)^{n-1} $ as $Stab_{\rho}$-modules.

By construction, $Z^1(F_n,{\operatorname{Ad}}\circ\rho) \cong {\mathfrak{sl}_2 (\C)}^n$. We have the splitting $$H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong H^1(F_n,h_0)\oplus
H^1(F_n,h_+)\oplus H^1(F_n,h_-).$$ A diagonal matrix $\pm \left(\!\begin{smallmatrix} a&0 \\ 0&a^{-1}
\end{smallmatrix}\!\right)$ acts trivially on $h_0$ and by multiplication by a factor $a^{\pm 2}$ on $h_{\pm}$. Therefore $B^1(F_n,h_0)\cong 0$ and $B^1(F_n,h_{\pm})\cong h_{\pm}$, and the lemma follows.

. Let $\rho\in R(\Gamma)$ be irreducible but ${\operatorname{Ad}}$-reducible. Up to conjugation the image of $\rho$ is contained in the group of diagonal and anti-diagonal matrices. There are two possibilities for the stabilizer $Stab_{\rho}$. If the image of $\rho$ has more than four elements, then $Stab_{\rho}$ has two elements: the identity and $\pm\left(\!\begin{smallmatrix} i&0\\ 0&-i
\end{smallmatrix}\!\right)$. Otherwise the image of $\rho$ is Klein’s 4-group (i.e. the group generated by $\pm\left(\!\begin{smallmatrix} i&0\\ 0&-i
\end{smallmatrix}\!\right)$ and $\pm\left(\!\begin{smallmatrix} 0&1\\
-1&0
\end{smallmatrix}\!\right)$ ). In this case $Stab_{\rho}$ equals the image of $\rho$. With the same argument as in Lemma \[lem:splitdiagonal\], one can prove:

Let $\rho$ be as above. Then both ${\operatorname{Ad}}\circ\rho$ and $Stab_{\rho}$ preserve the splitting ${\mathfrak{sl}_2 (\C)}=h_0\oplus
(h_+\oplus h_-)$.

\[lem:cohomologyantidiagonal\] Let $\rho\in R(F_n)$ be an irreducible but ${\operatorname{Ad}}$-reducible representation, then $H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong {\mathfrak{sl}_2 (\C)}^{n-1}$ as $Stab_{\rho}$-modules.

Again $Z^1(F_n,{\operatorname{Ad}}\circ\rho) \cong {\mathfrak{sl}_2 (\C)}^n$, and we have the decomposition $$H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong
H^1(F_n,h_0)\oplus H^1(F_n,h_+\oplus h_-).$$ The group $B^1(F_n,h_0)$ has dimension one, because the antidiagonal matrices act on $h_0$ by change of sign. In addition, $\dim(B^1(F_n,h_+\oplus h_-))=2$ is also maximal, because this is the case when we restrict it to diagonal representations (see the proof of Lemma \[lem:cohmologydiagonal\]).

Singular locus for free groups
------------------------------

We saw above that $X(F_2,{\mathrm{SL}_2 (\C)})\cong\C^3$ is smooth. We also showed that the singular points of $X(F_2 )$ are ${\operatorname{Ad}}$-reducible but irreducible characters.

For $n\geq 3$ the singular set of $X(F_n )$ is precisely the set of ${\operatorname{Ad}}$-reducible characters.

Since $R(F_n)\cong {\mathrm{PSL}_2 (\C)}^{n}$, $X(F_n)$ is irreducible and of dimension $3 n-3$. Thus $\chi\in X(F_n)$ is singular if and only if $$\dim T^{Zar}_{\chi} X(F_n)> 3 n -3.$$ This dimension is computed by means of Proposition \[prop:TZ\]: if the orbit of $\rho\in t^{-1}(\chi)$ is closed then $$\dim T^{Zar}_{\chi} X(F_n)=\dim
T_0^{Zar}(H^1(F_n,{\operatorname{Ad}}\circ\rho)/\!/Stab_{\rho}).$$ If $\rho\in R(F_n)$ is irreducible, by Lemma \[lem:cohomologyirreducible\] $\dim H^1(F_n,{\operatorname{Ad}}\circ\rho)=3n-3$. If in addition $\rho$ is ${\operatorname{Ad}}$-irreducible, then $Stab_{\rho}$ is trivial and therefore $\chi_{\rho}$ is smooth.

If $\rho$ is irreducible but ${\operatorname{Ad}}$-reducible, then $H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong {\mathfrak{sl}_2 (\C)}^{n-1} $ as $Stab_{\rho}$ modules, by Lemma \[lem:cohomologyantidiagonal\]. We may assume that the image of $\rho$ has more than 4 elements, because the adherence set of such characters is the whole set of irreducible but ${\operatorname{Ad}}$-reducible characters, and the singular set is closed. Hence $Stab_{\rho}$ is the group generated by the involution $\pm
\left(\!\begin{smallmatrix} i&0\\ 0&-i
\end{smallmatrix}\!
\right)$, that acts trivially on $h_0$ but as a change of sign on $h_+\oplus h_-$. Thus the action of $Stab_{\rho}$ on $H^1(F_n,{\operatorname{Ad}}\circ\rho)$ is equivalent to the involution on $\C^{3n-3}$ that fixes $(n-1)$ coordinates and changes the sign of the remaining $(2n-2)$ coordinates. The quotient of $\C^{3n-3}$ by this involution is not smooth, hence $\dim
T_0^{Zar}(H^1(F_n,{\operatorname{Ad}}\circ\rho)/\!/Stab_{\rho})>3n-3$.

When $\chi_{\rho}$ is reducible but non trivial, we may assume that $\rho$ is diagonal and its image has more that three elements (again the adherence set of those characters is the whole set of reducible ones). Thus $Stab_{\rho}$ is the group of diagonal matrices, and by Lemma \[lem:cohmologydiagonal\], $H^1(F_n,{\operatorname{Ad}}\circ\rho)\cong h_0^n\oplus (h_+\oplus h_-)^{n-1}$ as $Stab_{\rho}$-module. We have an isomorphism $Stab_{\rho}\cong
\C^*$ and $t\in \C^*$ acts on $h_{0}$ trivially and on $h_{\pm}$ by multiplication by $t^{\pm 1}$. An elementary computation shows that $(h_+\oplus
h_-)^{n-1}/\!/\C^*$ has dimension $2n-3$ and it is not smooth for $n>2$.

A similar argument yields that for $n\geq 3$ the singular part of $X(F_n,{\mathrm{SL}_2 (\C)})$ is precisely the set of reducible characters.

**Acknowledgment** The second author was partially supported by MCYT through grant BFM2000-0007.

<span style="font-variant:small-caps;">Laboratoire de Mathématiques Pures, Université Blaise Pascal, F-63177, Aubière Cedex, France</span>, heusener@math.univ-bpclermont.fr.

<span style="font-variant:small-caps;">Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain</span>, porti@mat.uab.es.

[\[CCGLS\]]{}

H.  Aslaksen, E.  Tan, C.  Zhu, Invariant theory of special orthogonal groups. *Pacific J. Math.* **168** (1995), 207–215.

A.F. Beardon. *The geometry of discrete groups*. GTM 91, Springer Verlag New York, Berlin Heidelberg, 1983.

M. Boileau, S. Maillot, J. Porti. *Three-dimensional orbifolds and their geometric structures*, Soc. Math. France, Paris, 2003.

S. Boyer, X. Zhang, On Culler-Shalen seminorms and Dehn filling. *Ann. of Math. (2)* **148** (1998), 737–801.

G. Burde. $SU(2)$-representation spaces for two-bridge knot groups. *Math. Ann.* **288** (1990), 103–119.

M. Culler. Lifting representations to covering groups. **59** (1986), 64–70.

M. Culler, P.B. Shalen. Varieties of group representations and splittings of 3-manifolds. **117** (1983), 109–146.

D. Cooper, M. Culler, H. Gillet, D.D. Long, P.B. Shalen. Plane curves associated to character varieties of $3$-manifolds. *Invent. Math.* **118** (1994), 47–84.

W. Fulton, J. Harris. *Representation Theory*. GTM 129, Springer Verlag New York, Berlin Heidelberg, 1991.

F. González-Acuña, J.M. Montesinos-Amilibia. On the character variety of group representations in ${\mathrm SL}(2,{\mathbf C})$ and ${\mathrm PSL}(2,{\mathbf C})$. *Math. Z.* **214** (1993), 627–652.

H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia. On the character variety of group representations of a $2$-bridge link $p/3$ into ${\rm PSL}(2, C)$. *Bol. Soc. Mat. Mexicana* **37** (1992), 241–262.

H.M. Hilden, M.T. Lozano, J.M. Montesinos-Amilibia. On the arithmetic $2$-bridge knots and link orbifolds and a new knot invariant. *J. Knot Theory Ramifications* **4** (1995), 81–114.

H.  Kraft. *Geometrische Methoden in der Invariantentheorie.* Vieweg Verlag, Braunschweig, 1985.

H.  Kraft, P.  Slodowy, T.A.  Springer. *Algebraische Transformationsgruppen und Invariantentheorie.* **13**. Birkhäuser Verlag, Basel, 1989.

V.L. Popov, È.B. Vinberg. *Invariant theory*. In [*Algebraic Geometry IV.*]{}

R. Riley, .

I.R. Shafarevich. . Springer Verlag, 1977.

A. Weil, .
