# Moduli of $\mathcal{G}$-bundles under nonconnected group schemes and
nondensity of essentially finite bundles

Ludvig Olsson, Stefan Reppen, Tuomas Tajakka  Department of Mathematics,
Stockholm University Graduate School of Mathematical Sciences, the University
of Tokyo<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of a projective good moduli space of principal
$\mathcal{G}$-bundles under nonconnected reductive group schemes $\mathcal{G}$
over a smooth projective curve $C$. We also prove that the moduli stack of
$\mathcal{G}$-bundles decomposes into finitely many substacks
$\operatorname{\mathrm{Bun}}_{\mathcal{P}}$ each admitting a torsor
$\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}}}\to\operatorname{\mathrm{Bun}}_{\mathcal{P}}$
under a finite group, for some connected reductive group schemes
$\mathcal{G}_{\mathcal{P}}$ over $C$. We use this for the second purpose of
the article: for any constant connected reductive group $G$, the subset of
essentially finite $G$-bundles in the moduli of degree 0 semistable
$G$-bundles over $C$ is not dense, unless $G$ is a torus or the genus of $C$
is smaller than 2. We do this by giving an upper bound on the dimension of the
closure of the subset of essentially finite $G$-bundles.

##  1\. Introduction

Let $C$ be a smooth projective curve over an algebraically closed field $k$
and let $\mathcal{G}$ be a reductive group scheme over $C$. In most studies on
$\mathcal{G}$-bundles over $C$ the group is usually assumed to be connected.
There are both group-theoretic and geometric complications that occur in the
nonconnected case. For example, Zariski-local triviality (see [DS95]) of
$\mathcal{G}$–bundles does not hold for nonconnected groups. Nonetheless,
nonconnected groups occur naturally both in their own right, such as the
orthogonal group corresponding to a vector bundle with a quadratic form, and
in the study of connected groups, for instance as normalizers of tori. Thus,
one of the purposes of this text is to study the moduli stack
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}$ of principal bundles under
nonconnected group schemes over $C$. In particular, we show that
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}$ decomposes into finitely many
substacks $\operatorname{\mathrm{Bun}}_{\mathcal{P}}$, each of which admits a
finite torsor
$\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}}}\to\operatorname{\mathrm{Bun}}_{\mathcal{P}}$,
where $\mathcal{G}_{\mathcal{P}}$ is a connected reductive group over $C$.
When $\operatorname{char}(k)=0$, we use this to prove that the substack of
semistable $\mathcal{G}$-bundles admits a projective good moduli space (see
Section 1.1 for some background on this topic).

We also use this decomposition for the second purpose of the article, which is
to settle a density question (as discussed in [DM10],[GR23],[Wei22]) regarding
essentially finite bundles under connected reductive groups over $k$, in the
characteristic 0 case (see Theorem 1.5).

###  1.1. Moduli of $\mathcal{G}$-bundles

The construction of moduli of $G$-bundles over $C$ has its roots in the work
of Ramanathan who constructed a projective good moduli space of semistable
$G$-bundles for any connected reductive group $G$ over $\mathbb{C}$ (see
[Ram96, Ram96a]). After decades of work addressing the existence of such
moduli spaces (see e.g. [Fal93], [BP03], [BS02]) the theory culminated in the
seminal work [GLS+10] by Gómez-Langer-Sols-Schmitt, where moduli of
$G$-bundles were constructed in arbitrary characteristic, for arbitrary
reductive groups over $k$, over an arbitrary smooth projective base
scheme.111We remark that to obtain projectivity in some of the previous works,
some version of the semistable reduction theorem was needed, and this was
proved by Heinloth in [Hei08] for large classes of reductive groups.

In recent years, there has been a growing interest in constructions of moduli
of $\mathcal{G}$-bundles for more general groups than reductive group schemes
over $k$, in particular for so-called Bruhat-Tits group schemes over $C$.
These are smooth affine group schemes $\mathcal{G}$ over $C$ with
geometrically connected fibers such that there is an open subset $U\subset C$
such that $\mathcal{G}|_{U}$ is connected reductive and for all $x\in
C\setminus U$, the group scheme
$\mathcal{G}|_{\operatorname{Spec}\mathcal{O}_{C,x}}$ is a connected parahoric
Bruhat-Tits group. In particular, $\mathcal{G}$ could be a connected reductive
group scheme over $C$. When $\operatorname{char}(k)=0$, Balaji and Seshadri
proved in [BS15] the existence of a projective good moduli space of
$\mathcal{G}$-bundles in the case where $\mathcal{G}|_{U}$ is semisimple and
simply connected, and for general Bruhat-Tits group schemes Alper–Halpern-
Leistner–Heinloth proved in [AHH18] the existence of a proper good moduli
space for arbitrary Bruhat-Tits group schemes.

In this paper we study moduli of principal $\mathcal{G}$-bundles in the case
where the group scheme $\mathcal{G}$ is reductive over all of $C$, but not
necessarily connected. We declare a $\mathcal{G}$-bundle $\mathcal{E}$ to be
semistable if the adjoint bundle $\operatorname{Ad}_{*}\mathcal{E}$ is
semistable (see Section 3.2), and we prove the following.

######  Theorem 1.1.

Suppose that $\operatorname{char}(k)=0$, and let $\mathcal{G}$ be a (not
necessarily connected) reductive group scheme over $C$. The stack
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}^{\mathrm{ss}}$ of semistable
$\mathcal{G}$-bundles admits a projective good moduli space.

We prove this result by reducing to the case of connected groups via the
following key statement.

######  Theorem 1.2.

Suppose that $\mathcal{G}$ is a (not necessarily connected) reductive group
over $C$. If $\operatorname{char}(k)>0$ assume furthermore that that
$\mathcal{G}/\mathcal{G}^{0}$ is finite and that $\operatorname{char}(k)$ does
not divide $|\mathcal{G}/\mathcal{G}^{\circ}|$. Then there exists a finite
decomposition
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}\cong\coprod_{\mathcal{P}}\operatorname{\mathrm{Bun}}_{\mathcal{P}}$
by open and closed substacks such that each
$\operatorname{\mathrm{Bun}}_{\mathcal{P}}$ admits a torsor
$\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}}}\to\operatorname{\mathrm{Bun}}_{\mathcal{P}}$
under a finite group, where $\mathcal{G}_{\mathcal{P}}$ is a connected
reductive group scheme over $C$.

We believe that this decomposition could be useful in extending various
results about $\operatorname{\mathrm{Bun}}_{\mathcal{G}}$ from the case of
connected groups to the case of disconnected groups, as illustrated in its use
to prove Theorem 1.1. It also seems useful in the study of moduli of principal
bundles under connected groups. This is manifested in its crucial use in our
proof of nondensity of essentially finite $G$-bundles for connected reductive
groups $G$ over $k$ (see below).

To obtain projectivity of the moduli space we use the following auxiliary
result which can be deduced from [Beh90, Theorem 4.4.3] and might be worth
stating explicitly since it has occurred in the literature in the case of
moduli of semistable vector bundles (see [Wei22, Theorem 4.2] cf. [Hei06,
Theorem 4.2], [Fal93, Theorem I.4]).

######  Proposition 1.3.

Let $\mathcal{G}$ be an affine smooth group scheme of finite type over $C$ and
suppose that there is a notion of semistability on
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}$ for which the substack of
semistable bundles admits a proper good moduli space
$M_{\mathcal{G},C}^{\mathrm{ss}}$. For any finite cover $f\colon D\to C$ such
that a $\mathcal{G}$–bundle $\mathcal{E}$ is semistable if and only if
$f^{*}\mathcal{E}$ is semistable, $f$ induces via pullback a finite morphism
$f^{*}\colon M_{\mathcal{G},C}^{\mathrm{ss}}\to
M_{f^{*}\mathcal{G},D}^{\mathrm{ss}}$.

###  1.2. Essentially finite bundles

Building on work by Weil in the 1930’s, Nori introduced in [Nor76] the notion
of an essentially finite (EF) vector bundle.222He studied these bundles over
more general bases and over fields which are not necessarily algebraically
closed. A vector bundle is EF if it admits a reduction to a finite group.
Essentially finite vector bundles form a Tannakian category and the
corresponding fundamental group $\pi_{1}^{N}(C)$ classifies covers of $C$ by
finite group schemes (omitting base points from the notation). For example, if
$\operatorname{char}(k)=0$ then
$\pi_{1}^{N}(C)(k)=\pi_{1}^{\mathrm{\acute{e}t}}(C)$. EF bundles have since
then been studied extensively from various perspectives (e.g. [EHS08],
[BdS11], [BV15]). If $G$ is any algebraic group one may extend the definition
to $G$-bundles by saying that a $G$-bundle is EF if it admits a reduction to a
finite group. A heuristic question one may ask is the following.

######  Question 1.4.

How many essentially finite bundles are there?

Nori proved in [Nor82] that EF vector bundles are semistable of degree 0 and
in [GR23] the second author and Ghiasabadi proved more generally that EF
$G$-bundles are semistable of torsion degree, for any connected reductive
group $G$ over $k$. Hence, for any such group $G$, in the moduli space
$M_{G}^{\mathrm{ss},0}$ of semistable $G$-bundles of degree $0$, we have a
subset $M_{G}^{\operatorname{\text{ef}}}\subset M_{G}^{\mathrm{ss},0}$
consisting of EF $G$-bundles of degree 0. A first approximation of 1.4 is to
ask about the size of $M_{G}^{\operatorname{\text{ef}}}$. For instance, if
$G=\mathbb{G}_{m}$, then EF bundles correspond to torsion points on the
Jacobian. Since torsion points are dense in any abelian variety, we see that
$M_{\mathbb{G}_{m}}^{\operatorname{\text{ef}}}\subset
M_{\mathbb{G}_{m}}^{\mathrm{ss},0}$ is dense. In [DM10] Ducrohet and Mehta
showed that if $\operatorname{char}(k)=p>0$, then
$M_{\mathrm{GL}(n)}^{\operatorname{\text{ef}}}$ is dense for all $n\geq 1$.
However, Weissmann proved in [Wei22] that those EF bundles which are
trivialized by a prime to $p$ cover are not dense by giving an explicit upper
bound on the dimension of their Zariski closure in
$M_{\mathrm{GL}(n)}^{\mathrm{ss},0}$. In a similar direction, it is proved in
[GR23] that if $\operatorname{char}(k)=0$, then
$M_{G}^{\operatorname{\text{ef}}}$ is not dense for any connected reductive
group $G$ of semisimple rank 1. In this article we settle this question of
density completely in the characteristic 0 case.

######  Theorem 1.5.

Let $\operatorname{char}(k)=0$, let $g$ denote the genus of $C$ and assume
that $g\geq 2$. For any connected, reductive group $G$ we have that

(1.1) $\dim\overline{M_{G}^{\textnormal{ef}}}\leq g\operatorname{rk}(G).$

In particular, if $G$ is not a torus, then $M_{G}^{\textnormal{ef}}$ is not
dense in $M_{G}^{\mathrm{ss},0}$.

######  Remark 1.6.

If $g=0$, then $M_{G}^{\mathrm{ss}}(k)=\\{\ast\\}$, and if $g=1$, then [GR23,
Proposition 4.9] shows that $M_{G}^{\operatorname{\text{ef}}}$ is dense for
all connected reductive groups $G$. The same holds for all $g$ if
$G\cong\mathbb{G}_{m}^{n}$ is a torus, since then
$M_{G}^{\mathrm{ss}}\cong\prod_{i=1}^{n}M_{\mathbb{G}_{m}}^{\mathrm{ss}}$.

A key input in proving Theorem 1.5 is the following purely group theoretic
result inspired by Jordan’s classical result in [Jor77] on finite subgroups of
reductive groups, which could be of independent interest.

######  Theorem 1.7.

Let $\operatorname{char}(k)=0$, and let $G$ be a connected, reductive group
over $k$. There exist subgroups $H_{1},\ldots,H_{n}\subset G$ such that the
connected component of identity of each $H_{i}$ is a torus and such that all
finite subgroups of $G$, up to conjugation, are contained in some $H_{i}$.

###  1.3. Outline

In Section 2 we introduce notation and assumptions used throughout the text.
Section 3 concerns the moduli stack of $\mathcal{G}$-bundles: in Section 3.1
we prove that $\operatorname{\mathrm{Bun}}_{\mathcal{G}}$ admits a
decomposition as stated in Theorem 1.2 and after some discussion regarding
semistability in Section 3.2 we prove in Section 3.3 that the semistable locus
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}^{\mathrm{ss}}$ admits a projective
good moduli space. Section 4 is divided into two sections: in Section 4.1 we
study finite subgroups of reductive groups over $k$ and we prove Theorem 1.7.
In Section 4.2 we use the results from the previous sections to prove our main
theorem on essentially finite bundles. Finally, Appendix A is devoted to a
proof of a generalization of Jordan’s theorem on finite subgroups of
$\mathrm{GL}(n)$ to arbitrary reductive groups.

###  1.4. Acknowledgements

We would like to thank Magnus Carlson, Carlo Gasbarri, Siddharth Mathur, David
Rydh, and Dario Weissmann for enlightening conversations. Part of this work
was carried out while the second author was a JSPS International Research
Fellow.

##  2\. Notation

Let $k$ be an algebraically closed field and let $C$ be a smooth projective
connected curve over $k$. From Lemma 3.7 onward we assume that
$\operatorname{char}(k)=0$.

For an affine group $G$ over a field $K$, let $G^{\circ}$ denote the connected
component of identity. A reductive group over $K$ is a smooth affine group $G$
such that $G_{\overline{K}}^{\circ}$ has trivial unipotent radical. A
reductive (resp. connected reductive) group over $C$ is a smooth, affine group
scheme $\mathcal{G}$ over $C$ whose geometric fibers $\mathcal{G}_{x}$ are
reductive (resp. connected reductive) groups for all geometric points $x$ in
$C$. By a constant group scheme over $C$ we mean a group scheme isomorphic to
$G\times_{k}C$ for some $k$-group $G$. For a finite type group scheme
$\mathcal{G}$ over $C$ we denote by $\operatorname{Lie}(\mathcal{G})$ the Lie
algebra of $\mathcal{G}$ (a vector bundle over $C$) and by
$\operatorname{Ad}\colon\mathcal{G}\to\mathrm{GL}(\operatorname{Lie}(\mathcal{G}))$
the adjoint representation. If $G$ is an affine $k$-group we also use the
notation $\mathfrak{g}$ for its Lie algebra.

By [Con14, Proposition 3.1.3], for any reductive group scheme $\mathcal{G}$
over $C$, there is an open and closed subgroup scheme $\mathcal{G}^{\circ}$
which is a connected reductive group over $C$, and
$\mathcal{G}/\mathcal{G}^{\circ}$ is a separated, étale, quasi-finite
$C$-group.

A $\mathcal{G}$-bundle over $C$ is an fppf scheme $\mathcal{E}\to C$ with a
$\mathcal{G}$-action such that
$\mathcal{E}\times_{C}\mathcal{G}\to\mathcal{E}\times_{C}\mathcal{E}$,
$(p,g)\mapsto(p,pg)$ is an isomorphism. Since all groups we consider are
smooth, we note that every $\mathcal{G}$-bundle is locally trivial in the
étale topology. As is common in the literature we also refer to a
$\mathcal{G}$-bundle as a principal $\mathcal{G}$-bundle or a
$\mathcal{G}$-torsor. Further, since $\mathcal{G}$ is affine over $C$ we may
view a bundle $\mathcal{E}$ as a scheme over $C$ or as a sheaf. Conveniently
but also quite uncommonly, we denote by $\mathbbm{1}_{\mathcal{G}}$ the
trivial $\mathcal{G}$-torsor.

We denote by $\operatorname{\mathrm{Bun}}_{\mathcal{G},C}$ the stack on the
category of $k$-schemes whose fiber over $T$ is the groupoid of
$T\times_{k}\mathcal{G}$-torsors on $T\times_{k}C$. If the meaning is clear
from context, we usually write $\operatorname{\mathrm{Bun}}_{\mathcal{G}}$ in
place of $\operatorname{\mathrm{Bun}}_{\mathcal{G},C}$. For a
$\mathcal{G}$–bundle $\mathcal{E}$ over $C$, we write
$\underline{\operatorname{Aut}}(\mathcal{E})$ for the $C$-group scheme of
automorphisms of $\mathcal{E}$ and we write $\operatorname{Aut}(\mathcal{E})$
for the $k$-group which is the stabilizer of
$\mathcal{E}\in\operatorname{\mathrm{Bun}}_{\mathcal{G}}(k)$.

If $\mathcal{X}$ is a stack on the category of $C$-schemes, we denote by
$\operatorname{Res}_{C/k}\mathcal{X}$ the restriction of scalars of
$\mathcal{X}$ to $k$, defined as the stack whose fiber over a $k$-scheme $T$
is $\mathcal{X}(\mathrm{pr}_{2}:T\times_{k}C\to C)$. If the left diagram below
is a commutative (resp. cartesian) diagram of stacks over $C$, then the right
diagram is a commutative (resp. cartesian) diagram of stacks over $k$.

${\mathcal{X}^{\prime}}$${\mathcal{X}}$${\mathcal{Y}^{\prime}}$${\mathcal{Y}}$${\operatorname{Res}_{C/k}\mathcal{X}^{\prime}}$${\operatorname{Res}_{C/k}\mathcal{X}}$${\operatorname{Res}_{C/k}\mathcal{Y}^{\prime}}$${\operatorname{Res}_{C/k}\mathcal{Y}}$

We further denote by $\mathrm{B}\mathcal{G}$ the classifying stack of
$\mathcal{G}$. It follows from the definitions that
$\operatorname{\mathrm{Bun}}_{\mathcal{G},C}\cong\operatorname{Res}_{C/k}(\mathrm{B}\mathcal{G})$.
The stack $\operatorname{\mathrm{Bun}}_{\mathcal{G}}$ is smooth and locally of
finite type over $k$ (see e.g. [Hei10, Proposition 1.]).

If $\alpha\colon\mathcal{H}\to\mathcal{H}^{\prime}$ is a morphism of affine
algebraic groups, then for any $\mathcal{H}$-bundle $\mathcal{E}$ on $C$ we
denote by $\alpha_{*}\mathcal{E}$ the $\mathcal{H}^{\prime}$-bundle
$\mathcal{E}\times^{\mathcal{H}}\mathcal{H}^{\prime}$. We remind the reader
that for any morphism of $k$-schemes $f\colon D\to C$ we have a natural
isomorphism

$f^{*}\alpha_{*}\mathcal{E}\cong\alpha_{*}f^{*}\mathcal{E}$

of $\mathcal{H}^{\prime}$-bundles on $D$.

Finally, given a stack $\mathcal{X}$, e.g.
$\mathcal{X}=\operatorname{\mathrm{Bun}}_{\mathcal{G}}$, and a notion of
semistability on $\mathcal{X}$, we denote by $\mathcal{X}^{\mathrm{ss}}$ the
semistable locus.

##  3\. Moduli of principal bundles under nonconnected group schemes

###  3.1. Decomposition of $\operatorname{\mathrm{Bun}}_{\mathcal{G}}$

Fix a reductive group scheme $\mathcal{G}$ over $C$. In this section we
decompose $\operatorname{\mathrm{Bun}}_{\mathcal{G}}$ into finitely many open
and closed substacks $\operatorname{\mathrm{Bun}}_{\mathcal{P}}$, each
admitting a finite torsor
$\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}}}\to\operatorname{\mathrm{Bun}}_{\mathcal{P}}$
for some connected reductive group scheme $\mathcal{G}_{\mathcal{P}}$ over
$C$.

######  Proposition 3.1.

Let $\Gamma$ be a quasi-finite étale group scheme over $C$. If
$\operatorname{char}(k)>0$ assume furthermore that $\Gamma$ is finite and that
$\operatorname{char}(k)$ and $|\Gamma|$ are coprime. There is a finite
decomposition

$\operatorname{\mathrm{Bun}}_{\Gamma,C}\cong\coprod_{[\mathcal{P}]\in\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(C,\Gamma)}\mathrm{B}\operatorname{Aut}(\mathcal{P}).$

###### Proof.

Suppose first that $\Gamma$ is finite over $C$. Upon choosing a geometric base
point $c$ in $C$, we have that
$\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(C,\Gamma)=\mathrm{H}^{1}(\pi_{1}^{\mathrm{\acute{e}t}}(C,c),\Gamma_{c})$,
where $\Gamma_{c}$ is the fiber of $\Gamma$ at $c$ equipped with its
$\pi_{1}^{\mathrm{\acute{e}t}}(C,c)$-action. Since $\Gamma_{c}$ is finite and
$\pi_{1}^{\mathrm{\acute{e}t}}(C,c)$ is (pro)finitely generated, we see that
$\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(C,\Gamma)$ is finite.

If $\operatorname{char}(k)=0$ and $\Gamma$ is quasi-finite, then let $U\subset
C$ be the largest open dense subset such that $\Gamma|_{U}$ is finite, and let
$u$ be a geometric point in $U$. Then the functor

$P\mapsto(P_{u},\\{P_{x}\\}_{x\in C\setminus U})$

determines an equivalence of categories between the category of quasi-finite
étale schemes $P\to C$ such that $P|_{U}$ is finite, and the category
consisting of tuples $(\Sigma,\\{F_{x}\\}_{x\in C\setminus U})$ where $\Sigma$
is a finite $\pi_{1}^{\mathrm{\acute{e}t}}(U,u)$-set and
$F_{x}\subset\Sigma^{\operatorname{Gal}(k(C)^{\text{sep}}/K_{x}^{\text{h}})}$,
where $K_{x}^{\text{h}}$ denotes the fraction field of the henselization of
$\mathcal{O}_{X,x}$. This functor respects products and takes group objects to
group objects. The essential image of all $\Gamma$-bundles $P\to C$ thus
consists of tuples $(\Sigma,\\{F_{x}\\}_{x\in C\setminus U})$ where $\Sigma$
is a $(\pi_{1}^{\mathrm{\acute{e}t}}(U,u),\Gamma_{u})$-set and the $F_{x}$ are
$\Gamma_{x}$-sets, where the action of $\Gamma_{u}$ on $\Sigma$ (resp.
$\Gamma_{x}$ on $F_{x}$) is free and transitive. To give such a
$(\pi_{1}^{\mathrm{\acute{e}t}}(U,u),\Gamma_{u})$-set is equivalent to giving
a cocycle in $\mathrm{H}^{1}(\pi_{1}^{\mathrm{\acute{e}t}}(U,u),\Gamma_{u})$.
Since $\operatorname{char}(k)=0$, $\pi_{1}^{\mathrm{\acute{e}t}}(U,u)$ is
(pro)finitely generated and thus
$\mathrm{H}^{1}(\pi_{1}^{\mathrm{\acute{e}t}}(U,u),\Gamma_{u})$ is finite.

Hence, in either case there are only finitely many isomorphism classes of
$\Gamma$-bundles on $C$. Since a $k$-valued point of
$\operatorname{\mathrm{Bun}}_{\Gamma,C}$ is a $\Gamma$-bundle on $C$, we have
a surjection

$Y\coloneqq\coprod_{[\mathcal{P}]\in\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(C,\Gamma)}\operatorname{Spec}{k}\to\operatorname{\mathrm{Bun}}_{\Gamma,C}.$

In particular, $\operatorname{\mathrm{Bun}}_{\Gamma,C}$ is quasi-compact.
Moreover, since all stabilizer groups are finite,
$\operatorname{\mathrm{Bun}}_{\Gamma,C}$ is a smooth Deligne-Mumford stack
(see e.g. [Alp23, Theorem 3.6.1]). Thus, there exists an étale surjection
$U\to\operatorname{\mathrm{Bun}}_{\Gamma,C}$ from a smooth, quasi-compact
$k$-scheme $U$. Consider the cartesian square

${V}$${U}$${Y}$${\operatorname{\mathrm{Bun}}_{\Gamma,C}.}$

Since $V\to Y$ is étale, the scheme $V$ is a disjoint union of copies of
$\operatorname{Spec}{k}$, and since $V\to U$ is surjective, we see that $U$ is
a finite disjoint union of copies of $\operatorname{Spec}{k}$ as well. Thus,
$V\to U$ is étale, hence so is $Y\to\operatorname{\mathrm{Bun}}_{\Gamma,C}$.
The result follows. ∎

From now on, let $\Gamma\coloneqq\mathcal{G}/\mathcal{G}^{\circ}$, and let

$1\to\mathcal{G}^{\circ}\xrightarrow{\iota}\mathcal{G}\xrightarrow{\pi}\Gamma\to
1$

be the corresponding short exact sequence of affine algebraic group schemes
over $C$. The homomorphisms $\iota$ and $\pi$ induce morphisms of stacks
$\iota_{*}\colon\operatorname{\mathrm{Bun}}_{\mathcal{G}^{\circ}}\to\operatorname{\mathrm{Bun}}_{\mathcal{G}}$
and
$\pi_{*}\colon\operatorname{\mathrm{Bun}}_{\mathcal{G}}\to\operatorname{\mathrm{Bun}}_{\Gamma}$.

######  Notation 3.2.

For any $\Gamma$-torsor $\mathcal{P}$ over $C$, we define the stack
$\operatorname{\mathrm{Bun}}_{\mathcal{P}}$ to be the preimage of the residual
gerbe
$\mathrm{B}\operatorname{Aut}(\mathcal{P})\hookrightarrow\operatorname{\mathrm{Bun}}_{\Gamma}$
corresponding to $\mathcal{P}$. In other words, we have a cartesian diagram

${\operatorname{\mathrm{Bun}}_{\mathcal{P}}}$${\operatorname{\mathrm{Bun}}_{\mathcal{G}}}$${\mathrm{B}\operatorname{Aut}(\mathcal{P})}$${\operatorname{\mathrm{Bun}}_{\Gamma}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\mathcal{P}}$

where the horizontal maps are open and closed embeddings.

######  Lemma 3.3.

Let $\mathcal{P}$ be a $\Gamma$-torsor over $C$ admitting a lift to a
$\mathcal{G}$-torsor. There is a connected reductive group scheme
$\mathcal{G}_{\mathcal{P}}$ over $C$ and an
$\operatorname{Aut}(\mathcal{P})$-torsor
$\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}}}\to\operatorname{\mathrm{Bun}}_{\mathcal{P}}$.

###### Proof.

Suppose first that $\mathcal{P}\cong\mathbbm{1}_{\Gamma}$ is the trivial
$\Gamma$-torsor. The short exact sequence

$1\to\mathcal{G}^{\circ}\xrightarrow{\iota}\mathcal{G}\xrightarrow{\pi}\Gamma\to
1$

induces a cartesian diagram of stacks over $C$

(3.1)
${\mathrm{B}\mathcal{G}^{\circ}}$${\mathrm{B}\mathcal{G}}$${C}$${\mathrm{B}\Gamma}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\mathbbm{1}_{\Gamma}}$

Since restricting scalars from $C$ to $\operatorname{Spec}{k}$ preserves fiber
products, applying $\operatorname{Res}_{C/k}$ to the diagram above we obtain a
cartesian square

${\operatorname{\mathrm{Bun}}_{\mathcal{G}^{\circ}}}$${\operatorname{\mathrm{Bun}}_{\mathcal{G}}}$${\operatorname{Spec}k}$${\operatorname{\mathrm{Bun}}_{\Gamma}}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\mathbbm{1}_{\Gamma}}$

This is readily seen to sit inside the larger commutative diagram

${\operatorname{\mathrm{Bun}}_{\mathcal{G}^{\circ}}}$${\operatorname{\mathrm{Bun}}_{\mathbbm{1}_{\Gamma}}}$${\operatorname{\mathrm{Bun}}_{\mathcal{G}}}$${\operatorname{Spec}k}$${\mathrm{B}\Gamma}$${\operatorname{\mathrm{Bun}}_{\Gamma}.}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\mathbbm{1}_{\Gamma}}$

Since the outer square and the right hand square are cartesian, so is the left
hand square. Hence,
$\iota_{*}\colon\operatorname{\mathrm{Bun}}_{\mathcal{G}^{\circ}}\to\operatorname{\mathrm{Bun}}_{\mathbbm{1}_{\Gamma}}$
is a $\Gamma$-torsor.

Let now $\mathcal{P}$ be an arbitrary $\Gamma$-torsor over $C$ admitting a
lift to a $\mathcal{G}$-torsor $\mathcal{E}$. Using [Sta23, Lemma 06QF], we
deduce from (3.1) that the morphism $\mathrm{B}\mathcal{G}\to\mathrm{B}\Gamma$
is a gerbe, and so by [Sta23, Lemma 06QE], the stack $\mathcal{Z}$ defined by
the cartesian square

(3.2)
${\mathcal{Z}}$${\mathrm{B}\mathcal{G}}$${C}$${\mathrm{B}\Gamma}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\mathcal{P}}$

is also a gerbe. The assumption that $\mathcal{P}$ has a lift to a
$\mathcal{G}$-torsor implies that $\mathcal{Z}\to C$ admits a section. Hence,
by [Sta23, Lemma 06QG], we see that
$\mathcal{Z}\cong\mathrm{B}\mathcal{G}_{\mathcal{P}}$, where
$\mathcal{G}_{\mathcal{P}}$ is the subgroup scheme of
$\operatorname{\underline{Aut}}(\mathcal{E})$ whose sections are morphisms
$\alpha$ of $\mathcal{E}$ such that
$\pi_{*}\alpha=\operatorname{id}_{\mathcal{P}}$. Since $\mathcal{P}$ is étale
locally isomorphic to $\mathbbm{1}_{\Gamma}$, we see that
$\mathcal{G}_{\mathcal{P}}$ is étale locally isomorphic to
$\mathcal{G}^{\circ}$, and hence $\mathcal{G}_{\mathcal{P}}$ is connected and
reductive. From diagram (3.2) we see that for any $k$-scheme $T$, the groupoid
$\mathrm{B}\mathcal{G}_{\mathcal{P}}(C\times T)$ consists of the data of a
$\mathcal{G}$-torsor $\mathcal{F}$ over $C\times T$ together with an
isomorphism $\alpha\colon\mathrm{pr}_{1}^{*}\mathcal{P}\to\pi_{*}\mathcal{F}$,
where $\mathrm{pr}_{1}\colon C\times T\to C$ is the projection. Hence, we see
that

$\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}},C}\cong\operatorname{Res}_{C/k}(\mathrm{B}\mathcal{G}_{\mathcal{P}})\cong\operatorname{Spec}k\times_{\mathcal{P},\operatorname{\mathrm{Bun}}_{\Gamma,C},\pi_{*}}\operatorname{\mathrm{Bun}}_{\mathcal{G},C}.$

Since the morphism
$\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}},C}\to\operatorname{\mathrm{Bun}}_{\mathcal{G},C}$
is identified with the forgetful morphism
$(\mathcal{F},\alpha)\mapsto\mathcal{F}$, we see that it factors through
$\operatorname{\mathrm{Bun}}_{\mathcal{P}}$. That is, we have a commutative
diagram

${\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}},C}}$${\operatorname{\mathrm{Bun}}_{\mathcal{P}}}$${\operatorname{\mathrm{Bun}}_{\mathcal{G},C}}$${\operatorname{Spec}k}$${\mathrm{B}\operatorname{Aut}(\mathcal{P})}$${\operatorname{\mathrm{Bun}}_{\Gamma,C}.}$$\scriptstyle{\psi_{\mathcal{P}}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\mathcal{P}}$

Since the right and outer squares are cartesian, so is the left. Since
$\Gamma$ is finite, so is $\operatorname{Aut}(\mathcal{P})$, and since
$\operatorname{Spec}k\to\mathrm{B}\operatorname{Aut}(\mathcal{P})$ is an
$\operatorname{Aut}(\mathcal{P})$-torsor, so is $\psi_{\mathcal{P}}$. ∎

######  Theorem 3.4.

Let $\mathcal{G}$ be a reductive group scheme over $C$. If
$\operatorname{char}(k)>0$ assume furthermore that $\Gamma$ is finite and that
$\operatorname{char}(k)$ and $|\Gamma|$ are coprime. Then there exist a finite
decomposition
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}\cong\coprod_{\mathcal{P}}\operatorname{\mathrm{Bun}}_{\mathcal{P}}$
by open and closed substacks such that each
$\operatorname{\mathrm{Bun}}_{\mathcal{P}}$ admits a torsor
$\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}}}\to\operatorname{\mathrm{Bun}}_{\mathcal{P}}$
under a finite group, where $\mathcal{G}_{\mathcal{P}}$ is a connected
reductive group scheme over $C$.

###### Proof.

By Proposition 3.1 we have a finite decomposition

$\operatorname{\mathrm{Bun}}_{\mathcal{G}}\cong\coprod_{\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(C,\Gamma)}\operatorname{\mathrm{Bun}}_{\mathcal{P}}.$

By Lemma 3.3 the statement follows. ∎

######  Remark 3.5.

Theorem 3.4 allows us to reduce any question about nonconnected groups to the
connected case, as long as we have a good theory of descent along finite
torsors. In Section 3.3 we use it to construct a good moduli space for the
semistable locus. Thus, we first prove that existence of projective good
moduli spaces descend along finite torsors.

######  Notation 3.6.

From now on, we assume that $\operatorname{char}(k)=0$.

######  Lemma 3.7.

Let $A$ be a finite group, let $\mathcal{X}$ and $\mathcal{Y}$ be quasi-
compact algebraic stacks over $k$, locally of finite type with affine
diagonal, and let $\phi\colon\mathcal{X}\to\mathcal{Y}$ be an $A$-torsor. If
$\mathcal{X}$ admits a separated (resp. proper, resp. projective) good moduli
space, then so does $\mathcal{Y}$.

###### Proof.

Suppose that $\mathcal{X}$ admits a separated good moduli space $X$. Then
$\mathcal{X}$ is both $\Theta$-reductive and S-complete by [AHH18, Theorem A].
Since $\phi\colon\mathcal{X}\to\mathcal{Y}$ is an $A$-torsor, it is in
particular finite, étale, and surjective. Thus, by [AHH18, Propositions
3.19(2) and 3.43(2)] the stack $\mathcal{Y}$ is also $\Theta$-reductive and
S-complete, and so by [AHH18, Theorem A], it admits a separated good moduli
space $Y$.

If $X$ is proper, then since the surjection
$\phi\colon\mathcal{X}\to\mathcal{Y}$ induces a surjection $X\to Y$, we see
that $Y$ is proper as well.

Suppose now that $X$ is projective and let $L$ be an ample line bundle on it.
By the universal property of $X$ [Alp13, Theorem 6.6], the action of $A$ on
$\mathcal{X}$ descends to $X$. We can replace $L$ by $\otimes_{\gamma\in
A}(\gamma^{*}L)$ and assume that $L$ is $A$-invariant. Consider the
commutative diagram

${\mathcal{X}}$${\mathcal{Y}}$${X}$${Y.}$$\scriptstyle{\phi}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\pi}$

Since $f^{*}L$ is invariant under the action of $A$, it descends to a line
bundle $M$ on $\mathcal{Y}$.

We now show that $M^{\otimes n}$ descends to $Y$, where $n\coloneqq|A|$. From
the Cartesian diagram

${\mathcal{X}}$${\operatorname{Spec}k}$${\mathcal{Y}}$${\mathrm{B}A}$$\scriptstyle{\phi}$

we see that for any $k$-points $y\in\mathcal{Y}$ and $x\in\mathcal{X}$ such
that $\phi(x)=y$, the index of $\phi(\operatorname{Aut}(x))$ in
$\operatorname{Aut}(y)$ is at most $n$. Now $\operatorname{Aut}(x)$ acts
trivially on the fiber of $f^{*}L$ at $x$, and so we see that
$\phi(\operatorname{Aut}(x))$ is contained in the kernel of the action of
$\operatorname{Aut}(y)$ on the fiber of $M$ at $y$, hence
$\operatorname{Aut}(y)$ acts trivially on $M^{\otimes n}$. Thus by [Alp13,
Theorem 10.3], $M^{\otimes n}$ descends to a line bundle $N$ on $Y$. Now

$f^{*}\pi^{*}N\cong\phi^{*}g^{*}N\cong f^{*}L^{\otimes n},$

so it follows that $\pi^{*}N\cong L^{\otimes n}$. Since $\phi$ is affine, so
is $\pi$, and since $\pi$ is also proper, it is finite, and so $N$ is ample on
$Y$ by [Sta23, Lemma 0GFB]. ∎

###  3.2. Semistability

In [Ram96, Ram96a] Ramanathan defined a notion of semistability for
$G$-bundles under a connected reductive group $G$ over $k$. When
$G=\mathrm{GL}_{n}$, this notion agrees with $\mu$-semistability when
$\mathrm{GL}_{n}$-bundles are viewed as vector bundles of rank $n$ (see [HM04,
Corollary 1] for a proof), and Ramanathan proved that a $G$-bundle $E$ is
semistable if and only if $\operatorname{Ad}_{*}E$ is semistable (see [Ram96,
Corollary 3.18]).

Let now $\mathcal{G}$ be a connected reductive group scheme over $C$. In
[Beh95] Behrend introduced the following notion of semistability for
$\mathcal{G}$-bundles, which reduces to Ramanathan’s notion in the case of a
constant group scheme. A $\mathcal{G}$-bundle $\mathcal{E}$ is semistable if
for each parabolic subgroup
$\mathcal{P}\subset\operatorname{\underline{Aut}}(\mathcal{E})$, we have that

$\deg\text{Lie}(\mathcal{P})\leq 0.$

######  Lemma 3.8.

If $\mathcal{G}$ is a connected reductive group over $C$, then a
$\mathcal{G}$–bundle $\mathcal{E}$ is semistable if and only if
$\operatorname{Ad}_{*}\mathcal{E}$ is semistable.

###### Proof.

Assume first that $\mathcal{G}$ is an inner form of a constant connected
reductive group $G$. That is, there is a $G$-bundle $E$ such that
$\mathcal{G}\cong\operatorname{\underline{Aut}}(E)$. The functor
$\mathcal{E}\mapsto\mathcal{E}\times^{\mathcal{G}}E$ is then an equivalence of
categories between $\mathcal{G}$-bundles on $C$ and $G$-bundles on $C$. Under
this equivalence semistable $\mathcal{G}$-bundles correspond to semistable
$G$-bundles. Indeed, if $\mathcal{E}$ is a $\mathcal{G}$-bundle, then we have
that

$\operatorname{\underline{Aut}}(\mathcal{E})\cong\mathcal{E}\times^{\mathcal{G}}\mathcal{G}\cong\mathcal{E}\times^{\mathcal{G}}(E\times^{G}G)\cong(\mathcal{E}\times^{\mathcal{G}}E)\times^{G}G\cong\operatorname{\underline{Aut}}(\mathcal{E}\times^{\mathcal{G}}E).$

Furthermore, since $\operatorname{Lie}(\mathcal{G})\cong
E\times^{G}\operatorname{Lie}(G)$ we see that for any $\mathcal{G}$-bundle
$\mathcal{E}$ we have that

$\operatorname{Ad}_{*}(\mathcal{E})\cong\mathcal{E}\times^{\mathcal{G}}\operatorname{Lie}(\mathcal{G})\cong\mathcal{E}\times^{\mathcal{G}}E\times^{G}\operatorname{Lie}(G)\cong\operatorname{Ad}_{*}(\mathcal{E}\times^{G}E).$

Since $\mathcal{E}$ is semistable if and only if $\mathcal{E}\times^{G}E$ is
semistable, the statement follows from the corresponding statement for a
constant group.

In the general case, by [Beh95, Section 4.2], there is a finite étale cover
$f\colon D\to C$ such that $f^{*}\mathcal{G}$ is an inner form of a constant
group. By [Beh95, Corollary 7.4] any $\mathcal{G}$-bundle $\mathcal{E}$ is
semistable if and only if $f^{*}\mathcal{E}$ is semistable, and since
$\operatorname{Ad}_{*}f^{*}\mathcal{E}\cong
f^{*}\operatorname{Ad}_{*}\mathcal{E}$ the statement thus follows. ∎

We thus make the following definition of semistability for
$\mathcal{G}$–bundles to include the nonconnected case.

######  Definition 3.9.

Let $\mathcal{G}$ be a reductive group scheme over $C$. A $\mathcal{G}$-bundle
$\mathcal{E}$ is semistable if $\operatorname{Ad}_{*}\mathcal{E}$ is
semistable in the sense above.

From the previous discussions we immediately obtain the following statements.

######  Lemma 3.10.

Let $\mathcal{G}$ be a reductive group scheme over $C$ and let $\mathcal{E}$
be a $\mathcal{G}$-bundle. The following are equivalent.

  1. (1)

The bundle $\mathcal{E}$ is semistable;

  2. (2)

for any finite cover $f\colon D\to C$, $f^{*}\mathcal{E}$ is semistable; and

  3. (3)

there exists a finite cover $f\colon D\to C$ such that $f^{*}\mathcal{E}$ is
semistable.

###### Proof.

This is true applied to the $\mathrm{GL}(\text{Lie}(\mathcal{G}))$-bundle
$\operatorname{Ad}_{*}\mathcal{E}$. Since
$\operatorname{Ad}_{*}f^{*}\mathcal{E}\cong
f^{*}\operatorname{Ad}_{*}\mathcal{E}$ we see that the statement follows. ∎

######  Lemma 3.11.

Let $\iota\colon\mathcal{G}^{\circ}\to\mathcal{G}$ denote the inclusion. A
$\mathcal{G}^{\circ}$-bundle $\mathcal{E}$ is semistable if and only
$\iota_{*}\mathcal{E}$ is a semistable $\mathcal{G}$–bundle.

###### Proof.

This follows from Lemma 3.8. ∎

We denote by $\operatorname{\mathrm{Bun}}_{\mathcal{G}}^{\mathrm{ss}}$ the
substack of semistable $\mathcal{G}$-bundles. Since it is the preimage of
$\operatorname{\mathrm{Bun}}_{\mathrm{GL}(\text{Lie}(\mathcal{G}))}^{\mathrm{ss}}$
under $\operatorname{Ad}_{*}$, it is an open substack. For any $\Gamma$-torsor
$\mathcal{P}$ admitting a lift to $\mathcal{G}$, we similarly let
$\operatorname{\mathrm{Bun}}_{\mathcal{P}}^{\mathrm{ss}}$ denote the substack
of $\operatorname{\mathrm{Bun}}_{\mathcal{P}}$ consisting of semistable
bundles (recall 3.2).

###  3.3. Existence of a projective good moduli space

We recall the following theorem of Behrend.

######  Lemma 3.12.

([Beh90, Theorem 4.4.3] Let $f\colon D\to C$ be a projective flat cover of
$C$, and let $\mathcal{G}$ be an affine group scheme of finite type over $C$.
The induced morphism

$f^{*}\colon\operatorname{\mathrm{Bun}}_{\mathcal{G},C}\to\operatorname{\mathrm{Bun}}_{f^{*}\mathcal{G},D}$

is affine of finite presentation.

We deduce immediately the following.

######  Proposition 3.13.

Let $\mathcal{G}$ be an affine smooth group scheme of finite type over $C$ and
suppose that there is a notion of semistability on
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}$ for which the substack of
semistable bundles admits a proper good moduli space
$M_{\mathcal{G},C}^{\mathrm{ss}}$. For any finite cover $f\colon D\to C$ such
that a $\mathcal{G}$–bundle $\mathcal{E}$ is semistable if and only if
$f^{*}\mathcal{E}$ is semistable, $f$ induces via pullback a finite morphism
$f^{*}\colon M_{\mathcal{G},C}^{\mathrm{ss}}\to
M_{f^{*}\mathcal{G},D}^{\mathrm{ss}}$.

###### Proof.

By assumption we have that
$\operatorname{\mathrm{Bun}}_{\mathcal{G},C}^{\mathrm{ss}}=\operatorname{\mathrm{Bun}}_{\mathcal{G},C}\times_{\operatorname{\mathrm{Bun}}_{f^{*}\mathcal{G},D}}\operatorname{\mathrm{Bun}}_{f^{*}\mathcal{G},D}^{\mathrm{ss}}$.
Hence, by Lemma 3.12 $f^{*}$ induces an affine morphism between the semistable
loci, and thus also induces an affine morphism between the good moduli spaces
$f^{*}\colon M_{\mathcal{G},C}^{\mathrm{ss}}\to
M_{f^{*}\mathcal{G},D}^{\mathrm{ss}}$. Since $M_{\mathcal{G},C}^{\mathrm{ss}}$
and $M_{f^{*}\mathcal{G},D}^{\mathrm{ss}}$ are proper, so is $f^{*}$. Hence,
$f^{*}\colon M_{\mathcal{G},C}^{\mathrm{ss}}\to
M_{f^{*}\mathcal{G},D}^{\mathrm{ss}}$ is finite. ∎

######  Lemma 3.14.

Let $\mathcal{G}$ be a connected reductive group scheme over $C$. The stack
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}^{\mathrm{ss}}$ admits a projective
good moduli space.

###### Proof.

By [AHH18, Section 8]
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}^{\mathrm{ss}}$ admits a proper good
moduli space $M_{\mathcal{G}}^{\mathrm{ss}}$. Let $f\colon D\to C$ be a finite
étale cover such that $f^{*}\mathcal{G}$ is an inner form of a constant group
scheme $G$. By Proposition 3.13 we obtain a finite morphism

$f^{*}\colon M_{\mathcal{G},C}^{\mathrm{ss}}\to
M_{f^{*}\mathcal{G},D}^{\mathrm{ss}}$

between proper good moduli spaces. Since $f^{*}\mathcal{G}$ is an inner form
of $G$, $M_{f^{*}\mathcal{G},D}^{\mathrm{ss}}\cong M_{G,D}^{\mathrm{ss}}$,
which is projective ([Ram96, Ram96a]). Since $f^{*}$ is finite this implies
that $M_{\mathcal{G},C}^{\mathrm{ss}}$ is projective. ∎

######  Theorem 3.15.

Let $\mathcal{G}$ be a (not necessarily connected) reductive group scheme over
$C$. The semistable locus
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}^{\mathrm{ss}}$ admits a projective
good moduli space.

###### Proof.

By Theorem 3.4 it suffices to prove that for any $\Gamma$-torsor $\mathcal{P}$
admitting a lift to a $\mathcal{G}$-torsor, the semistable locus
$\operatorname{\mathrm{Bun}}_{\mathcal{P}}^{\mathrm{ss}}$ admits a projective
good moduli space. To this end, let $\mathcal{P}$ be a $\Gamma$-torsor
admitting a lift to a $\mathcal{G}$-torsor. Since
$\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(C,\Gamma)$ is finite (cf. proof of
Proposition 3.1), there exists a quasi-finite étale cover $C^{\prime}\to C$
trivialising all $\Gamma$-torsors. By restricting to an irreducible component
we may assume that $C^{\prime}$ is irreducible. Let $f\colon D\to C$ be the
finite cover of $C$ corresponding to the field extension $k(C^{\prime})/k(C)$.
Then $f$ factors through a rational morphism $D\dasharrow C^{\prime}\to C$.
Hence, for any $\Gamma$-bundle $\mathcal{P}$ over $C$, $f^{*}\mathcal{P}$ is
generically trivial, whence trivial (by [AEG20, Proposition 3.4]). The short
exact sequence

$1\to\mathcal{G}^{\circ}\to\mathcal{G}\to\Gamma\to 1$

induces a long exact sequence of pointed sets

$\cdots\to\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(C,\mathcal{G}^{\circ})\to\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(C,\mathcal{G})\to\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(C,\Gamma)$

which is functorial with respect to $f^{*}$. Hence, for any
$\mathcal{G}$–bundle $\mathcal{F}$, $f^{*}\mathcal{F}$ has a reduction to
$\mathcal{G}^{\circ}$. We thus have a commutative diagram

${\operatorname{\mathrm{Bun}}_{\mathcal{G}^{\circ},D}}$${\operatorname{\mathrm{Bun}}_{\mathcal{G},D}}$${\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}},C}}$${\operatorname{\mathrm{Bun}}_{\mathcal{P},C}}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\psi_{\mathcal{P}}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$

Let now $\mathcal{E}$ be a $\mathcal{G}_{\mathcal{P}}$-torsor. By Lemma 3.10
we see that $\mathcal{E}$ is semistable if and only if $f^{*}\mathcal{E}$ is
semistable, which by Lemma 3.11 is equivalent to
$\iota_{*}f^{*}\mathcal{E}\cong f^{*}\psi_{\mathcal{P}}(\mathcal{E})$ being
semistable. Again using Lemma 3.10 we thus see that $\mathcal{E}$ is
semistable if and only if $\psi_{\mathcal{P}}(\mathcal{E})$ is semistable.
Hence, by base change along
$\operatorname{\mathrm{Bun}}_{\mathcal{P},C}^{\mathrm{ss}}\hookrightarrow\operatorname{\mathrm{Bun}}_{\mathcal{P},C}$,
the morphism $\psi_{\mathcal{P}}$ induces an
$\operatorname{Aut}(\mathcal{P})$-torsor on the semistable loci

$\psi_{\mathcal{P}}\colon\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}},C}^{\mathrm{ss}}\to\operatorname{\mathrm{Bun}}_{\mathcal{P},C}^{\mathrm{ss}}.$

By Lemma 3.14,
$\operatorname{\mathrm{Bun}}_{\mathcal{G}_{\mathcal{P}},C}^{\mathrm{ss}}$
admits a projective good moduli space. Hence, by Lemma 3.7 we conclude that
$\operatorname{\mathrm{Bun}}_{\mathcal{P},C}^{\mathrm{ss}}$ admits a
projective good moduli space. ∎

We remark also an immediate consequence of Proposition 3.13 and Theorem 3.15
that has appeared in the literature in the case of vector bundles (see [Wei22,
Theorem 4.2] cf. [Hei06, Theorem 4.2], [Fal93, Theorem I.4]) and could be of
independent interest.

######  Corollary 3.16.

Let $\mathcal{G}$ be a reductive group scheme over $C$. For any finite cover
$f\colon D\to C$, the induced morphism $f^{*}\colon
M_{\mathcal{G},C}^{\mathrm{ss}}\to M_{f^{*}\mathcal{G},D}^{\mathrm{ss}}$ is
finite.

###### Proof.

By Theorem 3.15 and Lemma 3.10 this is a special case of Proposition 3.13. ∎

######  Remark 3.17.

Corollary 3.16 applies to any affine group scheme $\mathcal{G}$ such that
$\operatorname{\mathrm{Bun}}_{\mathcal{G}}^{\mathrm{ss}}$ admits a proper good
moduli space, and such that semistability can be checked on a finite cover. In
particular, it applies to any Bruhat-Tits group scheme of the form
$\mathcal{G}=\operatorname{Res}_{C^{\prime}/C}(\mathcal{G}^{\prime})^{\operatorname{Gal}(C^{\prime}/C)}$
for some Galois cover $f\colon C^{\prime}\to C$ and some connected reductive
group $\mathcal{G}^{\prime}$ over $C^{\prime}$.

##  4\. Nondensity of essentially finite bundles

###  4.1. Finite subgroups of reductive groups

We remind the reader that we continue with the assumption that
$\operatorname{char}(k)=0$.

Let $G$ be an arbitrary reductive group over $k$. The main purpose of this
section is to prove that there are finitely many subgroups $H_{1},...,H_{n}$
of $G$ such that each $H_{i}^{\circ}$ is a torus and any finite subgroup of
$G$ is contained in some $H_{i}$, up to conjugation. To do so, recall first
Jordan’s classical result in [Jor77] on finite subgroups of $\mathrm{GL}(n)$.

######  Theorem 4.1.

([Jor77]) For each integer $n>1$ there is an integer $f_{n}$ such that every
finite subgroup $\Gamma\subset\mathrm{GL}(n)$ contains an abelian normal
subgroup $A$ such that $[\Gamma:A]\leq f_{n}.$

Inspired by this result, we prove the following generalized version.

######  Theorem 4.2.

Let $G$ be a (not necessarily connected) reductive group. There exists a
constant $f_{G}\geq 1$ with the following property: for every finite subgroup
$\Gamma\subset G$ there is an abelian normal subgroup $A\subset\Gamma$
contained in a torus of $G^{\circ}$ such that $[\Gamma:A]\leq f_{G}$.

The proof of Theorem 4.2 closely follows Breuillard’s exposition in [Bre23] on
Jordan’s theorem. Mainly, we make sure that one can run the same arguments
upon making appropriate changes from the $\mathrm{GL}(n)$-case to the general
case of a reductive group (e.g. the use of the center instead of scalar
matrices, the use of the projection $G\to G/[G,G]$ instead of the determinant
map, etc.). As such, we relegate the proof to an appendix (see Appendix A).

Before proving the main result of this section, we prove the following lemma
and introduce some notation.

######  Lemma 4.3.

For a (not necessarily connected) reductive group $G$ and a finite subgroup
$\Gamma\subset G$, there is a finite number of elements in
$\operatorname{Hom}(\Gamma,G)$ up to conjugacy of $G$.

###### Proof.

We can give $\operatorname{Hom}(\Gamma,G)$ the structure of a variety over
$k$, and we can consider the categorical quotient quotient
$X(\Gamma,G)\coloneqq\operatorname{Hom}(\Gamma,G)//G$, the $G$-character
variety of $\Gamma$. It is an affine variety, and the points of $X(\Gamma,G)$
are completely reducible representations of $\Gamma$ up to conjugacy. If
$\Gamma$ is finite and $\operatorname{char}(k)=0$, this is just every
homomorphism up to conjugacy. If $\rho:\Gamma\to G$ is a homomorphism, then

$\dim T_{[\rho]}X(\Gamma,G)\leq H^{1}(\Gamma,\mathfrak{g}),$

where $T_{[\rho]}$ is the tangent space at the class $[\rho]\in X(\Gamma,G)$
and where $\Gamma$ acts on $\mathfrak{g}$ trough $\rho$. This follows from
[Sik10, Theorem 1]. Take a cocycle $\gamma\in H^{1}(\Gamma,\mathfrak{g})$, and
define for $g\in G$ the map $\varphi_{g}:\mathfrak{g}\to\mathfrak{g}$

$\varphi_{g}(X)=\rho(g)X+\gamma(g).$

This is an action of $\Gamma$ on $\mathfrak{g}$ by affine transformations. By
averaging over an orbit, we find a fixed point $X_{0}$. We therefore have

$\gamma(g)=\varphi_{g}(X_{0})-\rho(g)X_{0}=X_{0}-\rho(g)X_{0}.$

In other words, $\gamma$ is a coboundary and $H^{1}(\Gamma,\mathfrak{g})=0$.
We see that $X(\Gamma,G)$ is finite. ∎

Given a subset $S\subset G$ we denote by $C_{G}(S)$ and $N_{G}(S)$ the
centralizer and the normalizer of $S$ in $G$. Fix a maximal torus $T\subset
G$. For any representation $V$ of $G$ with central kernel, let $\Phi(V)$ be
the set of weights of $V$ relative to $T$. We say that a subgroup $H\subset T$
is a weight kernel if it is defined by a finite number of equations of the
form $\alpha(t)=\beta(t)$ for $\alpha,\beta\in\Phi(V)$ and $t\in T$. That is,
for any $S\subset\Phi^{2}$ we have a weight kernel

$H_{S}=\\{\;t\in T\;:\;\alpha(t)/\beta(t)=1\;\text{for all}\;(\alpha,\beta)\in
S\;\\}.$

Conversely, any weight kernel gives a subset $S\subset\Phi^{2}$. We see that

$S\subset S^{\prime}\iff H_{S}\supset H_{S^{\prime}}.$

If $A\subset T$ is a group, then we let $H_{A}$ be the minimal weight kernel
containing $A$. Since both $C_{G}(A)$ and $C_{G}(H_{A})$ consist exactly of
the elements preserving the decomposition $V$ into characters of $A$, we see
that $C_{G}(A)=C_{G}(H_{A})$. Similarly, since both $N_{G}(A)$ and
$N_{G}(H_{A})$ consists exactly of the elements permuting the characters of
$V$ we obtain that $N_{G}(A)=N_{G}(H_{A})$.

######  Theorem 4.4.

Let $G$ be a reductive group. There is a set $S(G)=\\{H_{1},\ldots,H_{n}\\}$
of closed subgroups of $G$ such that $H_{i}^{\circ}$ is a torus and for any
finite subgroup $\Gamma\subset G$ some conjugate is contained in one of the
$H_{i}$.

###### Proof.

We proceed by induction on the dimension of $G$. If $\dim G=1$, then
$G^{\circ}$ is a torus, so we are done. For the induction step, we assume the
statement of the theorem for any group reductive group $H$ such that $\dim
H<\dim G$. Let $\Gamma\subset G$ be a finite subgroup. By Theorem 4.2, there
is an abelian normal subgroup $A\subset\Gamma$ contained in a maximal torus
$T\subset G$.

First, assume that $A$ is not in the center of $G^{\circ}$. This means that
$C_{G}(A)\subsetneq G$, which implies that $\dim C_{G}(A)<\dim G$. Since
$C_{G}(A)$ has finite index in $N_{G}(A)$, we also see that $\dim
N_{G}(A)<\dim G$. Now $\Gamma\subset N_{G}(A)$. Up to conjugacy, the number of
groups of the form $N_{G}(A)$ as we let $\Gamma$ run over all the finite
subgroups of $G$ will be finite. Indeed, we can choose a faithful
representation $V$ of $G$ and a weight kernel $H_{A}$ relative $T$ such that
$N_{G}(H_{A})=N_{G}(A)$, and there are only finitely many such weight kernels.
In other words, if we add $S(N_{G}(H))$ to $S(G)$ where $H$ is a weight kernel
we get by induction that $\Gamma\subset H_{i}$ for some $H_{i}\in S(G)$.

Suppose now that $A\subset Z(G^{\circ})$. By the structure theory of reductive
groups, $G^{\circ}=ZG^{\prime}$ where $Z\coloneqq Z(G^{\circ})^{\circ}$,
$G^{\prime}\coloneqq[G^{\circ},G^{\circ}]$ and where the intersection $Z\cap
G^{\prime}$ is finite. Consider the map $\rho:G\to G/Z$. The group $B\coloneqq
Z((G/Z)^{\circ})=Z(G^{\circ})/Z$ is a finite group, and $\rho(A)\subset B$.
This means that

$|\rho(\Gamma)|=[\rho(\Gamma):\rho(A)]|\rho(A)|\leq f_{G}|B|.$

This means that $|\rho(\Gamma)|$ is bounded in terms of $G$. In particular, up
to isomorphism the set

$\\{\rho(\Gamma):\Gamma\subset G,A\subset\mathrm{Z}(G^{\circ})\\}$

is finite. By Lemma 4.3, for a given choice of $\Gamma$ there is a finite
number of conjugacy classes in $\operatorname{Hom}(\rho(\Gamma),G/Z)$. This
means there is a finite set of choices for $\rho(\Gamma)$. Each choice
determines a group $H=\rho^{-1}(\rho(\Gamma))$, which has connected component
a torus. We add each $H$ to $S(G)$. ∎

###  4.2. Nondensity of essentially finite bundles

In this section, $G$ is a connected reductive group over $k$. We recall that a
$G$-bundle is essentially finite if it admits a reduction to a finite group.
The same definition applies to nonconnected groups as well, and an important
remark is that if a bundle is essentially finite, then there is a finite étale
cover trivialising it (these notions are in fact equivalent but we won’t need
the other direction in what follows). We denote by
$M_{G}^{\operatorname{\text{ef}}}$ the set of essentially finite $G$-bundles
of degree 0 inside the moduli space of semistable $G$-bundles of degree 0,
denoted $M_{G}^{\mathrm{ss},0}$.

In this section we give an upper bound on the dimension of
$\overline{M_{G}^{\operatorname{\text{ef}}}}$ and thus prove that
$M_{G}^{\operatorname{\text{ef}}}$ is not dense in $M_{G}^{\mathrm{ss},0}$.
The idea is as follows. Let $H_{i}$ be as in theorem 4.4 and denote by
$\varphi_{i}\colon H_{i}\hookrightarrow G$ the inclusion. We would like to say
that
$M_{G}^{\operatorname{\text{ef}}}\subset\cup_{i}\varphi_{i,*}M_{H_{i}}^{\mathrm{ss}}\subset
M_{G}^{\mathrm{ss},0}$ and then conclude by dimension reasons. However, if $E$
is a semistable $H_{i}$-bundle, then $\varphi_{i,*}E$ need not be a semistable
$G$-bundle, so we don’t have a morphism $\varphi_{i,*}\colon
M_{H_{i}}^{\mathrm{ss}}\to M_{G}^{\mathrm{ss}}$. Thus we prove first that
there is a sublocus of semistable $H_{i}$-bundles whose pushforward is
semistable. Then we show that every essentially finite $H_{i}$-bundle is
contained in this sublocus. Finally we make sure that we have the expected
relation between dimensions: $\dim M_{H_{i}}^{\mathrm{ss}}<\dim
M_{G}^{\mathrm{ss}}$. After introducing some notation, we prove these claims
in in a sequence of lemmas, leading up to the stated density result.

Let $\mathcal{T}$ be a torus over $C$. By [Hei10] we have that

$\pi_{0}(\operatorname{\mathrm{Bun}}_{\mathcal{T}})=X_{*}(\mathcal{T}_{k(C)^{\text{sep}}})_{\operatorname{Gal}(k(C)^{\text{sep}}/k(C))}.$

We denote by $\operatorname{\mathrm{Bun}}_{\mathcal{T}}^{d}$ the component of
$\operatorname{\mathrm{Bun}}_{\mathcal{T}}$ corresponding to
$d\in\pi_{0}(\operatorname{\mathrm{Bun}}_{\mathcal{T}})$, and for
$E\in\operatorname{\mathrm{Bun}}_{\mathcal{T}}^{d}(k)$, we say that $E$ is of
degree $d$. We use similar notation for the semistable locus. By [Hei17] for
all $d\in\pi_{0}(\operatorname{\mathrm{Bun}}_{\mathcal{T}})$,
$\operatorname{\mathrm{Bun}}_{\mathcal{T}}^{\mathrm{ss},d}$ is of finite type.
If $H$ is a non-connected reductive group such that $H^{\circ}=T$ is a torus,
then let

$\operatorname{\mathrm{Bun}}_{H}^{0}\coloneqq\operatorname{im}\Big{(}\coprod_{\mathcal{P}\in\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(C,H/H^{\circ})}\operatorname{\mathrm{Bun}}_{H_{\mathcal{P}}}^{0}\to\operatorname{\mathrm{Bun}}_{H}\Big{)},$

with the notation $H_{\mathcal{P}}$ analogous to that in Section 3.1. We
define $\operatorname{\mathrm{Bun}}_{H}^{\mathrm{ss},0}$ and
$M_{H}^{\mathrm{ss},0}$ similarly.

######  Lemma 4.5.

Let $\mathcal{T}$ be a torus over $C$, and let $f\colon C^{\prime}\to C$ be a
finite étale morphism. The induced map
$f^{*}\colon\pi_{0}(\operatorname{\mathrm{Bun}}_{\mathcal{T},C})\to\pi_{0}(\operatorname{\mathrm{Bun}}_{\mathcal{T},C^{\prime}})$
is given by multiplication with $\deg(f)$.

###### Proof.

Suppose first that $\mathcal{T}$ is induced, i.e. that there is some finite,
generically étale morphism $\pi\colon D\to C$ such that
$\mathcal{T}\cong\pi_{*}\mathbb{G}_{m}\coloneqq\operatorname{Res}_{D/C}(\mathbb{G}_{m})$.
The Leray spectral sequence

$\mathrm{H}^{p}_{\mathrm{\acute{e}t}}(C,\mathrm{R}^{q}\pi_{*}\mathbb{G}_{m})\implies\mathrm{H}^{p+q}_{\mathrm{\acute{e}t}}(D,\mathbb{G}_{m})$

induces a filtration on
$\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(D,\mathbb{G}_{m})$ whose graded pieces
are
$\mathrm{H}^{0}_{\mathrm{\acute{e}t}}(C,\mathrm{R}^{1}\pi_{*}\mathbb{G}_{m})$
and $\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(C,\pi_{*}\mathbb{G}_{m})$. For any
point $x\in C$, by proper base change (see e.g. [Den88]) and using that $\pi$
is finite, we see that
$(\mathrm{R}^{1}\pi_{*}\mathbb{G}_{m})_{x}\cong\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(D_{x},\mathbb{G}_{m})=0$.
Hence,
$\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(C,\mathcal{T})\cong\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(D,\mathbb{G}_{m})$
via the Leray spectral sequence and the result follows from this known case
for line bundles.

In the general case, following the proof of [Hei10, Lemma 16] (cf. [PR08,
Section 3. and 5.]) we can choose induced tori
$\mathcal{I}_{2}\to\mathcal{I}_{1}\twoheadrightarrow\mathcal{T}$ such that the
induced sequence

$\pi_{0}(\operatorname{\mathrm{Bun}}_{\mathcal{I}_{2}})\to\pi_{0}(\operatorname{\mathrm{Bun}}_{\mathcal{I}_{1}})\to\pi_{0}(\operatorname{\mathrm{Bun}}_{\mathcal{T}})\to
0$

is exact. The result now follows. ∎

######  Lemma 4.6.

Let $G$ be a connected, reductive group, let $T\subset G$ be a torus, and let
$H\subset G$ be a subgroup such that $H^{\circ}=T$. The inclusion
$\varphi\colon H\hookrightarrow G$ induces a morphism
$\varphi_{*}\colon\operatorname{\mathrm{Bun}}_{H}^{\mathrm{ss},0}\to\operatorname{\mathrm{Bun}}_{G}^{\mathrm{ss}}$.

###### Proof.

We need to show that for any $H$-bundle $E$ of degree 0, $\varphi_{*}E$ is
semistable. Let us first assume that $H=T$ and let $\rho\colon
H\to\mathrm{GL}(n)$ be an arbitrary representation. The image
$\operatorname{im}(\rho)$ is a torus in $\mathrm{GL}(n)$. Hence, since all
tori are conjugate, we may assume that $\operatorname{im}(\rho)$ lies in
$\operatorname{diag}_{n}$, the diagonal matrices. The induced map
$T\to\operatorname{diag}_{n}$ takes $E$ to a degree 0
$\operatorname{diag}_{n}$-bundle. The vector bundle $\rho_{*}E$ can then be
realised as $\rho_{*}E\cong L_{1}\oplus...\oplus L_{n}$ for some line bundles
$L_{1},...,L_{n}$, all of which are of degree 0. Hence $\rho_{*}E$ is
semistable.

Now let $H$ be an arbitrary subgroup of $G$ such that $H^{\circ}=T$ is a torus
and let $E$ be an $H$-bundle of degree 0, say coming from
$\operatorname{\mathrm{Bun}}_{H_{\mathcal{P}}}^{\mathrm{ss},0}$. Taking a
sufficiently large finite étale cover $f\colon D\to C$, by Lemma 4.5 we have a
commutative diagram

${\operatorname{\mathrm{Bun}}_{T,D}^{\mathrm{ss},0}}$${\operatorname{\mathrm{Bun}}_{H,D}^{\mathrm{ss},0}}$${\operatorname{\mathrm{Bun}}_{H_{\mathcal{P}},C}^{\mathrm{ss},0}}$${\operatorname{\mathrm{Bun}}_{H,C}^{\mathrm{ss},0}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$

Since $\varphi_{*}E$ is semistable if and only if $\varphi_{*}f^{*}E$ is
semistable, and since $f^{*}E$ has a reduction to a degree 0 $T$-bundle, upon
checking at the representation $\rho\colon T\to H\to
G\xrightarrow{\operatorname{Ad}}GL(\mathfrak{g})$ we see that $\varphi_{*}E$
is semistable. ∎

######  Lemma 4.7.

Let $H$ be a reductive group such that $H^{\circ}=T$ is a torus. Every
essentially finite $H$-bundle has degree 0.

###### Proof.

Suppose that $E$ is an essentially finite $H$-bundle. Using the notation as in
Section 3.3 with $\mathcal{P}\coloneqq\pi_{*}E$, we have a relative group
torus $H_{\mathcal{P}}$ over $C$ and a finite torsor
$\operatorname{\mathrm{Bun}}_{H_{\mathcal{P}}}\to\operatorname{\mathrm{Bun}}_{\mathcal{P}}$.
Let $\mathcal{E}$ be a $H_{\mathcal{P}}$-torsor whose image in
$\operatorname{\mathrm{Bun}}_{H}$ is $E$. We will show that the degree of
$\mathcal{E}$ is 0.

Let $\Gamma=H/H^{\circ}$ and let $f\colon C^{\prime}\to C$ be a finite étale
cover which trivialises all $\Gamma$-torsors. Since $E$ is essentially finite,
so is $f^{*}E$ and hence, there is a finite étale cover $g\colon
C^{\prime\prime}\to C^{\prime}$ trivialising it. Letting $h\coloneqq f\circ g$
we obtain the following commutative diagram

${\operatorname{\mathrm{Bun}}_{T,C^{\prime\prime}}}$${\operatorname{\mathrm{Bun}}_{\mathbbm{1}_{\Gamma},C^{\prime\prime}}}$${\operatorname{\mathrm{Bun}}_{H_{\mathcal{P}},C}}$${\operatorname{\mathrm{Bun}}_{\mathcal{P},C}}$$\scriptstyle{\iota_{*}}$$\scriptstyle{h^{*}}$$\scriptstyle{h^{*}}$

where $h^{*}E\cong\mathbbm{1}_{H}$. Hence,
$\iota_{*}h^{*}\mathcal{E}\cong\mathbbm{1}_{H}$. By Lemma 4.5 we have that
$\deg(h^{*}\mathcal{E})=\deg(h)\deg(\mathcal{E})$. To show that
$\deg(\mathcal{E})=0$, it thus suffices to show that for any $T$-torsor $F$
over $C^{\prime\prime}$, if $\iota_{*}F\cong\mathbbm{1}_{H}$, then $F$ has
degree 0. To this end, suppose that $F$ is such a $T$-torsor, and let
$(t_{ij})_{ij}$ be a cocycle representative, for some cover $(U_{i}\to
C^{\prime\prime})_{i}$. Since $T$ is commutative we obtain a $T$-torsor,
denoted $F^{2}$, given in cocycles by $(t_{ij}^{2})_{ij}$. Since $\iota_{*}F$
is trivial, there are $g_{i}\in H(U_{i})$ such that $g_{i}t_{ij}g_{j}^{-1}=1$.
Letting $\sim$ denote cohomological equivalence on cocycles, we then have that

$(t_{ij}^{2})_{ij}\sim(g_{i}t_{ij}^{2}g_{j}^{-1})_{ij}=(g_{i}t_{ij}g_{j}^{-1}g_{j}g_{i}^{-1}g_{i}t_{ij}g_{j}^{-1})_{ij}=(g_{j}g_{i}^{-1})_{ij}\sim(1)_{ij}$

Thus, $\iota_{*}F^{2}\cong\mathbbm{1}_{H}$. Similarly, if $F^{n}$ denotes the
$T$-torsor given in cocycles by $(t_{ij}^{n})_{ij}$, for all $n\in\mathbb{N}$,
then $\iota_{*}F^{n}\cong\mathbbm{1}_{H}$. Since
$\operatorname{\mathrm{Bun}}_{T,C^{\prime\prime}}\to\operatorname{\mathrm{Bun}}_{\iota,C^{\prime\prime}}$
is a $\Gamma$-torsor, the fibre over $\mathbbm{1}_{H}$ is finite. Thus, for
some $N\in\mathbb{N}$, $F^{N}\cong\mathbbm{1}_{T}$. This implies that $F$ is
essentially finite and hence of degree 0. Applying this to
$F=h^{*}\mathcal{E}$ we see that $\deg(\mathcal{E})=0$. ∎

######  Lemma 4.8.

Let $g$ denote the genus of $C$ and assume that $g\geq 2$. Let $G$ be a
connected, reductive group not equal to a torus, and let $H\subset G$ be a
subgroup such that $H^{\circ}=T$, for some torus $T\subset G$. Then we have
that

(4.1) $\dim M_{H}^{\mathrm{ss}}\leq g\dim T.$

In particular, $\dim M_{H}^{\mathrm{ss}}<\dim M_{G}^{\mathrm{ss}}$.

###### Proof.

Let $\overline{x}\in M_{H}^{\mathrm{ss}}$ be a closed point, let
$x\in\operatorname{\mathrm{Bun}}_{H}^{\mathrm{ss}}$ be the unique closed point
in the fiber over $\overline{x}$ and let $H_{x}$ denote the stabiliser of $x$.
By the local structure theorem for good moduli spaces [Alp23, Corollary 6.5.3]
we have a Cartesian diagram

${\left[\operatorname{Spec}A/H_{x}\right]}$${\operatorname{\mathrm{Bun}}_{H}^{\mathrm{ss}}}$${\operatorname{Spec}A^{H_{x}}}$${M_{H}^{\mathrm{ss}}}$

where the horizontal maps are étale. Hence, $\dim
M_{H}^{\mathrm{ss}}=\dim\operatorname{Spec}A^{H_{x}}$ and
$\dim\operatorname{\mathrm{Bun}}_{H}^{\mathrm{ss}}=\dim\operatorname{Spec}A-\dim
H_{x}$. To get an upper bound on $\dim\operatorname{Spec}A^{H_{x}}$ we may
assume that $H_{x}$ is connected, and hence irreducible. Since $H_{x}$ is
irreducible, it preserves the irreducible components of $\operatorname{Spec}A$
so we may furthermore assume that $\operatorname{Spec}A$, and thus
$\operatorname{Spec}A^{H_{x}}$, are irreducible. Letting $d$ be the minimum of
the dimensions of the stabilisers of points in $\operatorname{Spec}A$, then
(see e.g. [Dol03, Chapter 6, Corollay 6.2]) we have that

$\displaystyle\dim\operatorname{Spec}A^{H_{x}}$
$\displaystyle=\dim\operatorname{Spec}A-\dim G+d$
$\displaystyle=\dim[\operatorname{Spec}A/H_{x}]+d$
$\displaystyle=\dim\operatorname{\mathrm{Bun}}_{H}^{\mathrm{ss}}+d.$

Since $d\leq\dim H_{x}\leq\dim H$, we thus see that

(4.2) $\displaystyle\dim M_{H}^{\mathrm{ss}}$
$\displaystyle\leq\dim\operatorname{\mathrm{Bun}}_{H}^{\mathrm{ss}}+\dim H$
$\displaystyle=(g-1)\dim H+\dim H$ $\displaystyle=g\dim T.$

Since $\dim M_{G}^{\mathrm{ss}}=(g-1)\dim G+\dim\mathrm{Z}(G)$, to see that
$\dim M_{H}^{\mathrm{ss}}<\dim M_{G}^{\mathrm{ss}}$ it suffices to show that

$\frac{\dim G-\dim\mathrm{Z}(G)}{\dim G-\dim T}\leq g.$

Without loss of generality, we may assume that $T$ is a maximal torus. Let
$\Delta$ be a set of simple roots with respect to $T$. Since $\dim
G=\dim\mathfrak{g}$ and $G$ is not a torus we see that $\dim G>\dim
T+|\Delta|$. Since $\mathrm{Z}(G)=\bigcap_{\alpha\in\Delta}\ker(\alpha)$ we
see that $\dim T-\dim\mathrm{Z}(G)\leq|\Delta|$ and we obtain that

$\frac{\dim G-\dim\mathrm{Z}(G)}{\dim G-\dim T}=1+\frac{\dim
T-\dim\mathrm{Z}(G)}{\dim G-\dim T}<1+\frac{|\Delta|}{|\Delta|}=2\leq g.$

∎

######  Theorem 4.9.

For any connected, reductive group $G$ not equal to a torus, we have that

(4.3) $\dim\overline{M_{G}^{\textnormal{ef}}}\leq g\operatorname{rk}(G).$

In particular, $M_{G}^{\textnormal{ef}}\subseteq M_{G}^{\mathrm{ss},0}$ is not
dense.

###### Proof.

Let $H_{1},...,H_{n}\subset G$ be as in Theorem 4.4, and let
$\varphi_{i}\colon H_{i}\hookrightarrow G$ denote the inclusions. By Theorem
4.4 any essentially finite $G$-bundle has a reduction to some $H_{i}$, and by
lemma 4.6 we have a morphism $\varphi_{i,*}\colon M_{H_{i}}^{\mathrm{ss},0}\to
M_{G}^{\mathrm{ss}}$. By Lemma 4.7, all essentially finite $H_{i}$-bundles are
of degree 0, from which it follows that

$M_{G}^{\operatorname{\text{ef}}}\subset\bigcup_{i=1}^{n}\varphi_{i,*}M_{H_{i}}^{\mathrm{ss},0}\subset
M_{G}^{\mathrm{ss},0}.$

By (4.2) we have that $\dim M_{H_{i}}^{\mathrm{ss},0}\leq g\dim H_{i}^{0}\leq
g\operatorname{rk}(G)$ for all $i$, from which we obtain the dimension
estimate in (4.3). By lemma 4.8 we see that
$M_{G}^{\operatorname{\text{ef}},0}\subseteq M_{G}^{\mathrm{ss},0}$ is not
dense. ∎

## Appendix A

The purpose of this appendix is to prove the following.

######  Theorem A.1.

Let $G$ be a (not necessarily connected) reductive group and let
$\Gamma\subset G$ be a finite subgroup. Then there is a constant $f_{G}\geq 1$
with the following property: For every finite subgroup $\Gamma\subset G$ there
is an abelian, normal subgroup $A\subset\Gamma$ contained in a torus of
$G^{\circ}$ such that $[\Gamma:A]\leq f_{G}$.

We follow [Bre23], only making necessary adjustments to make the theorem work
for a general reductive group.

######  Definition A.2.

Let $T$ be a torus. Let $\Gamma\subset G$ be a finite group and $M$ be a
positive integer greater than 2. We say that a subgroup $F\subset\Gamma$ is an
$M$-fan if it is conjugate to a subgroup $F^{\prime}$ of $T$ such that for all
$\alpha,\beta\in\Phi$, $\alpha/\beta$ either attains $M$ different values on
$F^{\prime}$ or is reduced to 1 on $F^{\prime}$.

######  Remark A.3.

To clarify, any $M$-fan is by definition also an $M^{\prime}$-fan for any
$M^{\prime}\leq M$. Further, it follows from the definition that if $F$ is an
$M$-fan then $H_{F}$ is the weight kernel defined by setting the same
$\alpha/\beta=1$ as those which reduce to 1 on $F$. We also have that
$H_{H_{A}}=H_{A}$.

######  Lemma A.4.

Let $G$ be a reductive group. The following are equivalent.

  1. (1)

There is a constant $f_{G}\geq 1$ with the following property: For every
finite subgroup $\Gamma\subset G$ there is an abelian, normal subgroup
$A\subset\Gamma$ contained in a torus of $G^{\circ}$ such that $[\Gamma:A]\leq
f_{G}$.

  2. (2)

There is a positive integer $N=N(G)$ such that for any positive integer $M>N$
any finite subgroup $\Gamma\subset G$ contains a unique maximal $M$-fan
$\mathcal{F}$ such that $[\Gamma:\mathcal{F}]\leq N$.

###### Proof.

Let $\Gamma\subset G$ be an arbitrary finite subgroup.

If (2) holds, then $\Gamma$ contains a unique maximal $M$-fan, which is normal
and toral. Hence, with $f_{G}\coloneqq N$ we obtain (1).

Conversely, assume that (1) holds. Let $N\coloneqq f_{G}$, and let
$A\subset\Gamma$ be a normal toral subgroup such that $[\Gamma:A]\leq N$. Then
$\Gamma\subset N_{G}(A)=N_{G}(H_{A})$, where $H_{A}$ is a weight kernel for
the adjoint representation. We let $M>N$. If $F$ is an $M$-fan in $\Gamma$ and
$f\in F$, then we have that $f^{N}\in A\subset H_{A}$ since $[F:F\cap
A]\leq[\Gamma:A]\leq N$. Hence, for any
$\alpha_{1},\alpha_{2}\in\Phi(\mathfrak{g})$, if $\alpha_{1}/\alpha_{2}$
attains $M$ different values on $F$, then $(\alpha_{1}/\alpha_{2})(f^{N})$
attains at least two different values as we go trough $f\in F$. Furthermore,
since $M>N$, $\alpha_{1}/\alpha_{2}$ reduces to 1 on $F$ if and only if it
does so on $F\cap H_{A}$. This means that the weights determining $F\cap
H_{A}$ as a 2-fan in $\Gamma$ are the same as the weights determining $F$ as
an $M$-fan in $\Gamma$. Hence, $H_{F\cap H_{A}}=H_{F}$, which means that

$F\subset H_{F}=H_{F\cap H_{A}}\subset H_{A}.$

Since $F$ was arbitrary this implies that any $M$-fan in $\Gamma$ is contained
in $H_{A}$. Let $\mathcal{F}$ be the group generated by all $M$-fans in
$\Gamma$. This is an $M$-fan. Since $A\subset\Gamma\cap H_{A}$ we thus obtain
that

$[\Gamma:\mathcal{F}]\leq[\Gamma:\Gamma\cap H_{A}]\leq[\Gamma:A]\leq N.$

This means that (2) follows with $\mathcal{F}=\Gamma\cap H_{A}$. ∎

###  A.1. Proof of Theorem A.1

We proceed by induction on $\dim G$. If $\dim G=1$, then $G^{\circ}$ is a
torus which means that we can choose $A\coloneqq\Gamma\cap G^{\circ}$.

We now let $G$ be an arbitrary reductive group with $\dim G>1$ and we assume
Theorem A.1 for all reductive groups $H$ such that $\dim H<\dim G$. We keep
this assumption for the remainder of the appendix. Let also for the remainder
of the text $\Gamma\subset G$ be a fixed arbitrary finite subgroup. Let
$\mathcal{H}$ denote the set of all reductive subgroups $H$ of $G$ with same
rank as $G$. By Lemma A.4 the induction assumption also implies that we assume
the existence of $N(H)$ for any reductive group $H$ with $\dim H<\dim G$. We
define $N$ as

$\displaystyle N$ $\displaystyle\coloneqq\max_{H\in\mathcal{H}}\\{N(H)\\}.$

We see that $N$ is finite since there are a finite number of conjugacy classes
of reductive equal rank subgroups. To clarify this, there are a finite number
of conjugacy classes of connected equal rank subgroups, since these correspond
to closed subsystems of the root system of $G$. Furthermore, for any such
connected $H_{0}$, there are a finite number of groups $H$ such that
$H^{\circ}=H_{0}$, since $H$ must normalize $Z(H_{0})$, and
$[N_{G}(Z(H_{0})):C_{G}(Z(H_{0}))]$ is finite. We also fix an integer $M>N$,
which we may assume to be equal to $N+1$.

######  Lemma A.5.

An $M$-fan $F$ in $\Gamma$ not contained in $Z(G)$ is contained in a unique
maximal $M$-fan in $\Gamma$.

###### Proof.

Let $F$ be an $M$-fan in $\Gamma$ not contained in $\mathrm{Z}(G)$. Pick any
$f\in F$ not contained in $\mathrm{Z}(G)$. Since $F$ is abelian, we have that
$F\subset C_{\Gamma}(f)$. Thus, $F$ is an $M$-fan in $C_{\Gamma}(f)$. Since
$f\notin\mathrm{Z}(G)$ we have that $\dim C_{G}(f)<\dim G$. Thus, by the
ongoing induction assumption on $\dim G$ we see that $F$ is contained in a
unique maximal $M$-fan $\mathcal{F}$ in $C_{\Gamma}(f)$. Since $M$-fans are
abelian, any $M$-fan containing $F$ also centralizes $f$, which means it is
contained in $\mathcal{F}$. ∎

We say that an $M$-fan $F$ is associated to a noncentral element $g\in\Gamma$
if it is the unique maximal $M$-fan $C_{\Gamma}(g)$. We write $F_{g}$ for the
$M$-fan associated to $g$. In the sequel, we let

$Z\coloneqq Z(G)\cap\Gamma.$

This is an $M$-fan.

In the notation of the proof of Lemma A.5, since $F\subset\mathcal{F}$ and
both are abelian, we see that $\mathcal{F}\subset C_{\Gamma}(F)$. The group
$C_{\Gamma}(F)$ is reductive and of the same rank as $G$, which means by
induction that $\mathcal{F}$ is maximal in $C_{\Gamma}(F)$ and thus

(A.1) $[C_{\Gamma}(F):\mathcal{F}]\leq N(C_{\Gamma}(F))\leq N$

We can now prove some simple statements.

######  Lemma A.6.

Let $F$ be an $M$-fan such that $Z(G)\subsetneq F\subsetneq\mathcal{F}$, where
$\mathcal{F}$ is a maximal $M$-fan in $\Gamma$. The number $n_{F}$ of elements
$g$ of $\Gamma$ associated with $F$ is divisible by $|\mathcal{F}|$ and

$\frac{n_{F}}{|\mathcal{F}|}\leq N.$

###### Proof.

We can assume $n_{F}\geq 1$, since the statement is trivial otherwise. All
elements associated with $F$ lie in $C_{\Gamma}(F)$. Furthermore, if $g$ is
associated to $F$ and if $f\in\mathcal{F}$, then the element $gf$ is still
associated with $F$. To see this, note that we cannot have $gf\in Z(G)$, since
this would mean $C_{\Gamma}(f)=C_{\Gamma}(g)$. Since $\mathcal{F}\subset
C_{\Gamma}(f)$ and since $F$ is the maximal $M$-fan of $C_{\Gamma}(g)$, we see
that this cannot happen. Consider $F_{gf}$, the maximal $M$-fan of
$C_{G}(gf)$. We have that $F\subset C_{\Gamma}(gf)$, which means $F\subset
F_{gf}$. Since $\mathcal{F}$ is the unique maximal $M$-fan containing $F$, we
must have that $F_{gf}\subset\mathcal{F}$. This means that $F_{gf}$ must
commute with $f$ and therefore also $g$. We thus see that $F_{gf}\subset
C_{\Gamma}(g)$, whence $F_{gf}=F$. That is, $gf$ is associated with $F$.

The set of elements associated to $F$ are thus unions of cosets of
$\mathcal{F}$, and they all lie in $C_{\Gamma}(F)$. By Equation A.1, the index
of $\mathcal{F}$ in $C_{\Gamma}(F)$ is at most $N$, from which the statement
of the lemma follows. ∎

We get also get a counting formula for maximal $M$-fans in $\Gamma$.

######  Lemma A.7.

Let $\mathcal{F}$ be a maximal $M$-fan in $\Gamma$. The number of elements of
$\Gamma\setminus Z$ associated to $\mathcal{F}$ is

$n_{\mathcal{F}}=|C_{\Gamma}(\mathcal{F})|-|Z|.$

and $[C_{\Gamma}(\mathcal{F}):\mathcal{F}]\leq N$.

###### Proof.

The inequality on the index follows by the same argument as for Equation A.1.
Too see that $n_{\mathcal{F}}=|C_{\Gamma}(\mathcal{F})|-|Z|$, note that an
element $g\in\Gamma$ is associated to $\mathcal{F}$ if and only if
$\mathcal{F}\subset C_{\Gamma}(g)$, which is equivalent to $g\in
C_{\Gamma}(\mathcal{F})$. A maximal $M$-fan will contain $Z$, since if $F$ is
an $M$-fan in $\Gamma$, $ZF$ is an $M$-fan as well. Since $Z\subset
C_{\Gamma}(\mathcal{F})$, the statement of the lemma follows. ∎

We now begin to produce a formula for the number of elements in $\Gamma$,
which will allow us to finalize the proof of Theorem A.1. To do this, we split
the elements in $\Gamma$ into four sets,

$\Gamma=Z\cup\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}.$

Here $\Gamma_{1}$ consists of the noncentral elements associated to $Z$,
$\Gamma_{2}$ consists of the elements associated to an $M$-fan $F$ such that
$Z\subsetneq F\subsetneq\mathcal{F}$, where $\mathcal{F}$ is a maximal $M$-fan
containing $F$, and $\Gamma_{3}$ consists of the elements associated to a
maximal $M$-fan $\mathcal{F}$.

We begin by enumerating $\Gamma_{1}$. It is a union of cosets of $Z$. Indeed,
if $g\in\Gamma$ is associated to $Z$, then so is $zg$ for $z\in Z$, since
$C_{\Gamma}(g)=C_{\Gamma}(zg)$. Furthermore, $\Gamma$ permutes these cosets by
conjugation, since $\gamma gZ\gamma^{-1}=\gamma g\gamma^{-1}Z$, for all
$g,\gamma\in\Gamma$.

Let $S=G/[G,G]$ and let $\rho:G\to S$ be the quotient map. Let

$n\coloneqq|[G:G]\cap Z(G)|,$

a finite number. We claim that for any $g\in\Gamma$,

(A.2) $[N_{\Gamma}(gZ):C_{\Gamma}(g)]\leq n.$

To see this, note that for any $h\in N_{\Gamma}(gZ)$ there is a unique $z\in
Z$ such that $hgh^{-1}=gz$. This implies that $\rho(z)=1$, so $z\in[G:G]\cap
Z$. This explains the inequality in (A.2). Counting all elements in the
$\Gamma$-conjugacy class of the coset $gZ$, we see that

$|\text{$\Gamma$-orbit of
$gZ$}||Z|=\frac{|\Gamma||Z|}{N_{\Gamma}(gZ)}=|\Gamma|\frac{|C_{\Gamma}(g)|}{|N_{\Gamma}(gZ)|}\frac{|Z|}{|C_{\Gamma}(g)|}$
$=|\Gamma|\frac{1}{[N_{\Gamma}(gZ):C_{\Gamma}(g)][C_{\Gamma}(g):Z]}=|\Gamma|\frac{1}{\lambda},$

where
$\lambda\coloneqq\frac{1}{[N_{\Gamma}(gZ):C_{\Gamma}(g)][C_{\Gamma}(g):Z]}$.
By combining all such conjugacy classes we see that

$|\Gamma_{1}|=|\Gamma|\left(\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}+...+\frac{1}{\lambda_{k_{1}}}\right)$

where, for each $i=1,...,k_{1}$, we have that $\lambda_{i}\leq nN$.

We continue by counting the elements of $\Gamma_{2}$. If $F$ is an $M$-fan in
$\Gamma$ strictly contained in a maximal $M$-fan $\mathcal{F}$, we know that
if $n_{F}$ is the number of elements associated to $F$, then by Lemma A.6
$\frac{n_{F}}{|\mathcal{F}|}$ is an integer less than or equal to $N$. The
number of $M$-fans conjugate to $F$ is $\frac{|\Gamma|}{|N_{\Gamma}(F)|}$. We
have that

$[N_{\Gamma}(F):C_{\Gamma}(F)]\leq(\dim G)!$

since $N_{\Gamma}(F)$ permute the eigenspaces of $F$ and $C_{\Gamma}(F)$ are
the elements in $\Gamma$ preserving the eigenspace decomposition of $F$, and
$\dim\mathfrak{g}=\dim G$. Combining what we have above we get that the number
of $g\in\Gamma$ associated to an $M$-fan in the conjugacy class of $F$ is

$|\text{$\Gamma$-orbit of
$F$}|n_{F}=\frac{|\Gamma|n_{F}}{|N_{\Gamma}(F)|}=|\Gamma|\frac{|C_{\Gamma}(F)|}{|N_{\Gamma}(F)|}\frac{n_{F}}{|\mathcal{F}|[C_{\Gamma}(F):\mathcal{F}]}=|\Gamma|\frac{\nu}{\mu}$

where $\nu=\frac{n_{F}}{|\mathcal{F}|}$ and
$\mu=[N_{\Gamma}(F):C_{\Gamma}(F)][C_{\Gamma}(F):\mathcal{F}]$. Summming over
all conjugacy classes of $M$-fans we see that there are integers
$\nu_{i},\mu_{i}$ with $\nu_{i}\leq N$ and $\mu_{i}\leq N(\dim G)!$ such that

$|\Gamma_{2}|=|\Gamma|\left(\frac{\nu_{1}}{\mu_{1}}+\frac{\nu_{2}}{\mu_{2}}+...+\frac{\nu_{k_{2}}}{\mu_{k_{2}}}\right).$

Lastly, we count the elements in $\Gamma_{3}$. Let $\mathcal{F}$ be a maximal
$M$-fan. By Lemma A.7 we know the number of elements $n_{\mathcal{F}}$
associated to $\mathcal{F}$ is $|C_{\Gamma}(\mathcal{F})|-|Z|$. If we let
$\omega=[N_{\Gamma}(\mathcal{F}):C_{\Gamma}(\mathcal{F})]$ and
$q=[N_{\Gamma}(\mathcal{F}):Z]$, we see that the number of elements associated
to the conjugacy class of $\mathcal{F}$ is

$|\text{$\Gamma$-orbit of
$\mathcal{F}$}|(|C_{\Gamma}(\mathcal{F})|-|Z|)=|\Gamma|\left(\frac{|C_{\Gamma}(\mathcal{F})|}{|N_{\Gamma}(\mathcal{F})|}-\frac{|Z|}{|N_{\Gamma}(\mathcal{F})|}\right)=|\Gamma|\left(\frac{1}{w}-\frac{1}{q}\right).$

Summing over all conjugacy classes of all maximal $M$-fans we see that there
are integers $\omega_{i},q_{i}$, with $q_{i}$ dividing $\frac{|\Gamma|}{|Z|}$,
such that $\omega_{i}\leq(\dim G)!$,
$q_{i}=\omega_{i}[C_{\Gamma}(\mathcal{F}):Z]\geq 2\omega_{i}$ and

$|\Gamma_{3}|=|\Gamma|\left(\frac{1}{\omega_{1}}-\frac{1}{q_{1}}+\frac{1}{\omega_{2}}-\frac{1}{q_{2}}+...+\frac{1}{\omega_{k_{3}}}-\frac{1}{q_{k_{3}}}\right).$

We now combine our three enumerations to get the following formula for the
order of $\Gamma$

(A.3)
$|\Gamma|=|Z|+|\Gamma|\left(\sum_{i=1}^{k_{1}}\frac{1}{\lambda_{i}}+\sum_{i=1}^{k_{2}}\frac{\nu_{i}}{\mu_{i}}+\sum_{i=1}^{k_{3}}\left(\frac{1}{\omega_{i}}-\frac{1}{q_{i}}\right)\right).$

Here, $\lambda_{i},\nu_{i},\mu_{i},\omega_{i}$ are all positive integers
bounded above by $C=2nN(\dim G)!$ and $q_{i}$ divides $\frac{|\Gamma|}{|Z|}$.
Furthermore, since $1/\omega_{i}-1/q_{i}\geq 1/2\omega_{i}$ we see that
$k_{1}+k_{2}+k_{3}\leq C$. In particular, the $k_{i}$ are also bounded above
by $C$. We gather terms and obtain the following statement.

######  Lemma A.8.

If $\gamma\coloneqq|\Gamma|/|Z|$, then there are positive integers
$q_{1},...,q_{k}$ for some $k_{3}\geq 1$ such that

$\frac{1}{\gamma}=\left(\sum_{i=1}^{k_{3}}\frac{1}{q_{i}}\right)-\frac{b}{a},$

where $b/a$ is a reduced fraction and $b,a$ are integers both bounded by a
constant depending only on $G$.

###### Proof.

This is the conclusion of the counting arguments preceding the lemma. ∎

######  Lemma A.9.

Suppose that we have positive integers $a,b,\gamma,q_{i}$ with
$q_{i}\leq\gamma$ an equation of the form

$\frac{1}{\gamma}=\left(\sum_{i=1}^{k}\frac{1}{q_{i}}\right)-\frac{b}{a}.$

Then $\gamma\leq f(k,a)$, where $f:\mathbb{N}^{2}\to\mathbb{N}$ is some
function.

###### Proof.

This is [Bre23, Lemma 4.1]. ∎

###### Proof of Theorem A.1.

Let

$f_{G}\coloneqq\max\\{a,N(\dim G)!,f(k_{3},a)\\},$

where $f\colon\mathbb{N}^{2}\to\mathbb{N}$ is the function obtained from Lemma
A.8 and Lemma A.9.

Consider Equation A.3.

If $k_{3}=0$, then in Lemma A.8 we obtain that $1/\gamma=-b/a$, which implies
that $b=-1$ and thus $\gamma=\frac{|\Gamma|}{|Z|}=a$ which is bounded only in
terms of $G$. This means that $Z$ is of bounded index in $\Gamma$, and we can
thus choose $A=Z$.

If instead $k_{3}>0$ and $q_{i}=\gamma$ for some $i$, then for some maximal
$M$-beam $\mathcal{F}_{i}$ we have that
$[N_{\Gamma}(\mathcal{F}_{i}):Z]=q_{i}=\gamma=[\Gamma:Z]$, which means that
$\Gamma=N_{\Gamma}(\mathcal{F}_{i})$. We thus choose $A=\mathcal{F}_{i}$,
which is uniformly bounded since

$[\Gamma:\mathcal{F}_{i}]=[N_{\Gamma}(\mathcal{F}_{i}):C_{\Gamma}(\mathcal{F}_{i})][C_{\Gamma}(\mathcal{F}_{i}):\mathcal{F}_{i}]\leq
N(\dim G)!.$

Now assume that $q_{i}\neq\gamma$ for all $i$ and consider the non-negative
integer

$\gamma\left(\sum_{i=1}^{k_{3}}\frac{1}{q_{i}}\right)-1.$

If this is zero, then $\sum_{i=1}^{k_{3}}\frac{1}{q_{i}}=\frac{1}{\gamma}$
which, since $q_{i}\leq\gamma$ implies that $k_{3}=1$ and $q_{1}=\gamma$, a
contradiction. If it instead is positive, then $b>0$ and we can apply Lemma
A.9, in which case $\gamma$ is again bounded only in terms of $G$, this time
by $f(k_{3},a)$, and we again choose $A=Z$. ∎

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