# Multi-Modal Multi-Agent Optimization for LIMMS,
A Modular Robotics Approach to Delivery Automation

Xuan Lin1, Gabriel I. Fernandez1, Yeting Liu1, Taoyuanmin Zhu1, Yuki Shirai1,
and Dennis Hong1 1Xuan Lin, Gabriel I. Fernandez, Yeting Liu, Taoyuanmin Zhu,
Yuki Shirai and Dennis Hong are with RoMeLa (the Robotics and Mechanisms
Laboratory) at the University of California, Los Angeles, USA. {maynight,
gabriel808, liu1995, tymzhu, yukishirai4869<EMAIL_ADDRESS>
###### Abstract

In this paper we present a motion planner for LIMMS, a modular multi-agent,
multi-modal package delivery platform. A single LIMMS unit is a robot that can
operate as an arm or leg depending on how and what it is attached to, e.g., a
manipulator when it is anchored to walls within a delivery vehicle or a
quadruped robot when 4 are attached to a box. Coordinating amongst multiple
LIMMS, when each one can take on vastly different roles, can quickly become
complex. For such a planning problem we first compose the necessary logic and
constraints. The formulation is then solved for skill exploration and can be
implemented on hardware after refinement. To solve this optimization problem
we use alternating direction method of multipliers (ADMM). The proposed
planner is experimented under various scenarios which shows the capability of
LIMMS to enter into different modes or combinations of them to achieve their
goal of moving shipping boxes.

##  I Introduction

Modular self-reconfigurable robot systems have attracted the attention of
researchers [1, 2, 3]. The ability to transform relatively simple robot
modules to realize various forms and functions can be applied in areas such as
aerial or space robotics [4, 5] and more recently last-mile delivery [6].
Self-reconfigurable robots present challenging mechanical design, motion
planning, and control problems. As the number of robots scales up, issues
arise in communication, coordination, and decision making partially because it
can no longer be assumed to have accurate global information.

In recent work, we introduced LIMMS (Latching Intelligent Modular Mobility
System) as a modular approach to last-mile delivery [6], shown in Fig. 1.
LIMMS system is composed of individual modules that resemble a 6 degree-of-
freedom (DoF) arm but with wheels and a latching mechanism at both ends. As
such either side of LIMMS can act as the base depending on the need. In the
case of last-mile delivery, within the delivery truck, it is assumed that the
surface, as well as boxes, have anchor points LIMMS can attach the latch to.
By attaching one end to the walls within the vehicle LIMMS can carry and move
boxes by latching its free end to them like other manipulators. To actually
transport the box to the recipient’s door, four LIMMS can attach to a box and
use it as its body to walk itself there. Finally, to return LIMMS can self-
balance and wheel itself back. LIMMS introduces a challenging motion planning
problem with each LIMMS having multiple DoF. This requires kinematic
constraints which were rarely done in previous works [7, 8]. Moreover, each
LIMMS operate in different modes: a wheeled robot, a leg, or an arm. The
delivery package can also be transported in various ways: manipulation or
locomotion via acting as the body for a quadruped robot. This introduces
complicated mixing of discrete decisions and continuous constraints.

Figure 1: Top shows the various operational modes for LIMMS which will be
considered for the optimization formulation. Upper Left: 4 LIMMS attached to a
box in quadruped mode. Upper Right: 1 LIMMS self-balancing to move. This is
called free mode. Lower Left: 2 LIMMS anchored to surfaces in manipulator
mode. Lower Right: LIMMS attached to each other. This can be used in both
quadruped and manipulator mode. Bottom shows the joint axis and the latching
mechnanism to be integrated at the end effector.

Various approaches have been explored to resolve the reconfiguration planning
problem, such as graph search methods [7], reinforcement learning [9], or
optimization-based approaches [10]. We adopt optimization-based methods for
LIMMS. Among the optimization-based methods, mixed-integer programs (MIP) are
useful for discrete decision-making within multi-agent systems [11, 12].
However, the LIMMS motion planning problem includes nonlinear constraints such
as kinematics. Therefore, the problem becomes a mixed-integer nonlinear (non-
convex) program (MINLP). MINLPs are known to be computationally difficult when
the problem scale is large. Aside from directly applying commercialized
solvers, e.g., SCIP, there are generally two approaches that transform the
MINLP problems into MIPs using linear approximations of the nonlinear
constraints, or into nonlinear programs (NLPs) using complementary constraints
to replace binary variables. Unfortunately, as the problem scales up,
nonlinear constraints yield a large amount of piece-wise linear approximations
making the problem very slow [13], and complementary constraints tend to cause
infeasibility without a good initial guess [14].

In this paper, we implemented the alternating direction method of multipliers
(ADMM) to solve the MINLP problem, inspired by [15, 16]. ADMM decomposes the
problem into two sub-problems: an MIP problem and an NLP problem. By
decomposition, the problem scale of both MIP and NLP is reduced and becomes
more tractable. The logic constraints about the connections of LIMMS are
formulated into MIP while the nonlinear kinematics constraints are formulated
into NLP. We establish general rules for such a system to operate under and
use ADMM to explore possible approaches to resolve the given scenario, showing
the feasibility of our proposed method as well as the richness of the system.
To summarize, our contributions are as follows:

  1. 1.

Solved a unique type of problem that incorporates multi-agent locomotion and
manipulation.

  2. 2.

Proposed a formulation that can be generalized to solve such types of multi-
agent motion planning problems.

  3. 3.

Solved the proposed formulation with ADMM and demonstrated it on hardware.

##  II System Description

Hardware. To gain an intuition of the constraints in the optimization
formulation, the physical limitations and design of the first LIMMS hardware
prototype are detailed in this section and depicted in Fig. 1. LIMMS is a
symmetric 6 DoF robot. At full length, a single LIMMS unit stretches to about
0.75 m and weighs roughly 4.14 kg including batteries. From preliminary tests
in simulation with a target package payload of 2 kg, we determined off-the-
shelf Dynamixel motors from ROBOTIS were sufficient for the prototype. With
our custom gearbox, it reaches a peak velocity of 2 rad/s and a peak torque of
31 Nm. For interested readers previous recent work in [6] goes further in-
depth on the overall design.

One of the crucial aspects of LIMMS is its latching mechanism, which allows it
to attach either end to anchor points or itself. Latching will be used
frequently since it is how LIMMS transitions to different modes and grab
objects for manipulation. The current prototype proposed in [17] consists of a
radially symmetric multi-blade design. Fig. 1 (bottom right) depicts the latch
prototype. By the geometry of the blade and the anchor point hole pattern, the
latch mechanically self-aligns when it rotates.

The mechanism is designed to maximize robustness to misalignment in position
and orientation about its center axis, see [17]. This has a dual effect of
easing control effort and allowing for the latches to pull the box or itself
into the desired position and orientation without being fully positioned. If
LIMMS is kinematically constrained and cannot fully reach the target position
and orientation, there is slack created by the latch’s mechanical design. In
this sense when latching to an anchor point, the alignment does not need to
strictly satisfy its constraint.

Modes of Operation. Through latching, LIMMS can enter many different modes to
complete its tasks. For our proposed planner we only consider three different
modes as these will be sufficient for delivering packages. However, LIMMS has
the potential for many other modes, some of which are described in [6]. Fig. 1
depicts the 3 modes considered in our optimization formulation: 4 LIMMS
attached to a box are in quadruped mode, LIMMS attached to walls or surfaces
are in manipulator mode, a single LIMMS which wheels around or move like a
snake is in free mode. The last sub-figure in the top half, lower right shows
two LIMMS attached to the end of each other. This can be viewed as a sub-skill
accessible to all modes.

##  III Problem Formulation

In this section, we demonstrate the optimization formulation for LIMMS motion
planning. The objective function of the optimization problem is to minimize
the distance from the box center position to the target box position, i.e.
deliver the box to the goal. The constraints consist of two parts: logic,
which is formulated into constraints using integer variables, and kinematics
or dynamics, which is formulated into linear or nonlinear equations of motion.
As a result, the problem is an MINLP, which will be separated into an MIP and
NLP to be then solved with ADMM. Assume there are $B$ boxes and $L$ LIMMS.
Each box has $S_{B}=4$ anchor points at the center of each face. The
optimization is run from $t=1,...,T$ time steps. We use upper case letters to
indicate constants such as the number of boxes $B$, the number of anchor
points $S_{B}$, and use the lower case letters to index the quantities such as
$b=1,...,B$, $s_{b}=1,...,S_{B}$. $i$ is used to index the binary variables.
Upper case letters are also used as variable names. For example, $z_{B,i}$ are
binary variables associated with boxes. We state the assumptions made for this
formulation:

  1. 1.

The box does not rotate and the momentum is assumed to be balanced. This
simplifies the dynamics constraints. In practice, this minimizes the damage to
the contents in boxes during shipping.

  2. 2.

Multi-body dynamics of LIMMS are not enforced. This is to simplify the
constraints.

| Var | Dim | Description
---|---|---|---
blue!10 | $z_{B,i}$ | $[B,T]$ |  | Mode for box $b$ at time $t$. $i=\\{1:\text{stable object}$,
---
$2:\text{free object}$, $3:\text{manipulated}$, $4:\text{quadruped}\\}$
blue!10 | $z_{L,i}$ | $[L,T]$ |  | Mode for limb $l$ at time $t$. $i=\\{1:\text{free}$,
---
$2:\text{arm}$, $3:\text{add arm}$ $4:\text{leg}$, $5:\text{add leg}\\}$
blue!10 | $z_{S,i}$ | $[B,S_{B},L,T]$ |  | Mode of connection for anchor point $s_{b}$ on box $b$
---
to limb $l$ at time $t$
$i=\\{1:\text{empty}$, $2:\text{to arm}$, $3:\text{to leg}\\}$
blue!10 | $z_{W,i}$ | $[S_{w},L,T]$ |  | Mode of connection for anchor point $s_{w}$ on wall
---
to limb $l$ at time $t$ $i=\\{1:\text{empty}$, $2:\text{to arm}\\}$
blue!10 | $z_{Ac,i}$ | $[L_{pr},L_{po},T]$ |  | Mode for connection s.t. limb $l_{po}$
---
connects as an additional limb to $l_{pr}$
$i=\\{1:\text{not connected}$, $2:\text{connected}\\}$
blue!10 | $z_{Lc,i}$ | $[L_{pr},L_{po},T]$ |  | Mode for connection s.t. limb $l_{po}$
---
connects as an additional leg to $l_{pr}$
$i=\\{1:\text{not connected}$, $2:\text{connected}\\}$
blue!10 | $\textbf{p}_{B}$ | $[B,T]$ | Position of center of box $b$ at time $t$
blue!10 | $\textbf{R}_{B}$ | $[B,T]$ |  | Orientation of box $b$
---
blue!10 | $\textbf{c}_{B}$ | $[B,C,T]$ | Position of corner $c$ for box $b$ at time $t$
blue!10 | $\textbf{p}_{L}$ | $[J,L,T]$ | Position of joint $j$ of limb $l$ at time $t$
blue!10 | $\textbf{R}_{L}$ | $[J,L,T]$ |  | Rotation matrix of joint $j$ of limb $l$
---
blue!10 | $\textbf{f}_{L}$ | $[B,S,L,T]$ |  | Contact force at anchor point $s$ of box $b$
---
from limb $l$ at time $t$
blue!10 | a | $[L_{1},L_{2},T]$ |  | Normal vector of separating plane for $l_{1}$ and $l_{2}$
---
blue!10  Continuous | $b$ | $[L_{1},L_{2},T]$ |  | Offset of separating plane for $l_{1}$ and $l_{2}$
---
TABLE I: Table of optimization variables

All binary and continuous variables are summarized in Table I except those
that pertain to enforcing collision avoidance with the environment for
simplicity (i.e. binary variables $\delta_{Ba,i}$, $\delta_{Bg,i}$,
$\delta_{La,i}$, $\delta_{Lg,i}$, and the corresponding $\lambda\in[0,1]$
variables for convex combination). Note pr is short for previous, and po is
short for post.

###  III-A Integral Logic Constraints

Mode for Boxes. We define 4 modes for each box represented by 4 binary
variables: $z_{Bi}[b,t],i=1,...,4$ for the mode of box $b$ at $t$. Mode 1 is
stable object mode, where the box is supported by the ground. Mode 2 is free
object mode where the box is in the air subject to gravity. Mode 3 is
manipulated object mode where the box is connected to a manipulator. Mode 4 is
quadruped mode where the box is used as a robot body. We currently only allow
quadruped robot for walking. This can be relaxed to incorporate more solutions
such as simultaneously bipedal walking while manipulating boxes as in [18]. At
each $t$, a box is subject to 1 mode, such that: $\sum_{i=1}^{4}z_{Bi}[b,t]=1\
\forall b,\ \forall t$.

Mode for LIMMS. We define 5 modes for each LIMMS represented by 5 binary
variables: $z_{Li}[l,t],\text{where }i=1,...,5$ indicating the mode of LIMMS
$l$ at $t$. Mode 1 is free (wheeled) mode, where the corresponding LIMMS unit
moves on the ground like a Segway robot. Mode 2 is manipulation mode, where
LIMMS connects to one connection site on the wall and may connect to one box
to manipulate it. Mode 3 is add arm mode, where a LIMMS can connect to another
LIMMS to extend the length of the arm for a larger workspace. Mode 4 is leg
mode, where LIMMS connects to a box and serves as a leg. Mode 5 is add leg
mode, where it can connect to another LIMMS to extend the length of the leg
similar to mode 3. At any time step, LIMMS can only be in one mode, such that:
$\sum_{i=1}^{5}z_{Li}[l,t]=1\ \forall l,\forall t$.

TABLE II: Table of logic rules | # | Logic Rule Description | Mathematical Formulation
---|---|---|---
blue!10 | 1 | Box $b$ in quadruped mode, all 4 $s_{b}$ is connected to leg mode LIMMS. | $z_{B,4}[b,t]=1\implies\sum_{l}z_{S,3}[b,s_{b},l,t]=1\ \forall s_{b}$
blue!10 | 2 |  | Anchor point $s_{b}$ on box $b$ in leg mode, box $b$ is in quadruped mode.
---
$z_{S,3}[b,s_{b},l,t]=1\ \exists l,\ \exists S_{B}\implies z_{B,4}[b,t]=1$
blue!10 | 3 |  | Anchor point $s_{b}$ is connected to LIMMS $l$ as a leg, $l$ is in leg mode.
---
$z_{S,3}[b,s_{b},l,t]=1\implies z_{L,4}[l,t]=1$
blue!10  Quadruped | 4 |  | LIMMS $l$ in leg mode, it’s connected as a leg to one anchor point.
---
$z_{L,4}[l,t]=1\implies\sum_{s}\sum_{b}z_{S,3}[b,s_{b},l,t]=1$
blue!10 | 5 |  $b$ in manipulated mode, at least 1 $s_{b}$ is connected to 1 arm mode $l$. | $z_{B,3}[b,t]=1\implies\sum_{l}\sum_{s_{b}}z_{S,2}[b,s_{b},l,t]\geq 1$
blue!10 | 6 |  $s_{b}$ is connected to $l$ in arm mode, $l$ is in arm or add arm mode. |  $z_{S,2}[b,s_{b},l,t]=1\implies z_{L,2}{[}l,t{]}=1$ or $z_{L,3}{[}l,t{]}=1$
blue!10 | 7 |  $l$ is in arm mode, $l$ is connected to one $s_{w}$ on the wall or ground. | $z_{L,2}[l,t]=1\implies\sum_{s_{w}}z_{W,2}[s_{w},l,t]=1$
blue!10 | 8 |  $s_{w}$ on wall or ground, $s_{w}$ is connected to $l$, $l$ is in arm mode. | $z_{W,2}[s_{w},l,t]=1\implies z_{L,2}[1,t]=1$
blue!10 | 9 | LIMMS $l$ is in add arm mode, it’s connected to one other LIMMS. | $z_{L,3}[l,t]=1\implies\sum_{l_{pr}}z_{Ac}[l_{pr},l,t]=1$
blue!10 | 10 |  | LIMMS $l_{pr}$ is connected to $l_{po}$, $l_{pr}$ is in arm or add arm mode,
---
$l_{po}$ is in add arm mode.
| $z_{Ac}[l_{pr},l_{po},t]=1\implies\begin{cases}&z_{L,2}[l_{pr},t]=1\\\
&z_{L,3}[l_{po},t]=1\end{cases}$
---
blue!10  Manipulation | 11 |  | LIMMS $l_{pr}$ is connected to $l_{po}$, $l_{pr}$ cannot connect to any box.
---
$z_{Ac}[l_{pr},l_{po},t]=1\implies z_{S,2}[b,s_{b},l_{pr},t]=0$
blue!10  Free | 12 |  | Box $b$ is in stable or free object mode, all anchor points $s_{b}$ are empty.
---
$z_{B,1}[b,t]=1\ or\ z_{B,2}[b,t]=1\implies z_{S,1}[b,s_{b},l,t]=1\ \forall
s_{b}\ \forall l$

Mode for Anchor Points. Each box has 4 anchor points on each side face. We
define 3 modes for each anchor point by 3 binary variables
$z_{Si}[b,s_{b},l,t],\text{where }i=1,...,3$, denoting the connection mode for
anchor point $s_{b}$ on $b$ to $l$ at $t$. Mode 1 is empty mode, where $s_{b}$
on $b$ is not connected to $l$. Mode 2 is arm mode where $l$ connects to
$s_{b}$ as an arm. Mode 3 is leg mode where $l$ connects to $s_{b}$ as a leg.
Their summation has to be 1 at each time step:
$\sum_{i=1}^{3}z_{Si}[b,s_{b},l,t]=1\ \forall b,\ \forall s_{b},\ \forall l,\
\forall t$. In addition, at each time $s_{b}$ can connect to no more than 1
LIMMS, while a given LIMMS can connect to no more than 1 $s_{b}$ at its base
point. This introduces two more constraints:
$\sum_{b}\sum_{s_{b}}z_{Si}[b,s_{b},l,t]\leq 1\ \forall l,\ \forall t$ and
$\sum_{l}z_{Si}[b,s_{b},l,t]\leq 1\ \forall b,\ \forall s_{b},\ \forall t$.
$s_{b}$ on $b$ is associated with a physical connection. If latching is
enforced, the position and orientation of the base of LIMMS is constrained:

$\displaystyle\textbf{p}_{L}[j=0,l,t]=\textbf{p}_{B}[b,t]+\textbf{R}_{B}[b,t]\textbf{o}[s_{b}]$
(1)
$\displaystyle\textbf{R}_{L}[j=0,l,t]=\textbf{R}_{o}[s]\textbf{R}_{B}[b,t]$

Where $\textbf{o}[s]$ is the constant offset vector from the center of the box
$b$ to the anchor point $s_{b}$. $\textbf{R}_{o}[s_{b}]$ is the constant
rotation matrix from the box frame located at the geometric center of the box
to the $s_{b}$ frame located at anchor point $s_{b}$. This conditional
equality constraint can be enforced through big-M formulation such that if
$z_{S,i}=1$, (1) is enforced.

Each anchor site on the wall $s_{w}$ has two modes. Mode 1 is empty mode where
$s_{w}$ is empty, and mode 2 is manipulation mode where $l$ connects to
$s_{w}$ as an arm. Multiple $s_{w}$ can exist on the ground. Two binary
variables $z_{Wi}[s_{w},l,t],\text{where }i=1,2$ are used to represent those
modes. The associated mode constraints and physical connection constraints are
similar to the anchor points on the boxes. We also define binary variables for
additional connections between LIMMS as arms or legs:
$z_{Ac,i}[l_{pr},l_{po},t]$ or $z_{Lc,i}[l_{pr},l_{po},t]$, $i=1,2$. If
$z_{Ac,i}[l_{pr},l_{po},t]=1$, LIMMS $l_{po}$ connects as an additional arm to
LIMMS $l_{pr}$ at $t$. Similar for $z_{Lc,i}$.

Logic for Boxes as Robot Bodies. We define 4 logic rules for any box detailed
in rule $1-4$ in Table II. They constrain the boxes in quadruped mode and
LIMMS units connected to it. The gist is to ensure that several things happen
simultaneously: 1) $b$ is used as the robot body, 2) 4 LIMMS are its legs, and
3) connections happen between them. On the other hand, if no box is used as a
quadruped body, no LIMMS should be used as legs. This is enforced through
formulating the rules into a loop as shown in figure 2. If 1 LIMMS is used as
leg, it should connect to 1 anchor point on 1 of the boxes due to Logic 4, and
the corresponding box should be in quadruped mode due to Logic 2.

Each logic can be formulated as constraints between integer variables through
big-M formulation. For example, Logic $1$ can be written as
$1-M(1-z_{B,4}[b,t])\leq\sum_{l}z_{S,3}[b,s_{b},l,t]\leq 1+M(1-z_{B,4}[b,t])$
where $M$ is a large constant (usually $10^{5}$). Other constraints follow
similarly.

Figure 2: Implications of logic 1-12. Left half is associated with robot mode
while right half is associated with manipulation mode. Arrows are labeled with
specific logic rules. If not, the implication is mathematically correct. The
symbol $\Rightarrow\emptyset\Leftarrow$ means mutually exclusive.

Logic for Boxes as Manipulated Objects. We define 7 logic rules for boxes as
objects and are manipulated by LIMMS as arms. They again constrain the modes
for the box and LIMMS connected to it. The gist is to ensure that the
following happens simultaneously: 1) box is being manipulated by 1 or more
LIMMS, 2) 1 or more LIMMS operate in arm mode, 3) connections occur between
the arm and box, and 4) arm is connected to 1 anchor point on the wall or
ground. While most of the logic rules are single directional, the
bidirectional rules are enforced by formulating implicit loops. For example,
there is no explicit rules enforcing a box to be in manipulated object mode
(arrow from $8$ to $12$) if LIMMS arm is manipulating it (block $8$ in Fig.
2). However, block $8$ is mutually exclusive with block $4$ and $6$. This
means that if $8$ happens, $4$ and $6$ cannot happen, which means block $5$
and $7$ cannot happen (by reversing the direction of implication). This in
turn indicates that $12$, which is the complement of $5$ and $7$, has to
happen. Note that we did not include the constraints for additional arm or leg
mode in Fig. 2, but similar arguments carry over.

Logic for Boxes as Stable or Free Objects. Logic $12$ in Fig. II is defined
for boxes as stable or free object modes which states that if a box is stable
or free, all its anchors are empty.

###  III-B Continuous Constraints

Kinematics. Kinematics constraints are imposed for each LIMMS through a series
of linear constraints and bilinear constraints in the same fashion as [19]:

$\displaystyle\textbf{p}[j+1,l,t]=\textbf{p}_{L}[j,l,t]+\textbf{R}_{L}[j,l,t]\textbf{p}_{j+1,j}$
(2)
$\displaystyle\textbf{R}_{L}[j-1,l,t]\textbf{z}_{j-1,j}=\textbf{R}_{L}[j,l,t]\textbf{z}_{j,j}$
$\displaystyle\textbf{R}_{L}[j,l,t]\textbf{R}_{L}[j,l,t]^{T}=\textbf{I}$
$\displaystyle\textbf{R}_{L}[j,l,t]\ \text{represents a right handed frame}$

Where $\textbf{p}_{j+1,j}$ is the constant position vector of the next joint
as seen in the frame of the previous joint, and $\textbf{z}_{j-1,j}$ is the
constant orientation of the next joint as seen in the frame of the previous
joint, where ours are $\textbf{z}_{j-1,j}=[0,0,1]^{T}$.

Collision Avoidance with Environment. To enforce constraints such that LIMMS
and boxes does not collide with the environment, we model the environment into
discrete convex regions. All boxes and LIMMS have to stay within the convex
regions during the process. We also need to discriminate if the LIMMS or box
is making contact with the ground. This introduces additional binary variables
$\delta_{Ba,i}$, $\delta_{Bg,i}$, $\delta_{La,i}$, and $\delta_{Lg,i}$. Note
subscript $g$ stands for ground, and $a$ stands for air. If LIMMS and boxes
are within a convex region, the joint points of LIMMS and corners of boxes are
linear combinations of the vertices of the convex region:

$\textbf{p}=\sum_{v}\lambda_{v}\textbf{V}_{v},\ \ \sum_{v}\lambda_{v}=\delta,\
\ \lambda_{v}\in[0,1]$ (3)

Where p, $\lambda$ and $\delta$ are associated with either corner points of
the box $\textbf{c}_{B}$ or position of joints of LIMMS $\textbf{p}_{L}$ as
listed in Table I. $\lambda_{v}$’s represent the vertices of the convex
region. If one region is not selected, all $\lambda_{v}$’s are zero due to
$\delta=0$.

Collision Avoidance Between Agents. To enforce collision avoidance for LIMMS-
LIMMS and LIMMS-box contact, we use the formulation from [14] that uses
separating planes. For convex polygons, the two polygons do not overlap with
each other if and only if there exists a separating hyperplane
$\textbf{a}^{T}\textbf{x}=b$ in between [20]. That is, for any point
$\textbf{p}_{1}$ inside polygon 1 then $\textbf{a}^{T}\textbf{p}_{1}\leq b$,
and for any point $\textbf{p}_{2}$ inside polygon 2 then
$\textbf{a}^{T}\textbf{p}_{2}\geq b$. Our problem uses the following
constraints:

$\displaystyle\textbf{a}^{T}\textbf{p}_{L}[j,l_{1},t]\leq b,$
$\displaystyle\textbf{a}^{T}\textbf{p}_{L}[j,l_{2},t]\geq b$ (4)
$\displaystyle\textbf{a}^{T}\textbf{c}_{B}[c,b,t]\leq b,\ \
\textbf{a}^{T}\textbf{p}_{L}[j,l,t]$ $\displaystyle\geq b\ \forall t,\ \
\textbf{a}^{T}\textbf{a}\geq 0.5$

Where a and $b$ is the normal vector and offset for planes associated with the
specific pair. $0.5$ is just an arbitrary nonzero number that we choose, as a
does not necessarily need to be a unit vector. With this method, we enforce
collision avoidance purely through inequality constraints and avoid using
complementary constraints such as [21].

Dynamics for Box. The dynamics are required for the agent to generate strictly
feasible motions. However, enforcing dynamics for each LIMMS is expensive
given its high DoF. We only enforce dynamics for the boxes. This serves as two
purposes. First, it allows the system to generate dynamic motions such as
throwing or jumping. Second, it allows the system to select motion plans based
on the box weight.

Figure 3: Results for 5 experiments. The green rectangle represents the
initial region in which LIMMS is constrained in. LIMMS is trying to get the
box to the magenta region, where LIMMS is allowed to move in as well.: 1)
LIMMS picks up a box and throws it towards the goal. 2) LIMMS lifts box in
manipulator mode and switches to quadruped mode to jump out of the vehicle.
3a) LIMMS attempts to use manipulator mode to reach goal. 3b) LIMMS climbs
onto a step using quadruped mode. 4) LIMMS sends the box towards the goal
using dual double arm manipulation to enlarge the workspace.

When the box is in stable object mode, the gravity is compensated for by the
ground. Additionally, LIMMS can only apply reaction forces to the box when it
is connected through the anchor points on the box. We define the reaction
force on the end effector of a LIMMS $l$ to the box $b$ as
$\textbf{f}_{i}[b,s,l,t]$, where the index $s$ indicates that the force is
through the anchor point $s_{b}$. When LIMMS connects to the box as a leg,
$\textbf{f}_{i}$ serves as the contact force on the ground, while when LIMMS
connects to the box as an arm, $\textbf{f}_{i}$ serves as the contact force to
grasp the box. The box dynamics are:

$m\ddot{\textbf{p}}_{box}[b,t]=\sum_{s=1}^{4}\sum_{l=1}^{L}\textbf{f}_{i}[b,s,l,t]-m\textbf{g}(1-z_{B,1}[b,t])$
(5)

$\sum_{s=1}^{4}\sum_{l=1}^{L}(\textbf{p}_{L}[j=6,l,t]-\textbf{p}_{B}[b,t])\times\textbf{f}_{i}[b,s,l,t]=0$
(6)

$\textbf{f}_{i}[b,s,l,t]$ only exists when $l$ is connected to the box:

$\textbf{f}_{i}[b,s,l,t]=0\ \ \text{if}\ \ z_{S,1}[b,s_{b},l,t]=1$ (7)

When $l$ connects to the box as leg mode, f represents the contact force from
the ground. Therefore, f needs to satisfy the friction cone constraint:

$\textbf{f}\in Cone\ \ \text{if}\ \ z_{S,i}[b,s_{b},l,t]=1\ \text{and}\
\delta_{Lg}[j=6,l,\exists p,t]$ (8)

Since LIMMS has a loading capacity, we enforce the max norm constraint on any
contact force: $||\textbf{f}||\leq f_{max}$. Note that there is a contact
moment across the latching. Missing this moment results in the box incapable
of being manipulated with a single contact (moment balance will be violated).
For simplicity, we fix the box orientation: $\textbf{R}=\textbf{I}$. This can
be justified as in manipulated mode, the contact moment is sufficient to keep
the orientation of the box, and in quadruped mode, the four ground support
points is sufficient to keep the body orientation.

Support Polygon. The balance of moment constraint should guarantee the
stability of the quadruped body. However, there are already many nonlinear
constraints such as kinematics. To simplify the NLP formulation, one approach
is to enforce a simple stability constraint in replacement of the moment
balance constraint. Since the rotation matrix is fixed, we can simply enforce
that the end effector of LIMMS stays to one side of the body which guarantees
that the body’s center of mass lies within the support polygon of the foot.

Stability of LIMMS. One drawback of our formulation is that we do not include
the dynamics of LIMMS. As a compensation, there should be a constraint to
guarantee the stability when LIMMS is on the ground and tries to reach an
anchor point. We just assume that we have many latching points on the ground
so realizing this motion is relatively simple. Therefore, when LIMMS is in
free mode, we enforce that the base stays on the ground and give a speed
constraint:

$||\textbf{p}_{L}[j,l,t+1]-\textbf{p}_{L}[j,l,t]||\leq\Delta\textbf{P}\ \
\text{if}\ \ \delta_{Lg}[j=0,l,p,t]=1$ (9)

Continuity of Connection. One suboptimal solution to avoid is having LIMMS
frequently latch on and off an anchor point. We enforce equality constraints
for binary variables within a range $z_{S,i}[t]=...=z_{S,i}[t+n]$. We usually
choose $n$ between $3$ and $5$, decided based on the speed of latching.

##  IV ADMM Formulation

Collecting the constraints defined previously, the problem to solve becomes:

$\displaystyle\underset{z,\ \delta,\ \textbf{p},\ \textbf{R},\ \textbf{f},\
\lambda,\ \textbf{a},\ b}{\text{minimize}}\quad f_{obj}$ (10) subject to Mixed
integer constraints: Logic rules 1-12 collision avoidance with the environment
(3) Dynamics constraints: (5), (7), (8) Stability of LIMMS on ground (9)
Nonlinear constraints: Kinematics (2) Collision avoidance between agents (4)
Dynamics constraints (6)

Where $f_{obj}$ is a quadratic equation that minimizes the distance of the box
to the goal position. The variables and constraints of problem (10)
incorporate discrete and continuous variables with linear and nonconvex
(bilinear) constraints. This results in an MINLP. The commercial solvers tend
to perform non-satisfactory in this type of problem. There are generally two
approaches that convert this type of problem: an MICP using convex envelope
relaxations for nonlinear constraints [19] or conversion of the discrete
variables into continuous ones through complementary formulation [22]. MIPs
with convex envelopes tend to solve slowly when the problem scales up. Since
there are many discrete variables in this problem, complementary formulations
will be numerically difficult [16]. In this paper, we adopt the ADMM. ADMM
separates the problem into two sub-problems. Although those sub-problems have
different constraints, ADMM iterates between sub-problems such that constraint
$1$ which may not appear in sub-problem $2$ will be implicitly enforced as the
iteration proceeds. In the end, sub-problems will reach a consensus meaning
their solutions are close to each other. This procedure is detailed in
Algorithm 1. In our problem, the logic rule constraints are resolved through
MIPs, while the nonlinear kinematics and collision avoidance constraints are
resolved through NLPs. Similar to [15], we first make copies
$\textbf{var}_{2}$ of the variables
$\textbf{var}_{1}=[z,\delta,\textbf{p},\textbf{R},\textbf{f},\lambda,\textbf{a},b]$.
Represent the feasible set of mixed-integer constraints through $0-\infty$
indicator function by $\mathcal{I}_{M}$ and the nonlinear constraints by
$\mathcal{I}_{N}$. The consensus problem between MIP and NLP is:

$\displaystyle\underset{\textbf{var}_{1}\ \textbf{var}_{2}}{\text{minimize}\ \
}f_{obj}+\mathcal{I}_{M}(\textbf{var}_{1})+\mathcal{I}_{N}(\textbf{var}_{2})$
(11) $\displaystyle s.t.\ \ \textbf{var}_{1}=\textbf{var}_{2}$

The constraints are moved to the objective function through the indicator
function. Applying ADMM [23] to the Lagrangian $\mathcal{L}$ of (11) results
in three iterative operations:

$\displaystyle\textbf{var}_{1}^{i+1}=\text{argmin}_{\textbf{var}_{1}}\mathcal{L}(\textbf{var}_{1}^{i},\
\textbf{var}_{2}^{i},\ \textbf{w}^{i})$ (12a)
$\displaystyle\textbf{var}_{2}^{i+1}=\text{argmin}_{\textbf{var}_{2}}\mathcal{L}(\textbf{var}_{1}^{i+1},\
\textbf{var}_{2}^{i},\ \textbf{w}^{i})$ (12b)
$\displaystyle\textbf{w}^{i+1}=\textbf{w}^{i}+\textbf{var}_{1}^{i}-\textbf{var}_{1}^{i}$
(12c)

Where w is the dual variable of the Lagrangian of (11). In (12), (12a) solves
the MIP problem:

$\displaystyle\underset{\textbf{var}_{1}}{\text{minimize}}\quad||\textbf{var}_{1}^{i}-\textbf{var}_{2}^{i}+\textbf{w}^{i}||_{\textbf{W}_{\text{MIP}}^{k}}$
s.t. Mixed-integer constraints in (10)

In the next step, (12b), solves the NLP:

$\displaystyle\underset{\textbf{var}_{2}}{\text{minimize}}\quad||\textbf{var}_{2}^{i}-(\textbf{var}_{1}^{i+1}+\textbf{w}^{i}))||_{\textbf{W}_{\text{NLP}}^{k}}$
s.t. Nonlinear constraints in (10)

And the next step, (12c), updates the dual variable w. To finish one
iteration, the weights for MIP, $W_{\text{MIP}}$, the weights for NLP,
$W_{\text{NLP}}$, and the dual variable w, are updated with line $6-7$ in
Algorithm 1. Within one iteration, (12a), (12b), (12c) are solved in
succession. This iterative procedure continues until the discrepancy between
the MIP solutions and the NLP solutions
$\boldsymbol{\theta}=\textbf{var}_{1}^{i}-\textbf{var}_{1}^{i}$ are lower than
the user-set error threshold $\boldsymbol{\theta}_{th}$.

It is well known that ADMM has convergence guarantees for convex problems and
can significantly improve the solving speed. However, for complex MINLPs,
there is no convergence guarantee. In this problem, both MIP and NLP can be
slow and expensive. We avoid explicitly placing complementary constraints to
represent discrete modes as [15] did, since it hinders convergence for NLP.
However, NLP does need some information on discrete variables as it needs to
reason connections and turn variables such as f on or off accordingly. After
solving the MIPs, we directly use the solutions of $z_{S,i}$, $z_{W,i}$ in the
NLP step to enforce connections. This improves the precision of the NLP step
and the overall precision of consensus. The price to pay is an increase in
difficulty for NLP solvers to find solutions.

Algorithm 1 $\operatorname{ADMM\ for\ LIMMS}$

Input $\rho$, $\textbf{W}_{\text{MIP}}^{0}$, $\textbf{W}_{\text{NLP}}^{0}$,
$\textbf{w}^{0}$, $\textbf{var}_{2}^{0}$, $\boldsymbol{\theta}_{th}$

1: Initialization $i=1$

2: while $\boldsymbol{\theta}>\boldsymbol{\theta}_{th}$ and $i<i_{max}$ do

3: Compute $\textbf{var}_{1}^{i+1}$ via $~{}\eqref{eqn:MIP_step}$

4: Compute $\textbf{var}_{2}^{i+1}$ via $~{}\eqref{eqn:NLP_step}$

5:
$\textbf{w}^{i+1}\longleftarrow\textbf{w}^{i}+\textbf{var}_{1}^{i}-\textbf{var}_{1}^{i}$

6:
$\textbf{W}_{\text{MIP}}^{k+1}\longleftarrow\rho\textbf{W}_{\text{MIP}}^{k}$,
$\textbf{W}_{\text{NLP}}^{k+1}\longleftarrow\rho\textbf{W}_{\text{NLP}}^{k}$

7: $\textbf{w}^{i+1}\longleftarrow\textbf{w}^{i}/\rho$

8:
$\boldsymbol{\theta}\longleftarrow\textbf{var}_{1}^{i}-\textbf{var}_{1}^{i}$

9: $i=i+1$

10: return $\textbf{var}_{2}^{i}$

##  V Results

Figure 4: Convergence of mean and max residual $r_{pB}$, $r_{pL}$, $r_{RL}$
for experiment 1-4. Solid lines denote mean residuals and dashed lines denote
max residuals.

We performed 5 numerical experiments to evaluate the performance of the
proposed formulation and ADMM algorithm. For all experiments, the MIP
formulation was solved with Gurobi 9, and the NLP formulation was solved
through Ipopt on a Intel Core i7-7800X 3.5GHz $\times$ 12 machine. We solved
all the scenarios for 15 iterations. Other parameters we have chosen are:
$\Delta t=1sec$, $f_{max}=20N$, max moving speed for LIMMS on the ground of
$0.5m/s$, and a $30cm$ cube box. The 6 Dof LIMMS unit lengths are $5cm$,
$7.4cm$, $33cm$, $7.8cm$, $33cm$, $7.4cm$, $5cm$, respectively for each link
starting at one end. Fig. 3 shows the MATLAB visualization of the first 4
experiments. Animations of all tests are included in the attached video
available at https://www.youtube.com/watch?v=RH9gMOK24L0.

1) Throwing. In the first experiment, we placed 1 LIMMS and 1 box in the scene
and provided 1 anchor point on the wall. The goal is set higher than a LIMMS
unit can reach. The solver gives a solution where the LIMMS unit connects to
the wall and picks up the box, then throws the box to the goal. This simple
test shows the ability of the solver to generate manipulation motions using
dynamics.

2) Jumping. In the second experiment, we placed 4 LIMMS and 1 box on a raised
platform or inside a truck and set the goal position to be lower on the
ground. In addition, 2 anchor points are provided on the wall. The planner
provides the solution where 1 LIMMS latches itself onto the wall and lifts the
box. Then the other 3 LIMMS connects to the anchor points on the sides of the
box which enters quadruped mode. This quadruped robot then jumps down the step
to reach the goal. The solver automatically changes modes from manipulation to
quadruped.

3a,b) Weight Lifting. In the third experiment, we investigate the behavior
change due to a change in the weight of the box. We place 4 LIMMS and 1 box
down a ledge and set the goal to be above the ledge. Two anchor points on the
wall are provided. First, we set the box weight to be 0.5kg. The planner
provides a manipulation trajectory where two LIMMS connect to the wall and
lift the box to move it to the goal. We then increase the box weight to 7kg.
In this case, if we force LIMMS to manipulate the object, the planner returns
infeasible. The feasible trajectory returned by the solver set 3 LIMMS to be
anchored onto the ledge and box, while 1 leg stays down from the ledge to push
the body up onto the platform. As the box weight exceeds the capacity of dual
arm manipulation, a quadruped motion is necessary to lift the 7kg box.

4) Manipulation with Double-dual Arms. In the fourth experiment, we again
place 4 LIMMS and 1 box in the same scenario, below a desired platform or
step. Two anchor points are provided on the wall. However, we enlarged the
width of the truck such that 1 LIMMS cannot reach the box from the wall. In
this case, the solver connected a second LIMMS as an additional arm to the
first which is connected to the wall, allowing the arms to reach the box and
perform dual arm manipulation.

5) Quadruped Walking with Refinement. In the fifth experiment, we placed 4
LIMMS and a box on flat ground and set the goal to be at the same elevation
but separated by a distance. LIMMS moved to the box and assembled a quadruped
robot which then moved towards the goal. Furthermore, since there is no gait
optimization in our problem formulation, we used an additional planner similar
to [24] but included 6 DoF kinematics to provide the quadruped walking motion
to the goal. This experiment demonstrated that, although the planner may give
rough trajectories, they can be further refined with another planner to
correct kinematics discrepancies and gait cycles. This readies it to be
implemented on the hardware.

6) Box Lifting on Hardware In this experiment, one LIMMS is initially anchored
to the ground. There is another anchor point on the wall. The objective is to
lift the box higher. If LIMMS lifts the box with the other end anchored to the
ground, the kinematics quickly becomes infeasible. The planner instead lets
LIMMS first anchor to the box, then anchor to the wall from the box. The LIMMS
can easily lift the box higher with the other end anchored to the wall. The
hardware implementation is included in the video.

7) Convergence Define the residual to be the mismatch between the MIP and NLP
solutions. The mean residuals for $\textbf{p}_{B}$, $\textbf{p}_{L}$ and
$\textbf{R}_{L}$ are the mean value of all the norms:

$\displaystyle r_{pB}[i]=\underset{b}{\text{mean}}\
||\textbf{p}_{B,\text{MIP}}[b,i]-\textbf{p}_{B,\text{NLP}}[b,i]||$ (13a)
$\displaystyle r_{pL}[i]=\underset{j,\ l}{\text{mean}}\
||\textbf{p}_{L,\text{MIP}}[j,l,i]-\textbf{p}_{L,\text{NLP}}[j,l,i]||$ (13b)
$\displaystyle r_{RL}[i]=\underset{j,\ l}{\text{mean}}\
||\text{Vec}(\textbf{R}_{L,\text{MIP}}[j,l,i]-\textbf{R}_{L,\text{NLP}}[j,l,i])||$
(13c)

The max residual is the maximal value of all the norms. Figure 4 depicts the
change of mean and max residuals as a function of time. ADMM generally showed
decent average consensus after iteration 10, where $r_{pB}$ or $r_{pL}$
usually converges to cm-mm level and $r_{RL}$ usually less than 0.1 rad. The
maximal residual can sometimes be large. If we put the MIP solutions on the
real hardware, we can run another kinematics refiner to solve the kinematics
to ensure that the nonlinear constraints are strictly satisfied.

The number of variables, constraints and time cost for solving the experiments
above are listed in Table III. Generally, the NLP portion is the more
challenging portion of ADMM, since it includes kinematic and collision
avoidance constraints. To speed up the solving process, some linear
constraints in can be moved into the MIP formulations. This will speed up the
NLP solver but the residual may increase.

# | T | # of variables | # of constraints | Time in Minutes
---|---|---|---|---
MIP | NLP | MIP | NLP | T-MIP | T-NLP | T-Total
1 | 10 |  | 4857 cont.
---
670 bin.
1777 | 3623 |  | eq. 582
---
ineq. 5206
0.88 | 7.17 | 8.05
2 | 15 |  | 25917 cont.
---
3150 bin.
8787 | 34545 |  | eq. 621
---
ineq. 28990
129 | 24 | 153
3a) | 15 |  | 25917 cont.
---
3150 bin.
8787 | 33102 |  | eq. 1686
---
ineq. 28990
41 | 107 | 148
3b) | 10 |  | 17277 cont.
---
2100 bin.
5857 | 22032 |  | eq. 1131
---
ineq. 19180
5 | 50 | 55
4 | 15 |  | 25917 cont.
---
3690 bin.
8787 | 35924 |  | eq. 606
---
ineq. 31342
19 | 244 | 263
TABLE III: Solving time for experiment 1-4. Note: Far left column in ascending
order is: 1 for Throw, 2 for Jump, 3a for Lifting with Arm, 3b for Lifting
with Climb, and 4 for Double-dual Arm.

##  VI Conclusion and Future Work

This paper presents an optimization based motion planner for the multi-agent
modular robot system LIMMS. We demonstrated solving the proposed formulation
with ADMM. The results show how LIMMS autonomously coordinates between
different modes and generates trajectories of the system under different
situations. With proper refinement, the trajectories can be implemented on the
hardware.

It is worthwhile to mention that due to the separating plane for collision
avoidance, the NLP part of the problem takes a long time to solve. In the
previous work [14], we used data-driven methods to solve those constraints
fast online. A similar approach can be adopted here for future work.

Future work includes more testing of the ADMM design. Since the NLP portion of
our implementation of ADMM can be slow and sometimes difficult to converge,
further decompositions of the NLP to smaller problems may be helpful.
Additionally, some of the logic variables may be simplified. For example, the
mode of the box can be blended such that the box can have arms and legs
simultaneously. Other future work include development of a coordinated control
algorithm for larger scale implementations of LIMMS.

## Acknowledgements

The authors would like to thank LG Electronics for sponsoring this research
and giving useful feedback. We would also like to thank other lab members of
RoMeLa who have contributed to LIMMS development: Justin Quan, Colin Togashi,
and Samuel Gessow.

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# On Grothendieck’s section conjecture for curves of index $1$

Giulio Bresciani  Scuola Normale Superiore
Piazza dei Cavalieri 7
56126 Pisa
Italy<EMAIL_ADDRESS>
###### Abstract.

We prove that every hyperbolic curve with a faithful action of a non-cyclic
$p$-group (with a few exceptions if $p=2$) has a twisted form of index $1$
which satisfies Grothendieck’s section conjecture. Furthermore, we prove that
for every hyperbolic curve $S$ over a field $k$ finitely generated over
$\mathbb{Q}$ there exists a finite extension $K/k$ and a finite étale cover
$C\to S_{K}$ such that $C$ satisfies the conjecture.

Curves are smooth, projective and geometrically connected over a base field
$k$.

Given a hyperbolic curve $C$ over a field $k$ finitely generated over
$\mathbb{Q}$, there is a short exact sequence of étale fundamental groups

$1\to\pi_{1}(C_{\bar{k}})\to\pi_{1}(C)\to\operatorname{Gal}(\bar{k}/k)\to 1.$

Denote by $\Pi_{C/k}(k)$ the set of sections of this short exact sequence
modulo the action by conjugation of $\pi_{1}(C_{\bar{k}})$, there is a natural
map

$C(k)\to\Pi_{C/k}(k).$

In 1983, in a famous letter to Faltings [Gro97], Grothendieck conjectured that
this map is bijective; this statement is known as the _section conjecture_.
Injectivity was already known to Grothendieck and follows from the Mordell-
Weil theorem; the hard part is surjectivity. Some cases of the conjecture are
known, let us briefly recall them.

  * •

The conjecture holds if $k$ has an embedding $k\subset\mathbb{R}$ such that
$C(\mathbb{R})=\emptyset$. The literature contains several proofs starting
with Huisman in 2001 [Hui01, Proposition 4.2 (7.)] [Moc03, Corollary 3.13]
[Sti10, Appendix A] [Pá11] [Wic14]. The result was probably known even before
Huisman, though; exact attribution is unclear.

  * •

In 2008, J. Stix proved it in the case in which $k=\mathbb{Q}$ and $C$ maps to
a non-trivial Brauer-Severi variety [Sti10] (see [Bre22] for a
generalization).

  * •

Later in 2008, D. Harari and T. Szamuely proved it in the case in which the
cohomology class of $\underline{\operatorname{Pic}}^{1}_{C}$ in
$\operatorname{H}^{1}(k,\underline{\operatorname{Pic}}^{0}_{C})$ is not in the
maximal divisible subgroup [HS09].

  * •

In 2010, R. Hain proved it for the generic fiber of the universal curve over
$\mathcal{M}_{g}$, $g\geq 5$, over a field finitely generated over
$\mathbb{Q}$ [Hai11].

Recall that the index of a curve $C$ is the greatest common divisor of the
degrees of the residue fields of the closed points of $C$. All the results
above share a common feature: they are all about curves of index larger than
$1$. This is obvious for curves without real points and for the results of
Stix and Harari-Szamuely; it is the so-called Franchetta conjecture – proved
by Beauville (unpublished) and Arbarello-Cornalba [AC87] – for the result of
Hain. While A. Tamagawa showed that it is sufficient to prove the conjecture
for curves without rational points [Sti13, Corollary 101], we still need to
work with curves of index $1$.

It is not by chance that all the available results are about curves of index
larger than $1$: at least in the first three cases listed above, it can be
checked that the authors prove that the _abelianized_ fundamental group (that
is, the étale fundamental group of $\underline{\operatorname{Pic}}^{1}_{C}$)
has no Galois sections [Sti13, Chapter 3], even if they do not state it
explicitly. If a curve has index $1$, then
$\underline{\operatorname{Pic}}^{1}_{C}(k)\neq\emptyset$ and hence the
abelianized fundamental group has a Galois section.

In order to study the conjecture for curves of index $1$, we need to rely more
on the anabelian nature of the fundamental group.

### Curves of index 1

If we fix any finite group $G$, the conjecture for all curves with a free
action of $G$ over all finitely generated extensions of $\mathbb{Q}$ implies
the full conjecture; this is an easy consequence of well known facts [Sti13,
Corollary 107, Proposition 111]. Because of this, working with curves that
have many automorphisms is not restrictive.

We prove that every hyperbolic curve with a faithful action of a non-cyclic
$p$-group (with a few exceptions if $p=2$) has a twisted form of index $1$
which satisfies the conjecture.

######  Theorem 1.

Let $C$ be a hyperbolic curve over a field $k$ finitely generated over
$\mathbb{Q}$ with a faithful action of a finite, non-cyclic $p$-group $G$ for
some prime $p$.

If $p=2$, $G$ has a cyclic subgroup of index $2$ and the action is not free,
furthermore assume that $G$ is not dihedral and that the genus is odd.

There exists a finitely generated extension $K/k$ and a twisted form
$\mathfrak{C}$ of $C_{K}$ such that $\mathfrak{C}$ has index $1$ and
$\Pi_{\mathfrak{C}/K}(K)=\emptyset$.

There is a birational version of the section conjecture which is arguably more
approachable, and more is known about it [Koe05] [EW10] [Hos14] [Sti15] [ST21]
[Bred]. In particular, H. Esnault and O. Wittenberg have proved a result about
curves of index $1$, let us describe it.

Let $C$ be a curve over a number field $k$. Denote by $G_{k(C)}^{\rm[ab]}$ the
Galois group of $\bar{k}(C)^{\rm ab}/k(C)$, where $\bar{k}(C)^{\rm ab}$ is the
maximal abelian subextension of $\overline{k(C)}/\bar{k}(C)$. There is a
natural projection $G_{k(C)}^{\rm[ab]}\to\operatorname{Gal}(\bar{k}/k)$; we
might think of it as a birational, abelianized version of
$\pi_{1}(C)\to\operatorname{Gal}(\bar{k}/k)$. Assuming that the Tate-
Shafarevich group of the Jacobian of $C$ is finite, H. Esnault and O.
Wittenberg have proved that $C$ has index $1$ if and only if
$G_{k(C)}^{\rm[ab]}\to\operatorname{Gal}(\bar{k}/k)$ is split.

While at first glance Theorem 1 might seem in conflict with this result, they
are actually perfectly compatible. In fact, if we believe in Grothendieck’s
anabelian philosophy and $C$ is a hyperbolic curve of index $1$ with
$C(k)=\emptyset$, then we should expect
$\pi_{1}(C)\to\operatorname{Gal}(\bar{k}/k)$ to be non-split even though the
_abelianized_ fundamental group has a splitting induced by some point of
$\underline{\operatorname{Pic}}^{1}_{C}(k)\neq\emptyset$ (ignoring the
“birational” side of the story for clarity’s sake).

### Étale covers

Recall that, if $C\to S$ is a finite étale cover and the section conjecture
holds for $C_{K}$ _for all finite extensions $K/k$_, then the section
conjecture holds for $S$ [Sti13, Proposition 111]. Using the same methods as
in Theorem 1, we prove that every hyperbolic curve, after a finite extension
of the base field, has a finite étale cover which satisfies the conjecture.

######  Theorem 2.

Let $S$ be a hyperbolic curve over a field $k$ and $\ell$ a prime. Assume
either that $k$ is finitely generated over $\mathbb{Q}$, or that it is a
finite extension of $\mathbb{Q}_{p}$ with $p\neq\ell$. There exists a finite
extension $K/k$ and a finite étale $\left(\mathbb{Z}/{\ell}\right)^{2}$-cover
$\mathfrak{C}\to S_{K}$ such that $\Pi_{\mathfrak{C}/K}(K)=\emptyset$.

Notice that in the statement of Theorem 2 $K/k$ is a _finite_ extension, while
in Theorem 1 $K/k$ is _finitely generated_ : we obtain a better control over
$K/k$ in exchange for losing control over the index of $\mathfrak{C}$.

### Acknowledgements

I would like to thank A. Vistoli for many useful discussions: this paper would
not exist without the frequent brainstorming sessions I had with him. I would
also like to thank J. Stix for many useful comments which have substantially
improved the exposition of the paper. Among other things, he corrected a minor
mistake in the proof of Theorem 2, he told me about the isomorphism
$\mathbb{Z}/{2}\ast\mathbb{Z}/{2}\simeq\mathbb{Z}\rtimes\mathbb{Z}/{2}$ (see
Lemma 13), he provided an alternative proof of Proposition 5 (see Remark 6),
and a discussion with him inspired a much shorter proof using cohomological
dimension of Proposition 9 in the case of curves (unfortunately, I was not
able to adapt it to the case of orbicurves). Finally, I would like to thank H.
Esnault for some useful remarks.

### Étale fundamental gerbes

We use the formalism of _étale fundamental gerbes_ introduced by N. Borne and
A. Vistoli [BV15, §8] [Bre21b, Appendix A]: it is an alternative point of view
on the theory of étale fundamental groups which is particularly well-suited
for studying the section conjecture. From this point of view, the main object
is _not_ the fundamental group, but _the space of Galois sections itself_ ,
with a richer structure and constructed directly without passing through
fundamental groups.

If $S$ is a geometrically connected scheme (or algebraic stack) over a field
$k$, the étale fundamental gerbe of $S$ is a pro-finite étale gerbe
$\Pi_{S/k}$ over $k$ with a structural morphism $S\to\Pi_{S/k}$ which is
universal for this property: for every morphism $S\to\Phi$ where $\Phi$ is a
pro-finite étale gerbe, there exists a factorization $S\to\Pi_{S/k}\to\Phi$
unique up to a unique isomorphism.

The space of Galois sections of $S$ is in natural bijection with the set of
isomorphism classes of the groupoid $\Pi_{S/k}(k)$. If $S$ is a hyperbolic
curve over a field finitely generated over $\mathbb{Q}$, then it is known that
$\Pi_{S/k}(k)$ is already a set (as opposed to a groupoid), namely its
elements have no non-trivial automorphisms [Sti13, Proposition 104] [Bre21b,
Lemma 2.6].

##  1\. An obstruction for Galois sections of twisted forms

We give an obstruction for the existence of Galois sections of twisted forms.
We developed the obstruction for completely different purposes, namely proving
the existence of certain cycles on $\mathbb{P}^{2}$ not defined over their
field of moduli [Brea, Theorem 4] [Brec] [Breb], but the technique is
completely general and, as A. Vistoli pointed out to us, can be applied to the
study of the section conjecture. As J. Stix pointed out to us, a similar
obstruction has already been used by M. Stoll [Sto07, Lemma 5.5] (see also
[Sti13, Theorem 142]).

Let $S$ be a geometrically connected algebraic stack of finite type over a
field $k$. Our main example is the case in which $k$ is finitely generated
over $\mathbb{Q}$ and $S$ is a smooth, projective, hyperbolic orbicurve, but
the obstruction is completely general.

We stress the fact that allowing $S$ to be an algebraic stack, rather than a
scheme, is an important assumption. While we could work with schemes, we would
obtain significantly weaker results. For instance, restricting ourselves to
schemes would force us to assume that the action is free in Theorem 1.

Consider a short exact sequence of group schemes

$1\to N\to E\to G\to 1$

pro-finite étale over $k$. Suppose that we have a geometrically connected
$E$-torsor $U\to S$, i.e. an $E$-covering; denote by $C$ the quotient stack
$[U/N]$, it is a $G$-covering.

${U}$${C}$${S}$$\scriptstyle{N}$$\scriptstyle{E}$$\scriptstyle{G}$

######  Example 3.

If $E$ is a _constant_ finite group scheme, i.e. just a finite group, then $U$
is simply a Galois $E$-covering, while $C=[U/N]$ is a Galois $G$-covering.

######  Example 4.

Consider $s:\operatorname{Spec}k\to\Pi_{S/k}$ a Galois section and define $U$
as the fibered product

$U\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\operatorname{Spec}k\times_{\Pi_{S/k}}S.$

The morphism $U\to S$ is the projective limit of all the étale neighbourhoods
of $s$: it is a $k$-form of the universal covering of $S$, namely
$U_{k^{s}}\to S$ is the universal covering, where $k^{s}/k$ is a separable
closure. We have that $U$ is an
$E=\mathop{\underline{\mathrm{Aut}}}\nolimits_{\Pi_{S/k}}(s)$-torsor where $E$
is the group scheme of automorphisms of $s\in\Pi_{S/k}(k)$ and $C=[U/N]$ is
some étale neighbourhood of $s$. If $s$ is associated with a rational point
$p\in S(k)$, then $E=\underline{\pi}_{1}(S,p)$ is the étale fundamental group
scheme of $(S,p)$.

The $E$-torsor $U\to S$ induces morphisms

$S\to\mathcal{B}E\to\mathcal{B}G,$

where $\mathcal{B}E$ and $\mathcal{B}G$ are the classifying stacks of $E$ and
$G$ respectively. Recall that the set of rational points modulo isomorphism
$\mathcal{B}G(k)/\sim$ of $\mathcal{B}G$ is the set of $G$-torsors over
$\operatorname{Spec}k$ modulo isomorphism, which coincides with the non-
abelian cohomology set $\operatorname{H}^{1}(k,G)$, and similarly for $E$.

Choose $T\to\operatorname{Spec}k$ a $G$-torsor. We may use $T$ to define a
twisted form $\mathfrak{C}$ of $C$, namely

$\mathfrak{C}\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}C\times^{G}T=(C\times T)/G$

where $G$ acts on both sides of $C\times T$ simultaneously. Equivalently, we
may define $\mathfrak{C}$ as the fibered product

${\mathfrak{C}}$${\operatorname{Spec}k}$${S}$${\mathcal{B}G,}$

where the map $\operatorname{Spec}k\to\mathcal{B}G$ is the one given by $T$.

######  Proposition 5.

If $T$ does not lift to $E$, i.e. its class is not in the image of
$\operatorname{H}^{1}(k,E)\to\operatorname{H}^{1}(k,G)$, then
$\Pi_{\mathfrak{C}/k}(k)=\emptyset$.

The underlying idea is the following (we are going to give a slightly
different proof in order to avoid using the theory of group actions on
stacks). We have a morphism $C\to\mathcal{B}N$ associated with the covering
$U\to C$, this induces a morphism between twists
$\mathfrak{C}=C\times^{G}T\to\mathcal{B}N\times^{G}T$. By definition of
quotient stack, $\mathcal{B}N\times^{G}T=[\mathcal{B}N\times T/G]$ is the so-
called _gerbe of liftings_ of $T$ to $E$, which has no rational sections since
$T$ does not lift to $E$. By the universal property of the fundamental gerbe,
there is a splitting
$\mathfrak{C}\to\Pi_{\mathfrak{C}/k}\to\mathcal{B}N\times^{G}T$, hence
$\Pi_{\mathfrak{C}/k}(k)$ is empty as well.

###### Proof.

Consider the fibered product

${\Phi_{T}}$${\operatorname{Spec}k}$${\mathcal{B}E}$${\mathcal{B}G}$$\scriptstyle{T}$

where $\Phi_{T}$ is the gerbe of liftings of $T$ to $E$. Since the two
compositions $\mathfrak{C}\to\operatorname{Spec}k\to\mathcal{B}G$ and
$\mathfrak{C}\to S\to\mathcal{B}E\to\mathcal{B}G$ are equivalent by
construction, we get an induced map
$\mathfrak{C}\to\Phi_{T}=\operatorname{Spec}k\times_{\mathcal{B}G}\mathcal{B}E$.

Since $\mathcal{B}E$, $\mathcal{B}G$ are pro-finite étale gerbes and $E\to G$
is surjective, then $\Phi_{T}$ is a pro-finite étale gerbe over
$\operatorname{Spec}k$ as well, and hence we have an induced map
$\Pi_{\mathfrak{C}/k}\to\Phi_{T}$ by the universal property of the étale
fundamental gerbe. By construction, $\Phi_{T}(k)/\sim$ is the set of
$E$-torsors over $k$ which lift $T$, by hypothesis it is empty. It follows
that $\Pi_{\mathfrak{C}/k}(k)$ is empty as well. ∎

######  Remark 6.

Let us sketch an alternative proof of Proposition 5 in terms of étale
fundamental groups, for which I thank J. Stix. Let $k^{s}$ be a separable
closure of $k$ and $\Gamma=\operatorname{Gal}(k^{s}/k)$. The torsors $U\to
C\to S$ and $T\to\operatorname{Spec}k$ correspond to homomorphisms

$\pi_{1}(S)\to E(k^{s})\rtimes\Gamma\to G(k^{s})\rtimes\Gamma,$ $\Gamma\to
G(k^{s})\rtimes\Gamma$

over $\Gamma$, and $\pi_{1}(\mathfrak{C})$ identifies with the fibered product
of $\pi_{1}(S)$ and $\Gamma$ over $G(k^{s})\rtimes\Gamma$. A Galois section
$\Gamma\to\pi_{1}(\mathfrak{C})$ induces by composition a homomorphism
$\Gamma\to\pi_{1}(\mathfrak{C})\to\pi_{1}(S)\to E(k^{s})\rtimes\Gamma$. Since
$\pi_{1}(\mathfrak{C})$ is a fibered product over $G(k^{s})\rtimes\Gamma$,
this gives a lifting $\Gamma\to E(k^{s})\rtimes\Gamma$ of the homomorphism
$\Gamma\to G(k^{s})\rtimes\Gamma$ corresponding to $T$, which is absurd.

######  Remark 7.

In the particular case in which $U$ is the pro-finite limit of étale
neighbourhoods of a Galois section of $S$, and hence $\Pi_{C/k}=\mathcal{B}N$,
it is not difficult to see that the gerbe of liftings
$\Phi_{T}\simeq\mathcal{B}N\times^{G}T=\Pi_{C/k}\times^{G}T$ coincides with
the étale fundamental gerbe of the twist $\mathfrak{C}$.

##  2\. About the étale fundamental group of orbicurves

Let $S$ be a complete orbicurve over $\mathbb{C}$ with fundamental group
$\pi$. The aim of this section is to prove that, if $S$ is hyperbolic, a
pro-$p$ group of finite derived length with a closed embedding in $\hat{\pi}$
is pro-cyclic (with one exception if $p=2$), see Proposition 9. While this is
an essential tool for us, it is only used at the very last step of the proof
of Theorem 1, and its pourpose might seem not clear before reaching that
point. Furthermore, the proof of Proposition 9 is quite long. Because of this,
depending on taste the reader might want to read the proof of Theorem 1 in
Section 3 first.

We work over $\mathbb{C}$ for simplicity. Since the étale fundamental group is
invariant under base change of algebraically closed fields, the results hold
over any algebraically closed field of characteristic $0$.

Recall that the rational Euler characteristic of $S$ is defined as

$\chi(S)=2-2g-\sum_{s}\frac{\gamma_{s}-1}{\gamma_{s}},$

where the sum runs over non-trivial orbifold points $s$ of degree
$\gamma_{s}$, namely $S$ is locally isomorphic to
$[\mathbb{C}/\mu_{\gamma_{s}}]$ around $s$. The orbicurve $S$ is hyperbolic if
$\chi(S)<0$. The following is well known to experts, but we couldn’t find a
reference.

######  Lemma 8.

If $f:S^{\prime}\to S$ is a finite étale cover of degree $d$ of complete
orbicurves, then $\chi(S^{\prime})=d\chi(S)$.

###### Proof.

Denote by $g,g^{\prime}$ the genus of $S,S^{\prime}$ respectively. If
$s^{\prime}\in S^{\prime}$, denote by $e_{s^{\prime}}$ the ramification index
at $s^{\prime}$ of the underlying morphism of coarse moduli spaces. By
Riemann-Hurwitz, we get

$2g^{\prime}-2=d(2g-2)+\sum_{s^{\prime}\in S^{\prime}}(e_{s^{\prime}}-1),$

where the sum is finite since $e_{s^{\prime}}=1$ if $s^{\prime}$ is not a
ramification point. Hence,

$\chi(S^{\prime})=d(2-2g)-\sum_{s^{\prime}\in
S^{\prime}}e_{s^{\prime}}-1+\frac{\gamma_{s^{\prime}}-1}{\gamma_{s^{\prime}}}.$

Again, the sum is finite since $\gamma_{s^{\prime}}=1$ if $s^{\prime}$ is not
an orbifold point. Since $S^{\prime}\to S$ is étale, then
$\gamma_{f(s^{\prime})}=e_{s^{\prime}}\gamma_{s^{\prime}}$. We thus get

$\chi(S^{\prime})=d(2g-2)+\sum_{s^{\prime}\in
S^{\prime}}\frac{\gamma_{f(s^{\prime})}-\gamma_{s^{\prime}}+\gamma_{s^{\prime}}-1}{\gamma_{s^{\prime}}}=$
$=d(2g-2)+\sum_{s^{\prime}\in
S^{\prime}}e_{s^{\prime}}\frac{\gamma_{f(s^{\prime})}-1}{\gamma_{f(s^{\prime})}}=d(2g-2)+d\sum_{s\in
S}\frac{\gamma_{s}-1}{\gamma_{s}}$

where the last passage follows from the equality
$\sum_{f(s^{\prime})=s}e_{s^{\prime}}=d$.

∎

Consider the action by $-1$ of $\mathbb{Z}/{2}$ on $\mathbb{Z}_{2}$, and write
$D_{2^{\infty}}=\mathbb{Z}_{2}\rtimes\mathbb{Z}/{2}$ for the infinite,
$2$-adic dihedral group. Here is the main result of this section.

######  Proposition 9.

Assume that $S$ is hyperbolic, and let $G\subset\hat{\pi}$ be a closed
subgroup which is a $p$-group for some prime $p$.

If $G$ has finite derived length, it is either pro-cyclic or isomorphic to
$D_{2^{\infty}}$. Furthermore, if $G\simeq D_{2^{\infty}}$, then $S$ has at
least one orbifold point of even degree.

In order to prove Proposition 9, we need some technical lemmas.

######  Lemma 10.

A homomorphism $G\to H$ of pro-$p$-groups is surjective if and only if $G\to
H^{\rm ab}/p$ is surjective, where $H^{\rm ab}$ is the abelianization.

###### Proof.

This is a direct consequence of Burnside’s basis theorem. ∎

######  Lemma 11.

Let $p$ be a prime number and

$\pi=\left<a_{1},b_{1},\dots,a_{g},b_{g}\mid\prod[a_{i},b_{i}]=1\right>$

the fundamental group of a complete curve of genus $g\geq 2$. Let $r$ be
either $1$ or $2$, and $H\subset\pi^{\rm ab}/p$ a subgroup of rank $\geq r$.

There exists a surjective homomorphism

$q:\pi\to Q=\mathbb{Z}^{\ast r}$

such that the composition $H\subset\pi^{\rm ab}/p\to Q^{\rm
ab}/p\simeq\left(\mathbb{Z}/{p}\right)^{r}$ is surjective.

###### Proof.

For ease of notation, rename the generators $a_{1},b_{1},\dots,a_{g},b_{g}$ as
$x_{1},\dots,x_{2g}$ respectively.

If $r=1$ it is enough to define $\pi\to Q=\mathbb{Z}$ by mapping one suitably
chosen generator $x_{i}$ to $1\in\mathbb{Z}$ and all the others to $0$.

Assume $r=2$. Since $H\subset\pi^{\rm
ab}/p=\mathbb{F}_{p}x_{1}\oplus\dots\oplus\mathbb{F}_{p}x_{2g}$ has rank $\geq
2$, we may choose two linearly independent vectors $(u_{i})_{i},(v_{i})_{i}\in
H$.

Assume first that there are two indexes $1\leq j<j^{\prime}\leq 2g$ which are
_not_ of the form $j=2m-1$, $j^{\prime}=2m$ and such that the $2\times 2$
matrix $\begin{pmatrix}u_{j}&v_{j}\\\
u_{j^{\prime}}&v_{j^{\prime}}\end{pmatrix}$ has rank $2$. We may then define
$\pi\to Q\simeq\mathbb{Z}\ast\mathbb{Z}$ by

$Q=\left<x_{1},\dots,x_{2g}\mid x_{i}=1\text{ if }i\neq j,j^{\prime}\right>.$

If such indexes do not exist, then there exists $1\leq m\leq 2g$ such that
$\begin{pmatrix}u_{2m-1}&v_{2m-1}\\\ u_{2m}&v_{2m}\end{pmatrix}$ has rank $2$
_and_ for every $i\neq 2m-1,2m$ we have $u_{i}=v_{i}=0$, since $(u_{i},v_{i})$
must be a multiple of both $(u_{2m-1},v_{2m-1})$ and $(u_{2m},v_{2m})$. If
$m\leq g-1$, define

$Q=\left<x_{1},\dots,x_{2g}\mid x_{2m+1}=x_{2m},~{}x_{2m+2}=x_{2m-1},\right.$
$\left.x_{i}=1\text{ if }i\neq
2m-1,2m,2m+1,2m+2\right>=\mathbb{Z}\ast\mathbb{Z}.$

The case $m=2g$ is analogous. ∎

######  Lemma 12.

Let $S$ be a complete hyperbolic orbicurve with fundamental group $\pi$ and
$G\subset\hat{\pi}$ an infinite index closed subgroup such that $G^{\rm ab}$
has finite topological rank. For every $N$, there exists a finite index
subgroup $\pi^{\prime}\subset\pi$ such that
$\hat{\pi}^{\prime}\subset\hat{\pi}$ contains $G$ and such that the associated
covering $S^{\prime}\to S$ has genus larger than $N$.

###### Proof.

Let $r$ be the rank of $G^{\rm ab}$ and $m$ the least common multiple of the
degrees of the orbifold points of $S$, if we pass to étale coverings of $S$
then $m$ can only decrease.

Since $G$ has infinite index and $\chi(S)<0$, up to passing to a finite index
subgroup of $\pi$ we might assume that $-\chi(S)$ is arbitrarily large. If by
contradiction the genus of the coverings whose fundamental group contains $G$
is bounded by $N$, we might then assume that $S$ has an arbitrarily large
number of non-trivial orbifold points, say at least $2N+4rm^{2}$.

The torsion subgroup of $\hat{\pi}^{\rm ab}$ is
$\left(\prod_{s}\mathbb{Z}/{\gamma_{s}}\right)/(1,\dots,1)$, where $s$ runs
through all the orbifold points and $\gamma_{s}$ is the degree of $s$. If
$p\in S$ is an orbifold point, let $H_{p}\subset\hat{\pi}^{\rm ab}$ be the
subgroup generated by a loop around $p$, i.e. the image of
$\mathbb{Z}/{\gamma_{p}}\subset\prod_{s}\mathbb{Z}/{\gamma_{s}}$. Since
$(1,\dots,1)\in\prod_{s}\mathbb{Z}/{\gamma_{s}}$ has order $m$, for every non-
trivial torsion element $g\in\hat{\pi}^{\rm ab}$ there are at most $m$
orbifold points $s$ such that $g\in H_{s}$. Moreover, the intersection of the
image of $G^{\rm ab}$ and of the torsion subgroup of $\hat{\pi}^{\rm ab}$ is
bounded by $rm$. This implies that the number of orbifold points $s\in S$ such
that $H_{s}$ intersects non-trivially the image of $G^{\rm
ab}\to\hat{\pi}^{\rm ab}$ is bounded by $rm^{2}$. We call these bad points,
while the others are good points.

Choose $n\leq m$ good points $s_{1},\dots,s_{n}$ such that
$\operatorname{lcm}(\gamma_{s_{1}},\dots,\gamma_{s_{n}})=\operatorname{lcm}_{s\text{
good}}(\gamma_{s})$. Consider the standard presentation
$\pi=\left<a_{i},b_{i},x_{s}\mid
x_{s}^{\gamma_{s}}=1,\prod_{i}[a_{i},b_{i}]\cdot\prod_{s}x_{s}=1\right>$.
Define a homomorphism $f:\pi\to\mathbb{Z}/{m}$ as follows. All the generators
$a_{i},b_{i}$ map to $0$. If $s$ is a bad point, $x_{s}$ maps to $0$. If $s$
is a good point different from $s_{1},\dots,s_{n}$, $x_{s}$ maps to
$m/\gamma_{s}$. Finally, thanks to how we have chosen $s_{1},\dots,s_{n}$, we
may define $f(x_{s_{i}})$ so as to ensure that the relation
$\sum_{s}f(x_{s})=0\in\mathbb{Z}/{m}$ is satisfied. Let
$\mathbb{Z}/{m^{\prime}}\subset\mathbb{Z}/{m}$ be the image of $f$.

By construction, there are at least $2N+4rm^{2}-rm^{2}-n\geq 2N+2rm^{2}$ non-
trivial orbifold points $s$ such that the restriction of $f$ to $H_{s}$ is
injective. If $H_{s}\to\mathbb{Z}/{m^{\prime}}$ is injective, there exists a
unique point $s^{\prime}\in S^{\prime}$ over $s$ and the ramification index at
$s^{\prime}$ of the corresponding map of coarse moduli spaces is
$|H_{s}|-1\geq 1$. By Riemann-Hurwitz,

$g_{S^{\prime}}>m^{\prime}(g_{S}-1)+N+rm^{2}\geq N+rm^{2}-m^{\prime}\geq N.$

∎

Clearly $D_{2^{\infty}}$ is the $2$-adic completion of the infinite dihedral
group $D_{\infty}=\mathbb{Z}\rtimes\mathbb{Z}/{2}$, and there is a section
$D_{2^{\infty}}\subset\widehat{D_{\infty}}=\hat{\mathbb{Z}}\rtimes\mathbb{Z}/{2}$
of $\widehat{D_{\infty}}\to D_{2^{\infty}}$. Moreover, $D_{2^{\infty}}^{\rm
ab}\simeq\left(\mathbb{Z}/{2}\right)^{2}$ and
$[D_{2^{\infty}},D_{2^{\infty}}]\simeq\mathbb{Z}_{2}$.

The following fact, along with its proof, was kindly told to us by J. Stix.

######  Lemma 13.

The free product $\mathbb{Z}/{2}\ast\mathbb{Z}/{2}$ is isomorphic to the
infinite dihedral group $D_{\infty}$.

###### Proof.

Write $D_{\infty}=\left<r,s\mid s^{2}=(rs)^{2}=1\right>$. Consider the action
of $D_{\infty}$ on $\mathbb{R}$ where $r$ acts as $x\mapsto x+2$ and $s$ acts
as $x\mapsto-x$, the statement then follows from [Ser80, §I.4, Theorem 6].
Alternatively, if $\mathbb{Z}/{2}\ast\mathbb{Z}/{2}=\left<a,b\mid
a^{2}=b^{2}=1\right>$, the homomorphisms $a\mapsto s$, $b\mapsto rs$ and
$r\mapsto ba$, $s\mapsto a$ are inverses. ∎

It follows that $D_{2^{\infty}}$ is the $2$-adic completion of the fundamental
group $\mathbb{Z}/{2}\ast\mathbb{Z}/{2}$ of the disk with two orbifold points
of order $2$.

###### Proof of Proposition 9.

If $D_{2^{\infty}}\subset\hat{\pi}$, then up to passing to an étale cover of
$S$ we may assume that $\hat{\pi}^{\rm
ab}\supset\left(\mathbb{Z}/{2}\right)^{2}=D_{2^{\infty}}^{\rm ab}$ has non-
trivial torsion, hence $S$ has non-trivial orbifold points.

Let us now prove the main statement. We break down the proof in two steps. In
the first step, we prove it under the assumption that $[G,G]$ is pro-cyclic.
In the second step, we prove that $[G,G]$ cannot be isomorphic to
$D_{2^{\infty}}$.

By induction on the derived length, these two facts are sufficient. In fact,
if the derived length of $G$ is $1$ then $G$ is abelian and $[G,G]$ is
trivial, so $G$ is pro-cyclic by the first step. If the derived length is
$\geq 2$, then $[G,G]$ is pro-cyclic thanks to the induction hypothesis and
the second step; it follows that $G$ has derived length $2$ and it is
isomorphic to $D_{2^{\infty}}$ by the first step.

Step 1. Assume that $[G,G]$ is pro-cyclic, we want to show that $G$ is either
pro-cyclic or isomorphic to $D_{2^{\infty}}$. If $G^{\rm ab}$ has $p$-rank
$1$, then $G$ is pro-cyclic by Lemma 10. Assume that $G^{\rm ab}$ has $p$-rank
$\geq 2$, we want to show that $G\simeq D_{2^{\infty}}$.

It is enough to prove this in the case in which $G^{\rm ab}$ has finite
$p$-rank: if $G$ has infinite $p$-rank, since $[G,G]$ has finite $p$-rank we
may choose a closed subgroup $G^{\prime}\subset G$ such that $G^{\prime\rm
ab}$ has finite $p$-rank larger than $3>2=\operatorname{rk}D_{2^{\infty}}^{\rm
ab}$, giving a contradiction. Hence, we may assume that $G^{\rm ab}$ has
finite $p$-rank $\geq 2$. Up to passing to a finite index subgroup of $\pi$ we
might also assume that the image of $G$ in $\hat{\pi}^{\rm ab}/p$ has rank
equal to $\operatorname{rk}G^{\rm ab}\geq 2$.

Up to passing to another finite index subgroup of $\pi$, thanks to Lemma 12 we
may assume that the genus $g$ of $S$ is $\geq 2$. Consider the usual
presentation

$\pi=\left<a_{1},b_{1},\dots,a_{g},b_{g},c_{1},\dots,c_{n}\mid
c_{i}^{\gamma_{i}}=1,~{}\prod[a_{i},b_{i}]\cdot\prod c_{i}=1\right>.$

Case $p\neq 2$. Let us check that there exists a quotient $\pi\to\pi^{\prime}$
of the form

$\pi^{\prime}=\left<x_{1},x_{2},x_{3}\mid
x_{1}^{p}=x_{2}^{p}=x_{3}^{p}=x_{1}x_{2}x_{3}=1\right>$

such that $G\to\pi^{\prime\rm ab}/p\simeq\left(\mathbb{Z}/{p}\right)^{2}$ is
surjective.

We have quotients

$\tau=\left<a_{1},b_{1},\dots,a_{g},b_{g}\mid\prod[a_{i},b_{i}]=1\right>,$
$\tau^{\prime}=\left<c_{1},\dots,c_{n}\mid c_{i}^{\gamma_{i}}=1,~{}\prod
c_{i}=1\right>$

of $\pi$, and an induced surjective map $\pi\to\tau\ast\tau^{\prime}$ whose
abelianization is an isomorphism. If the image of $G$ in $\tau^{\rm ab}/p$ has
rank $\geq 2$, we may define the desired quotient $\tau\to\pi^{\prime}$ using
Lemma 11, since $\pi^{\prime}$ is a quotient of $\mathbb{Z}\ast\mathbb{Z}$. If
the image of $G$ in $\tau^{\prime\rm ab}/p$ has rank $\geq 2$, then we may
define the desired quotient $\tau^{\prime}\to\pi^{\prime}$ by mapping $3$
suitably chosen generators of $\tau^{\prime}$ to $x_{1},x_{2},x_{3}$, and the
others to the identity. The only remaining case is the one in which the images
of $G$ in $\tau^{\rm ab}/p$ and $\tau^{\prime\rm ab}/p$ have both rank $1$.
Analogously to the above, we may find quotients $\tau\to\mathbb{Z}/{p}$,
$\tau^{\prime}\to\mathbb{Z}/{p}$ such that the composition
$G\to\tau\ast\tau^{\prime}\to\mathbb{Z}/p\ast\mathbb{Z}/p\to\left(\mathbb{Z}/{p}\right)^{2}$
is surjective, and conclude since $\pi^{\prime}$ is a quotient of
$\mathbb{Z}/{p}\ast\mathbb{Z}/{p}$.

The group $\pi^{\prime}$ is the fundamental group of a complete orbicurve
$S^{\prime}$ of genus $0$ with three orbifold points of degree $p$. The
$\left(\mathbb{Z}/{p}\right)^{2}$ Galois covering $S^{\prime\prime}\to
S^{\prime}$ associated with the abelianization morphism
$\pi^{\prime}\to\left(\mathbb{Z}/{p}\right)^{2}$ defines a surface
$S^{\prime\prime}$ of genus $g^{\prime\prime}$. Because of this, the pro-$p$
completion $Q$ of
$\pi^{\prime}/[[\pi^{\prime},\pi^{\prime}],[\pi^{\prime},\pi^{\prime}]]$ is a
pro-$p$-group with $Q^{\rm ab}\simeq\left(\mathbb{Z}/{p}\right)^{2}$ and
$[Q,Q]=\mathbb{Z}_{p}^{2g^{\prime\prime}}$. By Lemma 10, $G\to Q$ is
surjective. Since $p\geq 3$ then
$\chi(S^{\prime\prime})=p^{2}\chi(S^{\prime})\leq 0$, hence
$g^{\prime\prime}\geq 1$ and $[Q,Q]$ has rank $\geq 2$, which is in
contradiction with the fact that $[G,G]$ is pro-cyclic.

Case $p=2$, $G^{\rm ab}\not\simeq\left(\mathbb{Z}/{2}\right)^{2}$. If $G^{\rm
ab}\simeq\left(\mathbb{Z}/{2}\right)^{m}$ for some $m\geq 3$, with an argument
analogous to the above we may find a quotient $\pi^{\prime}$ of $\pi$ of the
form

$\left<x_{1},x_{2},x_{3},x_{4}\mid\forall
i:x_{i}^{2}=1,~{}x_{1}x_{2}x_{3}x_{4}=1\right>$

and such that $G\to\hat{\pi}^{\prime\rm ab}$ is surjective. Otherwise, using
another analogous argument we might choose $\pi^{\prime}$ of the form

$\left<x_{1},x_{2},x_{3}\mid
x_{1}^{4}=x_{2}^{4}=x_{3}^{2}=x_{1}x_{2}x_{3}=1\right>.$

In both cases, $\pi^{\prime}$ is the fundamental group of an orbicurve
$S^{\prime}$ with $\chi(S^{\prime})=0$. In both cases, the abelianization of
$\pi^{\prime}$ is a finite $2$-group and defines a Galois cover which is a
surface, hence we might get a contradiction as in the case $p\geq 3$.

Case $p=2$, $G^{\rm ab}\simeq\left(\mathbb{Z}/{2}\right)^{2}$. Up to passing
to a finite index subgroup of $\pi$ we might assume that $G^{\rm
ab}\subset\hat{\pi}^{\rm ab}/2$. In particular, $S$ has at least $3$ orbifold
points of even degree. Since the genus is positive and $G^{\rm ab}$ is
torsion, the pullback of any abelian, non-trivial étale cover of the coarse
moduli space of $S$ defines an étale cover of $S$ whose étale fundamental
group contains $G$ and which has at least $6$ orbifold points of even order.
Hence, we may reduce to the case in which $S$ has at least $4$ orbifold points
of even degree.

If $S$ has at least $4$ orbifold points of even degree and since $G^{\rm
ab}\subset\hat{\pi}^{\rm ab}/2$, there exists a quotient $\pi^{\prime}$ of
$\pi$ of the form

$\left<x_{1},x_{2},x_{3},x_{4}\mid\forall
i:x_{i}^{2}=1,~{}x_{1}x_{2}x_{3}x_{4}=1\right>$

such that the image of $G$ in $\hat{\pi}^{\rm ab}=\pi^{\rm
ab}=\left(\mathbb{Z}/{2}\right)^{4}/(1,1,1,1)\simeq\left(\mathbb{Z}/{2}\right)^{3}$
is isomorphic to $\left(\mathbb{Z}/{2}\right)^{2}$. The group $\pi^{\prime}$
is the fundamental group of an orbicurve $S^{\prime}$ of genus $0$ with $4$
orbifold points of degree $2$.

Consider the homomorphism $\pi^{\prime}\to\mathbb{Z}/{2}$ which maps $x_{i}$
to $1$ for every $i$, the associated covering $S^{\prime\prime}\to S^{\prime}$
is a torus with fundamental group
$\pi^{\prime\prime}=\mathbb{Z}^{2}\subset\pi^{\prime}$. If the image of $G$ is
contained in $\hat{\pi}^{\prime\prime}\subset\hat{\pi}^{\prime}$, then $G^{\rm
ab}\simeq\left(\mathbb{Z}/{2}\right)^{2}$ embeds into
$\hat{\pi}^{\prime\prime}=\hat{\mathbb{Z}}^{2}$, which is absurd since the
latter is torsion free. Because of this and by rank reasons, there exists an
element in the kernel of
$\left(\mathbb{Z}/{2}\right)^{4}/(1,1,1,1)\to\mathbb{Z}/{2}$ which is not in
the image of $G$, say $(1,1,0,0)$ for simplicity.

Consider the homomorphism $\pi^{\prime}\to\mathbb{Z}/{2}\ast\mathbb{Z}/{2}$
which maps $x_{1},x_{2}$ to the first generator and $x_{3},x_{4}$ to the
second generator. The kernel of the induced map
$\left(\mathbb{Z}/{2}\right)^{4}/(1,1,1,1)\to\left(\mathbb{Z}/{2}\right)^{2}$
between abelianizations is generated by $(1,1,0,0)$, which is not contained in
the image of $G$. Because of this, the composition $G^{\rm
ab}=\left(\mathbb{Z}/{2}\right)^{2}\hookrightarrow\left(\mathbb{Z}/{2}\right)^{4}/(1,1,1,1)\to\left(\mathbb{Z}/{2}\right)^{2}$
is bijective, which in turn implies that the composition
$G\to\widehat{\mathbb{Z}/{2}\ast\mathbb{Z}/{2}}\to D_{2^{\infty}}$ is
surjective. Since $[G,G]$ is a pro-cyclic pro-$2$-group, the induced
surjective homomorphism
$[G,G]\to[D_{2^{\infty}},D_{2^{\infty}}]\simeq\mathbb{Z}_{2}$ must be
injective as well; this implies that $G\to D_{2^{\infty}}$ is injective and
hence an isomorphism.

Step 2. Finally, we have to check that $[G,G]$ cannot be isomorphic to
$D_{2^{\infty}}$. Clearly, we may assume that $p=2$ and that $G$ is not pro-
cyclic; in particular, $G^{\rm ab}$ is not cyclic. The construction of the
preceeding case gives us a surjective homomorphism $G\to D_{2^{\infty}}$,
which induces a surjective homomorphism
$[G,G]\to[D_{2^{\infty}},D_{2^{\infty}}]\simeq\mathbb{Z}_{2}$. We conclude
that $[G,G]\not\simeq D_{2^{\infty}}$ since $D_{2^{\infty}}^{\rm
ab}\simeq\left(\mathbb{Z}/{2}\right)^{2}$. ∎

######  Example 14.

The group $D_{2^{\infty}}$ can actually appear as a subgroup of the profinite
completion of the fundamental group of an hyperbolic orbicurve. Let $M$ be the
unit disk, and consider a complete curve $S$ of arbitrary genus with a
_continuous_ (not holomorphic) map to the disk $S\to M$ such that for some
open subdisk $U\subset M$ the restriction $S_{U}\to U$ is a disjoint union of
copies of $U$. Define $M_{2,2}\to M$ as an orbifold whose coarse moduli space
is the disk $M$ and which has two orbifold points of order $2$ over points of
$U\subset M$, and let $U_{2,2}\subset M_{2,2}$ be the inverse image of $U$.

Define $S^{\prime}$ as the fibered product $S\times_{M}M_{2,2}$, then
$S^{\prime}$ is an orbisurface whose coarse moduli space is $S$ with maps
$S^{\prime}\to M_{2,2}$, $U_{2,2}\to S^{\prime}$ such that the composition
$U_{2,2}\to M_{2,2}$ is the given embedding $U_{2,2}\subset M_{2,2}$ and thus
defines an isomorphism $\pi_{1}(U_{2,2})\simeq\pi_{1}(M_{2,2})$. Since
$D_{2^{\infty}}\subset\widehat{D_{\infty}}=\widehat{\pi_{1}(M_{2,2})}$, then
$\widehat{\pi_{1}(S^{\prime})}$ has a closed subgroup isomorphic to
$D_{2^{\infty}}$.

##  3\. Proof of Theorem 1

Let us now prove Theorem 1. Denote by $S$ the quotient stack $[C/G]$. Up to
passing to a finite extension of $k$, we might assume that $S(k)$ is non-
empty.

###  3.1. Definition of $\mathfrak{C}$

Recall that, by assumption, all curves are smooth, projective and
geometrically connected. Let $P=\underline{\operatorname{Pic}}^{1}_{C}$ be the
Picard scheme of line bundles of degree $1$ on $C$, the action of $G$ on $C$
induces an action on $P$.

Denote by $X$ the quotient stack $[P/G]$. Since the action on $C\subset P$ is
faithful then the action on $P$ is generically free, hence $X$ is generically
a scheme. Let $K=k(X)$ be its function field, then $k(P)/K$ is a Galois
extension with Galois group $G$ and
$\operatorname{Spec}k(P)\to\operatorname{Spec}K$ is a $G$-torsor associated
with a morphism $\operatorname{Spec}K\to\mathcal{B}G$. As in §1, define
$\mathfrak{C}$ as the twist of $C_{K}$ by this torsor, namely

$\mathfrak{C}\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}C_{K}\times^{G}\operatorname{Spec}k(P)=(C_{K}\times_{K}\operatorname{Spec}k(P))/G.$

###  3.2. The index of $\mathfrak{C}$

We want to show that $\mathfrak{C}$ has index $1$. By construction,
$\underline{\operatorname{Pic}}^{1}_{\mathfrak{C}}=P_{K}\times^{G}\operatorname{Spec}k(P)$
has a $K$-rational point and hence $\mathfrak{C}$ has period $1$. Let $I$ be
the index of $\mathfrak{C}$. By Lichtenbaum’s theorem [Lic69, Theorem 8], we
have either $I=1$ or $I=2$, and if $I=2$ then $g_{C}-1$ is odd, where $g_{C}$
is the genus of $C$. Since $S(k)\neq\emptyset$, the index divides $|G|$, hence
if $p\neq 2$ then $I=1$.

Assume $p=2$, it is sufficient to show that $g_{C}-1$ is even. If by
contradiction $g_{C}-1$ is odd, by the assumption in Theorem 1 either the
action is free or $G$ has no cyclic subgroups of index $2$. In both cases $4$
divides both $|G|$ and the cardinality of each orbit (since geometric
stabilizers are cyclic), hence by Riemann-Hurwitz $4$ divides $2g_{C}-2$ as
well, which is absurd.

###  3.3. Galois sections of $\mathfrak{C}$

It remains to prove that $\Pi_{\mathfrak{C}/K}(K)=\emptyset$. By Proposition
5, it is enough to show that
$T=\operatorname{Spec}k(P)\to\operatorname{Spec}K$ does not lift to a
$\underline{\pi}_{1}(S,p)$-torsor, where $p\in S(k)$ is some rational point.
Equivalently, we want to show that $\operatorname{Spec}K\to\mathcal{B}G$ does
not lift to $\Pi_{S/k}=\mathcal{B}\underline{\pi}_{1}(S,p)$.

Assume by contradiction that there exists a section
$\operatorname{Spec}K\to\Pi_{S/k}$ which lifts
$\operatorname{Spec}K\to\mathcal{B}G$. We would like to show that
$\operatorname{Spec}K\to\Pi_{S/k}$ extends to a morphism $X\to\Pi_{S/k}$;
since $X=[P/G]$ is smooth, it is well-known how to do this using a weight
argument. References are only available for schemes, though, while $X$ and $S$
are stacks; it is possible to sidestep this problem by working with $P$ and
$C$ instead of $X$ and $S$.

We have a $2$-cartesian diagram

${\operatorname{Spec}k(P)}$${\Pi_{C/k}}$${\operatorname{Spec}k}$${\operatorname{Spec}K}$${\Pi_{S/k}}$${\mathcal{B}G.}$

By a well-known weight argument (see e.g. [Bre21a, Corollary A.11], [Bree,
Lemma 3.3]) $\operatorname{Spec}k(P)\to\Pi_{C/k}$ extends to a morphism
$P\to\Pi_{C/k}$. By descent theory, this induces a morphism $X\to\Pi_{S/k}$.
It is possible to avoid using descent theory as follows.

Let $R\subset\operatorname{Gal}\left(\bar{K}/k(P)\right)$ be the kernel of
$\operatorname{Gal}\left(\bar{K}/k(P)\right)\to\pi_{1}(P)$. We have a
commutative diagram of short exact sequences

${1}$${\operatorname{Gal}\left(\bar{K}/k(P)\right)}$${\operatorname{Gal}\left(\bar{K}/k(X)\right)}$${G}$${1}$${1}$${\pi_{1}(P)}$${\pi_{1}(X)}$${G}$${1,}$

hence $\pi_{1}(X)=\operatorname{Gal}\left(\bar{K}/k(X)\right)/R$. Since
$\operatorname{Spec}k(P)\to\Pi_{S/k}$ extends to $P$, the associated
homomorphism $\operatorname{Gal}\left(\bar{K}/k(P)\right)\to\pi_{1}(S)$ maps
$R$ to the identity and hence we get a factorization

$\operatorname{Gal}\left(\bar{K}/k(X)\right)\to\pi_{1}(X)\to\pi_{1}(S)$

of the homomorphism $\operatorname{Gal}\left(\bar{K}/k(X)\right)\to\pi_{1}(S)$
associated with the section $\operatorname{Spec}K\to\Pi_{S/k}$.

Since $\operatorname{Spec}K\to\Pi_{S/k}$ lifts
$\operatorname{Spec}K\to\mathcal{B}G$, the composition

$\pi_{1}(X_{\bar{k}})\to\pi_{1}(S_{\bar{k}})\to G$

is a factorization of the surjective homomorphism $\pi_{1}(X_{\bar{k}})\to G$
induced by the $G$-covering $P\to X$.

Since $\pi_{1}(P_{\bar{k}})$ is abelian and $G$ is a finite $p$-group, the
image $H\subset\pi_{1}(S_{\bar{k}})$ of a $p$-Sylow of $\pi_{1}(X_{\bar{k}})$
is an extension of $G$ by an abelian pro-$p$-group; in particular $H$ has
finite derived length. By Proposition 9, such a subgroup of
$\pi_{1}(S_{\bar{k}})$ is either pro-cyclic or isomorphic to $D_{2^{\infty}}$.
If $H$ is pro-cyclic, then $G$ is cyclic, giving a contradiction. If $H\simeq
D_{2^{\infty}}$, then $p=2$ and $G$ is dihedral. By hypothesis this implies
that the action is free, which is contradiction with Proposition 9.

######  Remark 15.

If the section conjecture holds for $C_{k(P)}$ and the action is free, we only
need to assume that $G$ is non-trivial to obtain that the curve $\mathfrak{C}$
constructed above satisfies $\Pi_{\mathfrak{C}/K}(K)=\emptyset$. In fact, if
$\operatorname{Spec}K\to\mathcal{B}G$ lifts to $\Pi_{S/k}$, the induced
morphism $\operatorname{Spec}k(P)\to\Pi_{C/k}$ is associated with a point
$\operatorname{Spec}k(P)\to C$ which factorizes through $\operatorname{Spec}k$
since $C$ is hyperbolic and $P$ is an abelian variety. With notation as above,
this implies that $H\simeq G$ is finite, which is absurd since
$\pi_{1}(S_{\bar{k}})$ is torsion free.

On the other hand, this is false if the action is not free and $G$ is cyclic.
For instance, if $G$ is cyclic and fixes a rational point $c\in C(k)$, then
$c$ induces a $K$-rational point of $\mathfrak{C}$.

##  4\. Proof of Theorem 2

Consider the action $\phi$ of $\mathbb{Z}/{\ell}$ on $\mathbb{Z}/{\ell^{2}}$
given by $1\mapsto 1+\ell$, there exists a surjective homomorphism
$\mathbb{Z}/{\ell^{2}}\rtimes_{\phi}\mathbb{Z}/{\ell}\to\left(\mathbb{Z}/{\ell}\right)^{2}$.

######  Lemma 16.

Let $k,\ell$ be as in Theorem 2. There exists a finite extension $K/k$ with a
surjective homomorphism
$\operatorname{Gal}\left(\bar{K}/K\right)\to\left(\mathbb{Z}/{\ell}\right)^{2}$
which does not lift to $\mathbb{Z}/{\ell^{2}}\rtimes_{\phi}\mathbb{Z}/{\ell}$.

###### Proof.

Up to a finite extension, we may assume that $k$ contains a primitive
$\ell^{2}$-th root of $1$.

Assume first that $k$ is a finite extension of $\mathbb{Q}_{p}$, $p\neq\ell$.
Since $p\neq\ell$, the maximal $\ell$-adic quotient of
$\operatorname{Gal}(\bar{k}/k)$ is
$\mathbb{Z}_{\ell}\rtimes_{\rho}\mathbb{Z}_{\ell}$, where $\mathbb{Z}_{\ell}$
acts on itself with the cyclotomic character $\rho$. Since $k$ contains an
$\ell^{2}$-th root of $1$, the action of $\mathbb{Z}_{\ell}$ on the quotient
$\mathbb{Z}/{\ell^{2}}$ of $\mathbb{Z}_{\ell}$ is trivial, hence there exists
a surjective homomorphism
$\mathbb{Z}_{\ell}\rtimes_{\rho}\mathbb{Z}_{\ell}\to\left(\mathbb{Z}/{\ell^{2}}\right)^{2}$;
furthermore, every finite quotient of
$\mathbb{Z}_{\ell}\rtimes_{\rho}\mathbb{Z}_{\ell}$ of exponent $\ell^{2}$ is a
quotient of $\left(\mathbb{Z}/{\ell^{2}}\right)^{2}$.

It follows that there are no surjective homomorphisms
$\mathbb{Z}_{\ell}\rtimes_{\rho}\mathbb{Z}_{\ell}\to\mathbb{Z}/{\ell^{2}}\rtimes_{\phi}\mathbb{Z}/{\ell}$.
By Lemma 10, this implies that there are no liftings
$\mathbb{Z}_{\ell}\rtimes_{\rho}\mathbb{Z}_{\ell}\to\mathbb{Z}/{\ell^{2}}\rtimes_{\phi}\mathbb{Z}/{\ell}$
of the composition
$\mathbb{Z}_{\ell}\rtimes_{\rho}\mathbb{Z}_{\ell}\to\left(\mathbb{Z}/{\ell^{2}}\right)^{2}\to\left(\mathbb{Z}/{\ell}\right)^{2}$.
This concludes the proof for local fields.

If $k$ is finitely generated over $\mathbb{Q}$, let $h\subset k$ be the
algebraic closure of $\mathbb{Q}$ in $k$. I claim that there exists a finite
extension $h^{\prime}/h$ with a section
$\operatorname{Gal}(\bar{h}/h^{\prime})\to\operatorname{Gal}(\bar{k}/k)$. By
induction, it is enough to do the case in which $k$ has transcendence degree
$1$ over $h$. Let $V$ be a smooth curve over $h$ whose fraction field is $k$,
and $v\in V$ a closed point with residue field $h^{\prime}$. Let $k_{v}$ be
the completion of $k$ at the place defined by $v$, the residue field of
$k_{v}$ is $h^{\prime}$ and hence we get the desired splitting
$\operatorname{Gal}(\bar{h}/h^{\prime})\to\operatorname{Gal}(\bar{k}_{v}/k_{v})\to\operatorname{Gal}(\bar{k}/k)$.
Up to replacing $h$ with $h^{\prime}$ and $k$ with $h^{\prime}\otimes_{h}k$,
we might then assume that
$\operatorname{Gal}(\bar{k}/k)\to\operatorname{Gal}(\bar{h}/h)$ is split.

Let $h_{\nu}$ be the completion of $h$ at a prime $\nu$ dividing $p\neq\ell$.
Since $\operatorname{Gal}(\bar{k}/k)\to\operatorname{Gal}(\bar{h}/h)$ is split
and
$\operatorname{Gal}(\bar{h}_{\nu}/h_{\nu})\to\operatorname{Gal}(\bar{h}/h)$ is
injective, then $\operatorname{Gal}(\bar{k}/k)$ has a closed subgroup
isomorphic to $\operatorname{Gal}(\bar{h}_{\nu}/h_{\nu})$. Let
$\operatorname{Gal}(\bar{h}_{\nu}/h_{\nu})\to\left(\mathbb{Z}/{\ell}\right)^{2}$
be the only surjective homomorphism to $\left(\mathbb{Z}/{\ell}\right)^{2}$,
since $\operatorname{Gal}(\bar{h}_{\nu}/h_{\nu})$ is a closed subgroup of
$\operatorname{Gal}(\bar{k}/k)$ there exists a finite index subgroup
$\Gamma\subset\operatorname{Gal}(\bar{k}/k)$ containing
$\operatorname{Gal}(\bar{h}_{\nu}/h_{\nu})$ with a factorization
$\operatorname{Gal}(\bar{h}_{\nu}/h_{\nu})\to\Gamma\to\left(\mathbb{Z}/{\ell}\right)^{2}$.
Since $k$, and hence $h_{\nu}$, contains a primitive $\ell^{2}$-rooth of $1$,
then $\Gamma\to(\mathbb{Z}/\ell)^{2}$ does not lift to
$\mathbb{Z}/{\ell^{2}}\rtimes_{\phi}\mathbb{Z}/{\ell}$ and we may choose $K/k$
as the field fixed by $\Gamma$. ∎

Let us now prove Theorem 2. Since $S$ is hyperbolic, up to a finite extension
of $k$ by Lemma 11 we may assume that there exists a geometrically connected,
finite étale Galois covering $U\to S$ with Galois group
$\mathbb{Z}/{\ell^{2}}\rtimes_{\phi}\mathbb{Z}/{\ell}$; denote by $C\to S$ the
induced $\left(\mathbb{Z}/\ell\right)^{2}$-cover.

Let $K/k$, $\operatorname{Gal}(\bar{K}/K)\to\left(\mathbb{Z}/\ell\right)^{2}$
be as in Lemma 16, the homomorphism corresponds to a
$\left(\mathbb{Z}/\ell\right)^{2}$-torsor $T\to\operatorname{Spec}K$. The
twist $\mathfrak{C}=C_{K}\times^{\left(\mathbb{Z}/\ell\right)^{2}}T\to S_{K}$
is a torsor for the group scheme $\underline{\operatorname{Aut}}(T)$ of
$\left(\mathbb{Z}/\ell\right)^{2}$-equivariant automorphisms of $T$. Since
$\left(\mathbb{Z}/\ell\right)^{2}$ is abelian, then
$\underline{\operatorname{Aut}}(T)\simeq\left(\mathbb{Z}/\ell\right)^{2}$ and
hence $\mathfrak{C}\to S_{K}$ is a Galois
$\left(\mathbb{Z}/\ell\right)^{2}$-cover. By Proposition 5 the twist
$\mathfrak{C}$ satisfies the section conjecture.

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