# Negotiation Problem

Izat B. Baybusinov  Enrico Maria Fenoaltea  Yi-Cheng Zhang  yi-
<EMAIL_ADDRESS>Physics Department, University of Fribourg Chemin du
Musée 3, 1700 Fribourg, Switzerland

###### Abstract

We propose and solve a negotiation model of multiple players facing many
alternative solutions. The model can be generalized to many relevant
circumstances where stakeholders’ interests partially overlap and partially
oppose. We also show that the model can be mapped into the well-known directed
percolation and directed polymers problems. Moreover, many statistical
mechanics tools, such as the Replica method, can be fruitfully employed.
Studying our negotiation model can enlighten the links between social-economic
phenomena and traditional statistical mechanics and help to develop new
perspectives and tools in the fertile interdisciplinary field.

###### keywords:

Game Theory, Nash Equilibrium, Econophysics, Replica Method, Directed
Percolation, Directed Polymer.

††journal: Physica A

We propose and solve a non-zero sum game of negotiation.

We study its Nash equilibrium and ground state.

We employ both a combinatorial statistical approach and standard statistical
mechanics tools.

The model can be mapped into two well-known physics problems: directed
percolation and directed polymers.

##  1 Introduction

Often in social and economic processes we must reach an agreement with other
people. We may have partially overlapping interests and conflicts which, to
settle, we must negotiate. To study such behaviour, we propose a simple model
of negotiation which turns out to belong to a well-known class of physics
models, ranging from spin-glasses to random matching, and other models in
physics and beyond [1, 2, 3, 4, 5].

In principle, our problem follows the same philosophy of the bargaining
problem [6, 7]. The latter is a non-cooperative game where agents compete to
maximize their utilities. Each agent is fully rational and knows the
preferences of all the others. The main goal of the bargaining problem is
studying the conditions for a Nash equilibrium, i.e. a situation where, for
each agent, it is not convenient to change strategy [8]. In this approach,
each agent can choose among infinite possible strategies and evaluate the
opponents’ one before making a move.

By contrast, we consider bounded-rational players that cannot span across all
possible choices, but are limited to few, depending on their current knowledge
[9]. Indeed, here we want to model a more realistic scenario where people,
during the negotiation, have limited time to make choices and must find a
compromise between time spent and expected utility. We consider a non-zero sum
game: the utility of each player is independent on all the others, rather than
being anti-correlated [7]. This means that people have different tastes and
preferences that are not always conflicting. This approach is inspired by the
Stable Marriage Problem [5], where players have independent preference-lists.

In this paper, we will study analytically and numerically the statistical
properties of our negotiation model. The rest of the paper is organized as
follows: In section 2, we formally describe the model with 2 players, studying
its Nash-equilibrium solution and its ground state, i.e. the global best
solution; In section 3 we generalize our model to more than 2 players and show
its connection to other known physics models; Finally, in the last section, we
expose our conclusions.

##  2 Negotiation Model

Let us consider the case of two players, A and B, who must find an agreement
among $N$ alternatives. Each player ranks these alternatives according to his
preference, from the best to the worst. The preference-lists are randomly
drawn, and each alternative is acceptable to both players, even the worst,
with descending benefits.

We assume that players propose alternatives, in sequence, from their most
favourite down to the least favourite, by taking alternate turns. If player A
is the first mover (i.e. the first who proposes), then at time $t=1$ he
proposes his best choice to player B, and player B can either accept it,
ending the negotiation, or reject it. In the latter case, the negotiation
continues, and it will be player B’s turn to propose his first-ranked
alternative; if also player A rejects, then the negotiation proceeds to $t=2$,
and player A must propose his second-best choice, as his first one was already
rejected. The process continues with the same logic until a proposal is
accepted. This happens when, at a given time $t$, player A (or B) proposes his
next alternative to player B (or A), to whom it happens to be better (for the
first time) than the alternatives that he would propose next. So player B (or
A) accepts and the negotiation ends with a solution. Note that, by
construction, an alternative is accepted only if it is the first alternative
to have been proposed by both players. An example of negotiation is shown in
Fig.1.

Let us denote the time to reach an agreement by $t_{f}$, the finally agreed
alternative by $f$, and its respective ranking in the two players’ preference-
lists by $\epsilon_{{\scriptscriptstyle A}}$ and
$\epsilon_{{\scriptscriptstyle B}}$. Analogously to the matching problem [5],
we use the term energy to describe ranking: alternative $i$ has an assigned
energy $\epsilon_{i,{\scriptscriptstyle A}}$ for A, and
$\epsilon_{i,{\scriptscriptstyle B}}$ for B. For simplicity, energies assume
integer values, i.e. $\epsilon_{i}=1,2,...,N$.

Figure 1: Figurative representation of the negotiation process: the rows
represent the alternative proposed by player A (top row) and by player B
(bottom row). The proposals are made from left to right and the alternatives
are labeled with natural numbers from 1 to N (of course, these have nothing to
do with the energies). Here only the four best alternatives in the preference-
lists of the two players are shown: in particular, the first choice of player
A is the alternative 3, while the fourth is the alternative 7; for player B,
instead, the best choice is the alternative 1, and the fourth-best is the
alternative 5. Hence, player A starts by proposing the alternative 3, then
player B proposes the alternative 1 and so on. The process stops when player A
accepts the alternative 5 proposed by player B at time $t_{f}=4$. Indeed, for
player A, his subsequent proposals would have been worse than alternative 5.
Note that the alternative 5 is the only one proposed by both players until
that moment and, actually, this is the condition for an alternative to be
accepted. In this example the energies are $\epsilon_{5,A}=\epsilon_{A}=3$ and
$\epsilon_{5,B}=\epsilon_{B}=4$.

###  2.1 Negotiation Solution

In the negotiation solution we are interested in the energy of both players,
$\epsilon_{{\scriptscriptstyle A}}$ and $\epsilon_{{\scriptscriptstyle B}}$,
and in the energy gap, $d=|\epsilon_{{\scriptscriptstyle
A}}-\epsilon_{{\scriptscriptstyle B}}|$, which tells how fair the solution is.
We evaluate these quantities by sample averaging them over all configurations
of the preference-lists, and we shall denote the sample average of an
observable $o$ by $\overline{o}$.

Firstly we compute the probability $P(\epsilon_{{\scriptscriptstyle
A}},\epsilon_{{\scriptscriptstyle B}})$ of finding the negotiation solution
with energies $\epsilon_{{\scriptscriptstyle A}}$ and
$\epsilon_{{\scriptscriptstyle B}}$.

When $\epsilon_{{\scriptscriptstyle A}}>\epsilon_{{\scriptscriptstyle B}}$,
the alternative proposed by player A at time
$t_{f}=\epsilon_{{\scriptscriptstyle A}}$ is accepted, while all previous
proposals up to time $t_{f}-1$ are rejected. The number of times this
situation occurs is given by the number of ways of choosing $t_{f}-1$
alternatives out of the $t_{f}-\epsilon_{A}$ alternatives worse than
$\epsilon_{A}$, i.e. $\binom{N-\epsilon_{{\scriptscriptstyle
A}}}{\epsilon_{{\\!\scriptscriptstyle A}}-1}(\epsilon_{{\scriptscriptstyle
A}}-1)!$. Since there are $\binom{N}{\,\epsilon_{{\\!\scriptscriptstyle
A}}}\epsilon_{{\scriptscriptstyle A}}!$ ways of choosing
$\epsilon_{{\scriptscriptstyle A}}$ elements out of $N$, the probability is

$P(\epsilon_{{\scriptscriptstyle A}},\epsilon_{{\scriptscriptstyle
B}})=\frac{\binom{N-\epsilon_{{\\!\scriptscriptstyle
A}}}{\epsilon_{{\scriptscriptstyle
A}}-1}}{\binom{N}{\epsilon_{{\\!\scriptscriptstyle
A}}}\epsilon_{{\scriptscriptstyle A}}},\quad\epsilon_{{\scriptscriptstyle
A}}>\epsilon_{{\scriptscriptstyle B}}\;.$ (1)

Expanding the binomial coefficients, we can rewrite this as follows:

$P(\epsilon_{{\scriptscriptstyle A}},\epsilon_{{\scriptscriptstyle
B}})=\frac{1}{N-\epsilon_{{\scriptscriptstyle
A}}+1}\prod_{i=1}^{\epsilon_{{\scriptscriptstyle
A}}-1}\left(1-\frac{\epsilon_{{\scriptscriptstyle A}}}{N-i+1}\right).$ (2)

In this equation, the first factor is the probability that player A accepts a
proposal at time $t_{f}=\epsilon_{{\scriptscriptstyle A}}$, while the product
represents the probability that player A rejects all the proposals before
$t=t_{f}$.
By symmetry, the case $\epsilon_{{\scriptscriptstyle
B}}>\epsilon_{{\scriptscriptstyle A}}$ is obtained by exchanging $A$ and $B$.
Combining both cases, for large $N$ we obtain

$P(\epsilon_{{\scriptscriptstyle A}},\epsilon_{{\scriptscriptstyle
B}})\approx\frac{1}{N}\exp{\left\\{-\frac{t_{f}^{2}}{N}\right\\}}\;,$ (3)

where $t_{f}=\max\left(\epsilon_{{\scriptscriptstyle
A}},\epsilon_{{\scriptscriptstyle B}}\right)$. From this, we can compute the
sample averages:

$\displaystyle\overline{\epsilon_{{\scriptscriptstyle
A}}}=\overline{\epsilon_{{\scriptscriptstyle
B}}}=\frac{3\sqrt{\pi}}{8}\sqrt{N}\approx 0.665\sqrt{N}\;,$ (4)
$\overline{d}=\frac{\sqrt{\pi}}{4}\sqrt{N}\approx 0.443\sqrt{N}\;.$ (5)

The probability that the two players have the same energy is
$P(\epsilon_{{\scriptscriptstyle A}}=\epsilon_{{\scriptscriptstyle
B}})=\sqrt{\pi}/2\sqrt{N}$, showing that such cases are rare when
$N\to\infty$.

We also calculate the players’ energy distribution, i.e.
$P(\epsilon_{{\scriptscriptstyle A}})=\sum_{\epsilon_{\scriptscriptstyle
B}}P(\epsilon_{{\scriptscriptstyle A}},\epsilon_{{\scriptscriptstyle B}})$.
From (3) we have

$\sum_{\epsilon_{\scriptscriptstyle B}}P(\epsilon_{{\scriptscriptstyle
A}},\epsilon_{{\scriptscriptstyle
B}})\approx\frac{\displaystyle{\epsilon}_{{\\!\scriptscriptstyle
A}}}{N}e^{-\epsilon_{{\\!\scriptscriptstyle
A}}^{2}/{N}}+\int\limits_{\epsilon_{\scriptscriptstyle
A}+1}^{N}\frac{d\epsilon_{{\scriptscriptstyle
B}}}{N}\,e^{-\epsilon_{{\\!\scriptscriptstyle B}}^{2}/N}\;.$ (6)

Figure 2: Probability distribution of the total energy compared with that of
both the winner (the player with lower energy) and the loser (the player with
higher energy).

In the right hand side of (6), the first term is the probability of obtaining
a higher energy than the other player, and the second is the probability of
obtaining a lower energy (Fig. 2).

Note that, in the negotiation model, all rules are fair except for the first
move. Indeed, the player who moves first, on average, loses more (i.e. he
obtains higher energy than the opponent). One can show that the stopping time
$t_{f}$ is independent of the first mover, therefore the solution depends on
the first mover whenever both proposals of player A and player B at time
$t_{f}$ can be accepted by the other player. This happens with probability
$t_{f}/2N$, and the energy difference is, on average, $t_{f}/2$. It follows
that the average first mover disadvantage is given by:

$\overline{\epsilon_{A}-\epsilon_{B}}=\overline{\frac{t_{f}}{2N}\frac{t_{f}}{2}}=\frac{1}{4}+O(N^{-1/2})$
(7)

However, this is a negligible effect when $N\rightarrow\infty$.

Nevertheless, the first-mover disadvantage provides insight into the
negotiation model. Whoever moves first, or by extension moves more steps, will
bear additional costs. This is because whoever exposes more information, shows
more cards so to speak, is at a disadvantage. Therefore rational and selfish
players will show as little and as late as possible their cards. This, a
posteriori, justifies that our strategy is optimal for each player.

###  2.2 Ground State Solution

Now, from physicists’ point of view, the most interesting is the Ground State,
i.e. the solution that minimizes the total energy.

The minimal energy is usually found by defining a Hamiltonian, but we shall
follow the probabilistic approach first, then compare it to the traditional
Hamiltonian approach.

The ground state is defined as

$\displaystyle\epsilon_{\scriptscriptstyle
0}=\min\limits_{i=1,..,N}\\{\epsilon_{i,{\scriptscriptstyle
A}}+\epsilon_{i,{\scriptscriptstyle B}}\\}=\epsilon_{\scriptscriptstyle
0,A}+\epsilon_{\scriptscriptstyle 0,B}\;.$ (8)

To find it, we compute the cumulative distribution function of
$\epsilon_{{\scriptscriptstyle 0}}$:

$\displaystyle P(\epsilon_{{\scriptscriptstyle 0}}\geq
l)=\prod\limits_{i=1}^{N}P(\epsilon_{i,{\scriptscriptstyle
A}}+\epsilon_{i,{\scriptscriptstyle B}}\geq l)\;.$ (9)

We set $\epsilon_{i,{\scriptscriptstyle A}}=i$ and
$\epsilon_{i,{\scriptscriptstyle B}}=\pi_{N}(i)$, where $\pi_{N}$ is a random
permutation, obtaining $P(\epsilon_{i,{\scriptscriptstyle B}}\geq
l-i)=\frac{N-l+1}{N-i+1}$ for $i<l$. So (9) can be rewritten as

$\displaystyle P(\epsilon_{{\scriptscriptstyle 0}}\geq
l)=\prod\limits_{i=1}^{l-1}\frac{N-l+1}{N-i+1}=\prod_{i=1}^{l-1}\left(1-\frac{l-i}{N-i+1}\right)\;,$
(10)

which for $N\to\infty$ converges to:

$\displaystyle P(\epsilon_{{\scriptscriptstyle 0}}\geq
l)\approx\exp{\left(-\frac{l(l-1)}{2N}\right)}\;.$ (11)

From (11) we can compute the average as
$\overline{\epsilon_{{\scriptscriptstyle
0}}\mathstrut}=\sum\limits_{l=1}^{N+1}P(\epsilon_{{\scriptscriptstyle 0}}\geq
l)$, so the average energy per player in the ground state,
$\overline{\epsilon_{{\scriptscriptstyle
0}}}/2=\overline{\epsilon_{\scriptscriptstyle
0,A}}=\overline{\epsilon_{\scriptscriptstyle 0,B}}$, is

$\displaystyle\overline{\epsilon_{\scriptscriptstyle
0,A}\mathstrut}=\overline{\epsilon_{\scriptscriptstyle
0,B}\mathstrut}=\sqrt{\frac{\pi}{8}}\sqrt{N}\approx 0.627\sqrt{N}\;.$ (12)

The average energy gap is obtained by averaging all the possible
$\epsilon_{\scriptscriptstyle 0,A}$ and $\epsilon_{\scriptscriptstyle 0,B}$
for a given energy value of the ground state. Since the energies can take
values from $1$ to $\epsilon_{{\scriptscriptstyle 0}}-1$, the average gap is
equal to the average energy per player (true only if sample averaged):

$\displaystyle\overline{|\epsilon_{\scriptscriptstyle
0,A}-\epsilon_{\scriptscriptstyle
0,B}|}=\overline{\epsilon_{\scriptscriptstyle
0,A}\mathstrut}=\overline{\epsilon_{\scriptscriptstyle 0,B}\mathstrut}\;,$
(13)

The ground state energy in (12) is about $6\%$ lower than that of the
negotiation solution in (4). On the other hand, comparing (13) with (5), we
see that the energy gap in the ground state is about $42\%$ larger than that
in the negotiation solution. Indeed, the ground state has no fairness
constraint (as players have no decision-making process), and, though the total
energy is minimal, the energy gap is larger. Thus, a lack of equality is the
price to pay for reducing the total energy.

Incidentally, the two solutions often are identical. As it turns out, with
$\frac{\pi}{4}\approx 78\%$ of probability, the negotiation finds the ground
state exactly, when $N\to\infty$.

To see it, we compute the probability $P(\epsilon_{{\scriptscriptstyle
0}}=\epsilon_{{\scriptscriptstyle A}}+\epsilon_{{\scriptscriptstyle B}})$ that
the two solutions coincide. In the large $N$ limit, we have

$\displaystyle P(\epsilon_{{\scriptscriptstyle
0}}=\epsilon_{{\scriptscriptstyle A}}+\epsilon_{{\scriptscriptstyle
B}})\approx\frac{1}{N}\exp\left\\{\frac{-\epsilon_{{\\!\scriptscriptstyle
A}}^{2}-\epsilon_{{\scriptscriptstyle B}}^{2}+\epsilon_{{\\!\scriptscriptstyle
A}}+\epsilon_{{\scriptscriptstyle B}}}{N}\right\\}\;.$ (14)

Summing over all possible energies, the probability that the ground state and
the negotiation solution overlap is:

$\displaystyle P(\text{overlap})=\sum\limits_{\epsilon_{{\scriptscriptstyle
A}},\epsilon_{{\scriptscriptstyle B}}}P(\epsilon_{{\scriptscriptstyle
0}}=\epsilon_{{\scriptscriptstyle A}}+\epsilon_{{\scriptscriptstyle
B}})=\frac{\pi}{4}\;.$ (15)

##  3 Negotiation model for $m$ players

It is interesting to generalize the negotiation model to more players. Not
only it is a natural generalization like from Ising to Potts model [10, 11],
but it is also of wider applications. Indeed, seeking an agreement between
several parties is common in diplomatic and business negotiations: as an
example, European Union negotiations take place with 26 countries and each
agreement must be accepted by all.

###  3.1 Negotiation Solution

Consider $m$ players that must find an agreement among $N$ alternatives, but
each has a distinct preference-list. Again, each player takes turns to propose
and all the others may agree or reject, as in the two-players model. Note that
a partial agreement is not enough: even if only one player rejects a proposal,
the negotiation must continue.

Let us first study the case of three players, $A$, $B$ and $C$. Analogously to
the two-player model, we compute the probability
$P(\epsilon_{{\scriptscriptstyle A}},\epsilon_{{\scriptscriptstyle
B}},\epsilon_{{\scriptscriptstyle C}})$ of finding the negotiation solution
with energies $\epsilon_{{\scriptscriptstyle A}}$,
$\epsilon_{{\scriptscriptstyle B}}$ and $\epsilon_{{\scriptscriptstyle C}}$.
To simplify, we divide the energies by $N$ and we consider the case
$\epsilon_{{\scriptscriptstyle A}}=\max\left(\epsilon_{\scriptscriptstyle
A},\epsilon_{\scriptscriptstyle B},\epsilon_{\scriptscriptstyle C}\right)$.
Hence, $P(\epsilon_{{\scriptscriptstyle A}},\epsilon_{{\scriptscriptstyle
B}},\epsilon_{{\scriptscriptstyle C}})$ is equivalent to the probability that
both player $B$ and player $C$ accept the proposal of player $A$ after
$N\epsilon_{{\scriptscriptstyle A}}-1$ rejections. In the $N\to\infty$ limit,
we have

$\begin{split}P(\epsilon_{{\scriptscriptstyle
A}},\epsilon_{{\scriptscriptstyle B}},\epsilon_{{\scriptscriptstyle
C}})\approx N\exp\left\\{-N\epsilon_{{\scriptscriptstyle
A}}^{3}\right\\}\;.\end{split}$ (16)

In the general case of $m$ players, similar computations lead to

$\begin{split}P(\epsilon_{1},...,\epsilon_{m})\approx
N\exp\left\\{-Nt_{f}^{m}\right\\}\;,\end{split}$ (17)

where $\epsilon_{i}$ denotes the energy of player $i$ and
$t_{f}=\max\left(\epsilon_{1},...,\epsilon_{m}\right)$.

From (17) it follows that the average energy per player is:

$\begin{split}\overline{\epsilon}\equiv\frac{1}{m}\overline{\sum_{i=1}^{m}\epsilon_{i}}&=\frac{m+1}{2m}\Gamma\left(1+\frac{1}{m}\right)N^{-{1}/{m}}\;.\end{split}$
(18)

As expected, the larger $m$, the higher the average energy per player, as
there are more constraints in the negotiation. However, one can easily show
that a larger $m$ implies a smaller relative energy gap between the players.

###  3.2 Ground State Solution

We now compute the ground state $\epsilon_{{\scriptscriptstyle 0}}$ for $m$
players and we consider only the large $N$ limit, so that we can approximate
the preference-lists by $N$ random and independent variables in the interval
$[0,1]$. Hence, the extreme value distribution of $N$ random variables
$\epsilon_{i}$ is

$P(\epsilon_{{\scriptscriptstyle 0}}>l)=\left(1-F(l)\right)^{N}\;,$ (19)

where, in our case, $F(x)$ is the cumulative distribution function of the sum
of $m$ random variables. Since the largest contribution to (19) is for $l\sim
0$, the cumulative distribution can be approximated by
$F(l)\sim\frac{l^{m}}{m!}$, obtaining a Weibull distribution [12]:

$P(\epsilon_{{\scriptscriptstyle
0}}>l)=\exp\left\\{-N{l^{m}}/{m!}\right\\}\;.$ (20)

From (20), the average energy per player in the ground state is:

$\frac{\overline{\epsilon_{{\scriptscriptstyle
0}}}}{m}=\int_{0}^{\infty}dlP(\epsilon_{{\scriptscriptstyle
0}}>l)=\frac{m!^{1/m}}{m}\Gamma\left(1+\frac{1}{m}\right)N^{-1/m}\;.$ (21)

Figure 3:  Average energy per player against the number of players $m$, both
for negotiation and ground state solutions, with N=10000. In the inset: the
probability that negotiation and ground state solutions coincide against the
number of player $m$. As shown in (15), with two players this probability is
$\pi/4$.

Even in this case, the average energy per player increases with $m$. However,
comparing (18) and (21), we see that, as the number of players grows, the
ratio between the energy of the negotiation and that of the ground state
becomes larger, reaching the asymptotic limit $e$. It means that with more
players, i.e. more constraints, the average benefit found by a superpartes
matchmaker is larger than that with $m$ players negotiating between
themselves.

From these results, we also expect that, with more players, the probability of
reaching the ground state through the negotiation process is lower, as shown
in the inset of Fig. 3.

It is remarkable that our negotiation model can be mapped to two well-known
models of statistical mechanics: directed percolation and directed polymer
[13, 14]. Indeed, the $m$ players and the $N$ alternatives can be arranged as
a $m\times N$ matrix as shown in Fig. 4. The element $\epsilon_{ij}$ of this
matrix corresponds to the energy of alternative $i$ for player $j$.

let us examine the following transfer matrix relations:

$\displaystyle\text{directed percolation:}\quad\;\;\;\;\,\eta_{\;\\!i,t}$
$\displaystyle=\max(\eta_{\;\\!i,t-1},\epsilon_{i,t})\;,$
$\displaystyle\text{directed polymer:}\quad\quad\,\,\;\;\eta_{\;\\!i,t}$
$\displaystyle=\eta_{i,t-1}+\epsilon_{\;\\!i,t}\;.$

Here, the time $t=1,...,m$ represents the players, and the space $i=1,...,N$
represents the alternatives. At each time step, the energy is updated
following the relations above, up to $t=m$. At the end of the process,
minimizing $\eta_{\;\\!i,m}$ in the first relation gives the alternative $i$
corresponding to $f$ (i.e. the negotiation solution) and, in the second
relation, it gives the ground state energy. Indeed, in the directed
percolation, each value $\eta_{i,m}$ corresponds to the worst ranking of the
alternative $i$ in the preference-lists of the $m$ players. Since players make
their proposals starting from the top of their lists, the alternative $i^{*}$,
such that $\eta_{i^{*},m}=\min_{i}(\eta_{\;\\!i,m})$, coincides with the first
alternative to have been proposed by all players. As mentioned in section 2,
this is precisely the condition for an alternative to be accepted by all. In
the directed polymer, instead, $\eta_{i,m}$ is the sum of the rankings of the
alternative $i$ in the lists of the $m$ players. Intuitively, the alternative
$i^{*}$, such that $\eta_{i^{*},m}=\min_{i}(\eta_{\;\\!i,m})$, is the
alternative that minimizes the sum of energies, i.e. the ground state
solution. Hence, the two transfer matrix solutions correspond respectively to
directed percolation and directed polymer in our context.

Figure 4: Transfer matrix of the negotiation problem: the rows represent
players; the columns represent alternatives (ordered following A’s preference-
list). In the first row there are the energies assigned by player $A$ to each
of the $N$ alternatives; in the second row there are the energies assigned by
player $B$; and so on up to player $m$. In the transfer matrix, for each
column, we iterate step by step from the bottom to the top. In this example,
for the directed percolation, we have $\eta_{\;\\!1,A}=1$, and the first step
gives $\eta_{\;\\!1,B}=\max(\eta_{\;\\!1,A},19)=19$. By updating the energies
step by step, up to $t=m$, the agreed alternative can be found by
minimization, i.e. $\min_{i}(\eta_{\;\\!i,m})$

.

Even though the negotiation problem seems to be hard, it is now mapped to
well-known models that make it easily solvable, numerically, via the transfer
matrix methods. However, our analytical solutions are still valuable as
transfer matrix methods do not guarantee exact solutions. Combining our exact
solutions and the transfer matrix approaches, we obtain both an analytical
understanding and a fast algorithm to get numerical results. Moreover, the
transfer matrix method has the advantage to yield all the degenerate
solutions: each degenerate solution corresponds to a standard negotiation
process with a different first-mover. Therefore, with the transfer matrix
approach, one finds all the problem’s outcome with a single sweep.

Finally, we solve the ground state (21) by following a conventional
statistical mechanics approach. Let us define the partition function of the
system:

$Z_{\beta}=\sum_{i=1}^{N}e^{-\beta\epsilon_{i}}\;.$ (22)

Note that the partition function (22) is equivalent to that of the Random
Energy Model (REM) by Derrida [15].

To find the free energy, we are interested in the average of the logarithm of
the partition function over a distribution $\rho(\epsilon)$.

We have:
$F_{\beta}=\int\limits_{0}^{\infty}\frac{dl}{l}\left[e^{-l}-\left(\int\limits_{0}^{\infty}d\epsilon\rho(\epsilon)e^{-le^{-\beta\epsilon}}\right)^{N}\right]$.
Here the variables $\epsilon_{i}$ follow the distribution $\rho(\epsilon)=\int
dx_{1}..dx_{m}\delta\left(\epsilon-\sum_{i=1}^{m}x_{i}\right)$. Changing the
variable to $l=e^{\beta u}$, and knowing that
$\lim\limits_{\beta\rightarrow\infty}e^{e^{-\beta x}}=\theta(x)$, in the zero-
temperature limit we obtain:

$\begin{split}F_{\infty}=\int\limits_{0}^{\infty}du\left(1-\int\limits_{0}^{u}d\epsilon\rho(\epsilon)\right)^{N}=\int\limits_{0}^{\infty}duP(\epsilon_{{\scriptscriptstyle
0}}>u)=\overline{\epsilon_{{\scriptscriptstyle 0}}}\;,\end{split}$ (23)

that is the same ground state found in (21).

It is enlightening that, because of its simplicity, the traditional REM
obtained with the (non-broken symmetry) Replica method can be reproduced by a
procedural-combinatorial approach, making the negotiation problem particularly
interesting. The latter method enjoys the advantage to be intuitive at each
step, and the results can be verified numerically. On the other hand, the
physical insights in the former method remain obscure.

##  4 Conclusion

To conclude, we introduced a simple model of negotiation, where different
players with partially overlapping interests must find an agreement. We
studied its analytical and numerical solutions with our new probabilistic
procedure as well as with standard statistical mechanics approaches, showing
that it can be mapped to a class of well-known physics problems.

Moreover, it is an alternative model to those existing in the literature [16,
17, 18, 19], which enriches the understanding of the negotiation process by
proposing a different perspective. For example, it is possible to compare
negotiation processes with bounded rational and hyper-rational players. In
turn, this can help to find insights about people’s behaviour and reasoning.

We believe that the negotiation model is one of the simplest model dealing
with such complex situations and, beyond its potential applications and
extensions, it elucidates in a intuitive way the connection between social
sciences and statistical mechanics.

## Acknowledgement

The authors would like to thank Fei Jing, Guiyuan Shi, and Ruijie Wu who
supported this work with relevant discussions. This work was partially
supported by the Swiss National Science Foundation (grant no.
$200020\\_182498/1$).

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# Towards the Generalized Purely Wild Inertia Conjecture for product of
Alternating and Symmetric Groups

Soumyadip Das  Mathematics, Chennai Mathematical Institute, H1, SIPCOT IT
Park, Siruseri, Kelambakkam 603103, India<EMAIL_ADDRESS>
###### Abstract.

We obtain new evidence for the Purely Wild Inertia Conjecture posed by
Abhyankar and for its generalization. We show that this generalized conjecture
is true for any product of simple Alternating groups in odd characteristics,
and for any product of certain Symmetric or Alternating groups in
characteristic two. We also obtain important results towards the realization
of the inertia groups which can be applied to more general set up. We further
show that the Purely Wild Inertia Conjecture is true for any product of
perfect quasi $p$-groups (groups generated by their Sylow $p$-subgroups) if
the conjecture is established for individual groups.

###### Key words and phrases:

Galois covers of curves, Inertia Conjecture, ramification, formal patching

###### 2020 Mathematics Subject Classification:

14H30, 14G17 (Primary) 13B05, 11S15, 14B20 (Secondary)

##  1\. Introduction

The objective of this article is to establish the Generalized Purely Wild
Inertia Conjecture for arbitrary finite product of Alternating and Symmetric
groups in arbitrary prime characteristic. This is completely achieved in odd
characteristics, and with conditions on the degrees of these permutation
groups in even characteristic.

Let $p$ be a prime number and $k$ be an algebraically closed filed of
characteristic $p$. A finite group $G$ is said to be a quasi $p$-group if $G$
is generated by all its Sylow $p$-subgroups. In [1], Abhyankar posed a
conjecture stating that the quasi $p$-groups are precisely the groups
occurring as the Galois groups for connected Galois étale covers of the affine
$k$-line $\mathbb{A}^{1}$. This is a sharp contrast from the behavior of
finite covers defined over $\mathbb{C}$ as there is no non-trivial étale cover
of the Riemann sphere. The conjecture (known as Abhyankar’s Conjecture on the
affine line) was proved by Serre (for solvable quasi $p$-groups; [18]) and
Raynaud ([16]). It can be seen that any such Galois étale cover of
$\mathbb{A}^{1}$ extends uniquely to a Galois cover of the projective $k$-line
$\mathbb{P}^{1}$ that is branched over $\infty$ and is étale everywhere else.
Moreover, the inertia groups above $\infty$ are conjugate to each other. For a
quasi $p$-group $G$, and a subgroup $I\,\subset\,G$, we say that the pair
$(G,\,I)$ is realizable if there is a connected $G$-Galois cover of
$\mathbb{P}^{1}$ branched only at $\infty$, and at a point above $\infty$, the
inertia group is $I$. From the theory of extension of local fields, $I$ is
necessarily an extension of a $p$-group by a cyclic group of order prime-
to-$p$. The question arises that given a quasi $p$-group $G$ and a subgroup
$I\,\subset G$ which is potentially an inertia group (i.e., has the above
necessary form), whether the pair $(G,\,I)$ is realizable. Abhyankar
conjectured a necessary and sufficient group theoretic condition to this
question, now known as the Inertia Conjecture.

######  Conjecture 1.1 (The IC, [2, Section 16]).

Let $G$ be a finite quasi $p$-group. Let $I$ be a subgroup of $G$ which is an
extension of a $p$-group by a cyclic group of order prime-to $p$. Then the
pair $(G,\,I)$ is realizable if and only if the conjugates of $P$ in $G$
generate $G$ (in notation, $G=\,\langle\,P^{G}\,\rangle$).

It can be seen that the condition on $I$ in the conjecture is a necessary one.
When $I=P$ is a $p$-group, a special case of the above conjecture is known as
the Purely Wild Inertia Conjecture (henceforth referred to as the PWIC).

######  Conjecture 1.2 (The PWIC, [2, Section 16]).

Let $G$ be a finite quasi $p$-group. Let $P$ be a $p$-subgroup of $G$. Then
the pair $(G,\,P)$ is realizable if and only if $G=\langle\,P^{G}\,\rangle$.

It is known that the Inertia Conjecture is true when $G$ is a $p$-group. From
[8, Theorem 2], it follows that for any quasi $p$-group and a Sylow
$p$-subgroup $P$ of $G$, the pair $(G,\,P)$ is realizable. In particular, the
PWIC is true for any quasi $p$-group whose order is strictly divisible by $p$.
In other earlier works, the Inertia Conjecture was shown to be true for a few
groups. For a summary on these works and other important developments, see
[11, Section 4]. Recent works from [4], [5], [7] shed light on several
systematic ways to construct the Galois étale covers of the affine line. As a
consequence, a larger class of groups provide evidence towards the above
conjectures. In general, the status of the conjectures remain open at the
moment. In [5], the PWIC was generalized (the Generalized Purely Wild Inertia
Conjecture or GPWIC) to the case with multiple branched points in
$\mathbb{P}^{1}$.

######  Conjecture 1.3 (The GPWIC, [5, Conjecture 6.8]).

Let $G$ be a finite quasi $p$-group. Let $P_{1},\,\cdots,\,P_{r}$ be non-
trivial $p$-subgroup of $G$ for some $r\geq 1$ such that $G\,=\,\langle
P_{1}^{G},\,\cdots,\,P_{r}^{G}\rangle$. Let $B=\\{x_{1},\,\cdots,\,x_{r}\\}$
be a set of closed points in $\mathbb{P}^{1}$. Then there is a connected
$G$-Galois cover of $\mathbb{P}^{1}$ étale away from $B$ such that $P_{i}$
occurs as an inertia group above the point $x_{i}$ for $1\leq i\leq r$.

When $p$ is an odd prime, $A_{d}$ is a quasi $p$-group for all
$d\geq\text{max}\\{5,p\\}$; when $p=2$, $A_{d}$ for $d\geq 5$ is a quasi
$2$-group and $S_{d}$ for $d\geq 3$ is a quasi $2$-group. Moreover, any
product of quasi $p$-groups is again a quasi $p$-group. This way, we have a
large class of potential candidates for which the above conjectures can be
checked. Our goal in this paper is to establish the above mentioned conjecture
for the Alternating and Symmetric groups, and their products.

When $p$ is an odd prime, [5, Theorem 1.7] shows that the GPWIC is true for
any product $A_{d_{1}}\times\cdots\times A_{d_{n}}$, $n\geq 1$, where each
$d_{i}=p$ or $d_{i}\geq p+1$ is co-prime to $p$. In Corollary 3.4 we remove
this restriction on the $d_{i}$’s, hence proving the GPWIC for arbitrary
product of simple quasi $p$ Alternating groups. In view of the proof of [5,
Theorem 7.4], this boils down to showing the following important result.

######  Theorem 1.4 (Theorem 3.3).

Let $p$ be an odd prime, $r\geq 1$. Then the PWIC (Conjecture 1.2) is true for
$A_{rp}$.

In fact, we prove the following result which can be applied to a more general
set up.

######  Proposition 1.5 (Proposition 3.2).

Let $p$ be a prime. Let $G$ be a finite group, $G_{1}\times G_{2}\subset G$,
where $G_{1}$ and $G_{2}$ are quasi $p$-groups. Suppose that for $i=1,\,2$,
$\lambda_{i}$ is an element of order $p$ in $G_{i}$ such that the pair
$(G_{1}\times G_{2},\,\langle\,(\lambda_{1},\lambda_{2})\,\rangle)$ is
realizable. Let $c\in G$ be an element of order prime-to-$p$ that interchanges
$\lambda_{1}$ and $\lambda_{2}$ via conjugation, i.e.
$c^{-1}\,(\lambda_{1},\lambda_{2})\,c=(\lambda_{2},\lambda_{1})$. Consider the
subgroup $H\coloneqq\langle\,G_{1}\times G_{2},\,\langle\,c\,\rangle\,\rangle$
of $G$. Let $T$ be the maximal common quotient of $H$ and
$\langle\,c\,\rangle$. Then the pair
$(H\times_{T}\langle\,c\,\rangle,\,\langle\,(\lambda_{1},\lambda_{2})\,\rangle)$
is realizable.

The above result lets us potentially realize new Galois groups starting from
realization of a product of groups, while the inertia groups remain the same.
The approach is different from the previously employed methods to construct
purely wildly ramified covers. Namely, we first enlarge the Galois group,
enlarge the inertia by a prime-to-$p$ part and add new branched points. Then
we obtain our cover with the desired properties via a suitable pullback under
a Kummer cover. Such ideas were implicit in [5] to obtain Alternating group
covers with prescribed inertia groups using Abhyankar-type equations. To prove
the proposition, we apply a formal patching result Lemma 3.1 of Pries to the
given cover and a suitable Harbater-Katz-Gabber cover (see [15, Theorem
2.3.7]; one can ‘patch’ together two covers, one with inertia $\mathbb{Z}/p$
and the other with inertia of the form $\mathbb{Z}/p\rtimes\mathbb{Z}/m$,
$(p,m)=1$).

Another application (Corollary 3.7) of the above mentioned patching result is
the Inertia Conjecture (Conjecture 1.1) for $A_{p+1}$ for any prime $p\geq 5$
(this was proved in [5, Theorem 5.3] under the condition that $p\equiv
2\pmod{3}$).

In Remark 3.5, we note certain important general cases of product groups where
Proposition 1.5 can be applied. One important improvement from the earlier
known results (see [5, Theorem 7.5]) is the following.

######  Proposition 1.6 (Remark 3.5 (3)).

Let $p$ be a prime number. Let $G_{1}$ and $G_{2}$ be two perfect quasi
$p$-groups such that the PWIC holds for $G_{1}$ and $G_{2}$. Then the PWIC is
true for $G_{1}\times G_{2}$.

When $p=2$, we consider the product of Alternating and Symmetric groups. This
is the first non-trivial result towards the PWIC in characteristic $2$. We
establish the following evidence towards the GPWIC.

######  Theorem 1.7 (Corollary 3.17).

When $p=2$, the GPWIC (Conjecture 1.3) is true for any product
$G=G_{1}\times\cdots\times G_{n}$, $n\geq 1$, where $G_{i}=A_{d_{i}}$ for some
$d_{i}\geq 5$ with $4\nmid d_{i}$ or $G_{i}=S_{d_{i}}$ for an odd integer
$d_{i}\geq 5$.

The above result is proved in several steps. After we observe a group
theoretic characterization for the potential inertia groups in $A_{d}$ and
$S_{d}$ covers (Lemma 3.8), we prove the PWIC for $A_{d}$ when $4\nmid d$,
$d\geq 5$ (Theorem 3.11) and for $S_{d}$ when $d\geq 5$ is an odd integer
(Theorem 3.14). These results are consequence of formal patching results ([16,
Theorem 2.2.3]), [8, Theorem 2]); in fact, the covers are constructed out of
several simpler Galois covers where the Sylow $2$-subgroups are realized as
inertia groups. Further using the patching results to these $A_{d}$ and
$S_{d}$ covers, one obtains the general Theorem 1.7.

Throughout this article, we use consequences of formal and rigid patching
techniques. Except for the proof of Lemma 3.1 (which is essentially the same
as [15, Theorem 2.3.7]), the detailed method has not been used in this paper.
We omit the technical details and refer to [10], [6, Section 3.4] or [9].

### Acknowledgments

I would like to thank the referee for various suggestions in improving the
presentation of the paper. This work was completed while the author was at
Tata Institute of Fundamental Research, Mumbai. The author is supported by
NBHM Post-doctoral Fellowship.

##  2\. Notation, Convention and Definitions

We will use the following notation throughout this article without repeated
mention.

  1. (1)

Let $p$ be a prime number. Let $k$ be an algebraically closed field of
characteristic $p$.

  2. (2)

All the $k$-curves considered will be smooth connected curves over
$\text{Spec}(k)$, unless otherwise specified.

  3. (3)

In this article, we work with finite group covers. We will be mostly
interested in the Alternating and Symmetric groups of degree $\geq 5$.

  4. (4)

For a finite group $G$, let $p(G)$ denote the (necessarily normal) subgroup of
$G$ generated by all the Sylow $p$-subgroups of $G$. A finite group $G$ is
said to be a quasi $p$-group if $p(G)=G$.

  5. (5)

For any scheme $X$ and a point $x\in X$, the local ring at $x$ is denoted by
$\mathcal{O}_{X,x}$. We denote its completion at the maximal ideal by
$\widehat{\mathcal{O}}_{X,x}$. When $\widehat{\mathcal{O}}_{X,x}$ is a domain,
$K_{X,x}$ stands for its fraction field.

A cover of curves is defined to be a finite, generically separable morphism of
$k$-curves. We say that a cover $Y\,\longrightarrow\,X$ is connected
(respectively, integral or normal) if $X$ and $Y$ are both connected
(respectively, integral or normal). For a finite group $G$, a cover
$\phi\,\colon\,Y\,\longrightarrow\,X$ of integral $k$-curves is said to be
Galois with group $G$ or $G$-Galois if there is an inclusion of groups
$G\hookrightarrow\text{Aut}\left(Y/X\right)\coloneqq\\{\sigma\in\text{Aut}_{k}(Y)\,|\,f\circ\sigma=f\\}$
via which $G$ acts simply transitively on each generic geometric fibre. The
Galois theory and the Ramification theory of covers of Noetherian normal
schemes is standard in literature, see [17, Chapter IV], [6, Section 3]. Any
cover $f\,\colon\,Y\longrightarrow X$ of connected integral $k$-curves is
étale away from a (possibly empty) finite set of closed points. The closed
subset of points in $Y$ where $f$ is not étale is said to be the ramification
locus of $f$, and its image in $X$ is said to be the branch locus of $f$.

Our objects of study are the inertia groups over points in a $G$-Galois cover
$f\,\colon\,Y\longrightarrow X$ of smooth connected $k$-curves for a finite
group $G$. Since $k$ is an algebraically closed field, the inertia group at a
point $y\in Y$ is the stabilizer subgroup of $y$ under the natural transitive
$G$-action on the finite set $f^{-1}(f(y))\subset Y$. It turns out that the
inertia group at $y$ is also the Galois group of the Galois field extension
$K_{Y,y}/K_{X,f(y)}$. Moreover, for $y,\,y^{\prime}\in Y$ with
$f(y)=f(y^{\prime})$, the inertia groups at $y$ and $y^{\prime}$ are conjugate
in $G$. For a subgroup $I$ of $G$, we say that $I$ occurs as an inertia group
at a point $x\in X$ if there is a point $y\in Y$ with $f(y)=x$ such that
$I=\text{Gal}\left(K_{Y,y}/K_{X,x}\right)$. By [17, Chapter IV, Corollary 4],
every inertia group $I$ is of the form $I=P\rtimes\mathbb{Z}/m$ for some
$p$-group $P$ and a co-prime to $p$ integer $m$. We say that the inertia group
$I$ is purely wild or that the cover $f$ is purely wildly ramified over $x$ if
$I$ is a $p$-group. Similarly, $f$ is tamely ramified over $x$ or a tame
inertia group $I$ refers to the case when $I$ is a cyclic group of order
prime-to-$p$. Our main interest in this article is when
$X=\mathbb{P}^{1}_{k}$.

######  Definition 2.1.

Let $G$ be a finite group, $I\subset G$. We say that the pair $(G,\,I)$ is
realizable if there exists a $G$-Galois cover $Y\longrightarrow\mathbb{P}^{1}$
of smooth projective connected $k$-curves that is étale away from $\infty$,
and $I$ occurs as an inertia group above $\infty$.

In the above definition, $G$ is necessarily a quasi $p$-group (i.e. $G=p(G)$),
and the group $I$ has the necessary structure of an inertia group.

##  3\. The GPWIC for certain product groups

Throughout this section, $k$ is an algebraically closed field of a prime
characteristic $p$. Our goal is to establish the Generalized Purely Wild
Inertia Conjecture (the GPWIC, Conjecture 1.3) for any finite product of
Alternating groups for an arbitrary prime $p$ and also of Symmetric groups
when $p=2$. Recall that the GPWIC (Conjecture 1.3) asserts that given a quasi
$p$-group $G$ and a nonempty finite set $B\,\subset\,\mathbb{P}^{1}$ of closed
points, there is connected $G$-Galois cover of $\mathbb{P}^{1}_{k}$, étale
away from $B$, with prescribed $p$-groups (satisfying a necessary condition:
their conjugates generate $G$) as the inertia groups above the points in $B$.

When $p$ is an odd prime number, $A_{d}$ is a quasi $p$-group for all
$d\geq\text{\rm max}\\{5,p\\}$. For $p=2$, $A_{d}$ is a quasi $2$-group for
$d\geq 5$, and $S_{d}$ is a quasi $2$-group for $d\geq 3$. Throughout this
section, we will implicitly assume these bounds on $d$. Our strategy is to
first prove the PWIC (Conjecture 1.2) for $A_{d}$ (with some restrictions on
$d$ for $p=2$), namely, for any $p$-subgroup $P\subset A_{d}$, there is a
connected $A_{d}$-Galois cover of $\mathbb{P}^{1}$, étale away from $\infty$,
such that $P$ occurs as an inertia group above $\infty$ (i.e. the pair
$(A_{d},\,P)$ is realizable; Definition 2.1). Then the argument of [4, Theorem
7.4] shows that the GPWIC holds for arbitrary product of the above Alternating
groups. We will prove the GPWIC for product of Symmetric groups in Theorem
3.16.

###  3.1. Odd Characteristic Case

We suppose that $p\geq 3$. By [5, Theorem 1.7(3)], the GPWIC is true for any
product $A_{d_{1}}\times\cdots\times A_{d_{n}}$, $n\geq 1$, where each
$d_{i}=p$ or $d_{i}\geq p+1$ is co-prime-to $p$. This was proved using formal
patching techniques and by understanding the covers of the affine $k$-line
given by explicit affine equations, à la Abhyankar. We will remove
restrictions on the $d_{i}$’s:

$p\nmid d_{i}\text{ if }d_{i}>p$

which arose from the fact that the status of the PWIC remained unknown for the
groups $A_{rp}$, $r\geq 2$ (see [4, Corollary 5.5, Corollary 5.6]). This is
achieved as an application of the general result Proposition 3.2 which in turn
is obtained as a consequence of the following result on the construction of
covers using formal patching due to Pries ([15, Theorem 2.3.7, Remark 2.3.8];
although the original result assumes that $p$ strictly divides the order of
the group $G$ in its statement, we observe that the same argument works in the
following general set up as the inertia groups have Sylow $p$-subgroups of
order $p$). Although the proof is a bit technical, involving certain numerical
criteria followed by a standard application of formal patching and a Lefchetz
type argument, we include it for the sake of completeness.

######  Lemma 3.1 ([15, Theorem 2.3.7, Remark 2.3.8]).

Let $X$ be a smooth projective connected $k$-curve, $x\in X$ be a closed
point. Let $G$ be a finite group and $I\subset G$ be an extension of a
$p$-cyclic group $\langle\,\tau\,\rangle\cong\mathbb{Z}/p$ by a cyclic group
$\langle c\rangle$ of order $m$, $(m,p)\,=\,1$. Let $G_{1}$ and $G_{2}$ be
subgroups of $G$ such that $\tau\in G_{1},\,I\subset G_{2}$, and assume that
the following hold.

  1. (1)

There is a connected $G_{1}$-Galois cover $\psi_{1}$ of $X$ étale away from a
set $B\subset X$ of closed points with $x\in B$. Suppose that
$\langle\,\tau\,\rangle$ occurs as an inertia group above $x$. For each point
$y\neq x$ in $B$, let $I_{y}$ denote an inertia group above $y$.

  2. (2)

$\psi_{2}\,\colon\,Y_{2}\longrightarrow\mathbb{P}^{1}$ is a connected
$G_{2}$-Galois cover, étale away from a set
$\\{0=\eta_{0},\eta_{1},\ldots,\eta_{r}\\}$ such that $I$ occurs as an inertia
group above $0$, and for $1\leq i\leq r$, let $J_{i}$ denote an inertia group
above $\eta_{i}$.

If $G=\langle\,G_{1},G_{2}\,\rangle$, there is a set
$B^{\prime}=\\{x_{1},\ldots,x_{r}\\}$ of closed points in $X$, disjoint from
$B$, and a connected $G$-Galois cover $Y\longrightarrow X$, étale away from
$B\sqcup B^{\prime}$, such that $I$ occurs as an inertia group above $x$, for
each point $y\neq x$ in $B$, $I_{y}$ occurs as an inertia group above $y$, and
$J_{i}$ occurs as an inertia group above $x_{i}$ for $1\leq i\leq r$.

###### Proof.

We have $I=\langle\,\tau\,\rangle\rtimes\langle\,c\,\rangle$. Let $m^{\prime}$
be the order of the prime-to-$p$ part of the center of $I$. Let $u$ be a local
parameter in $\mathbb{P}^{1}$ at $0$. Let $y_{2}\in Y_{2}$ be a point above
$0$ such that $I=\text{Gal}\left(K_{Y_{2},y_{2}}/k((u))\right)$. Let $h_{2}$
be the conductor of this extension. Then $(h_{2},p)=1$,
$(h_{2},m)=m^{\prime}$, and $m\,|\,h_{2}(p-1)$. Also let $v$ be a local
parameter in $X$ at $x$. Thus we identify $K_{X,x}$ with $k((v))$. Suppose
that $h_{1}^{\prime}$ be the conductor for the local
$\langle\,\tau\,\rangle\cong\mathbb{Z}/p$-Galois field extension in $\psi_{1}$
near $x$. So, $(h_{1}^{\prime},p)=1$.

Choose $\gamma$ co-prime to $p$ such that $(\gamma,m)=1$, $p\nmid(\gamma
m+1)$, and $\gamma h_{2}\geq h_{1}^{\prime}$. Set $h_{1}\coloneqq\gamma
h_{2}$. By [15, Theorem 2.2.2], there is a connected $G_{1}$-Galois cover
$\phi_{1}\,\colon Y_{1}\longrightarrow X$ with the same inertia groups and
ramification behavior above the points $y\neq x$ in $X$, and there is a point
$y_{1}\in Y_{1}$ above $x$ such that the $\langle\,\tau\,\rangle$-Galois field
extension $K_{Y_{1},y_{1}}/K_{X,x}$ has conductor $h_{1}$.

Consider the integral $k[[t]]$-scheme
$S\,\coloneqq\text{Spec}\left(k[[u,v,t]]/(uv-t^{\gamma m+1})\right)$, and
denote its closed fibre (the subscheme given by the locus of $t=0$) by
$S^{\prime}$. By our choice of the local parameters, $S^{\prime}$ is the union
of $\text{Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)$ with
$\text{Spec}\left(\widehat{\mathcal{O}}_{\mathbb{P}^{1},0}\right)$, meeting at
the point $\sigma=(x,0)$ which we identify with $(v=0,u=0)$ and has equation
$uv=t^{\gamma m+1}$ near the point $\sigma$.

Set $e\coloneqq h_{1}m+h_{2}=(\gamma m+1)h_{2}$. Then
$(e,m)=(h_{2},m)=m^{\prime}$, $p\nmid e$, and
$(m,\frac{e}{(h_{1},h_{2})})=(m,\frac{e}{h_{2}})=(m,\gamma m+1)=1$, and [15,
Notation 2.3.2 and Numerical Hypothesis] hold. By [15, Theorem 2.3.4], there
is an irreducible $I$-Galois cover
$\widehat{\phi}\,\colon\,\widehat{Z}\longrightarrow S$ whose generic fibre is
irreducible, and the special fibre
$\widehat{\phi}_{k}\,\colon\,Z_{k}\longrightarrow S^{\prime}$ has the
following properties.

$\displaystyle Z_{k}\times_{S^{\prime}}\text{Spec}\left(K_{X,x}\right)$
$\displaystyle\cong$
$\displaystyle\text{Ind}_{\langle\,\tau\,\rangle}^{I}\,\text{Spec}\left(K_{Y_{1},y_{1}}\right),$
$\displaystyle Z_{k}\times_{S^{\prime}}\text{Spec}\left(k((u))\right)$
$\displaystyle\cong$ $\displaystyle\text{Spec}\left(K_{Y_{2},y_{2}}\right)$

as $I$-Galois étale covers of $\text{Spec}\left(K_{X,x}\right)$ and
$\text{Spec}\left(k((u))\right)$, respectively.

Let $T$ be a regular irreducible projective $k[[t]]$-curve whose geometric
fibre is $X_{k((t))}$, and the closed fibre is $X\cup\mathbb{P}^{1}$ meeting
at the point $\sigma=(x,0)$ (which we have identified with $(v=0,u=0)$ having
equation $uv=t^{\gamma m+1}$ near $\sigma$; so
$\text{Spec}\left(\widehat{\mathcal{O}}_{T,\sigma}\right)=S$). Let
$X-x=\text{Spec}\left(A\right)$,
$Y_{1}-\psi_{1}^{-1}(x)=\text{Spec}\left(B_{1}\right)$,
$Y_{2}-\psi_{2}^{-1}(0)=\text{Spec}\left(B_{2}\right)$. By [9, Proposition
2.3], there is an irreducible normal $\langle\,G_{1},G_{2}\,\rangle=G$-Galois
cover $h\,\colon\,V\longrightarrow T$ such that

$\displaystyle V\,\times_{T}\,\text{Spec}\left(A[[t]]\right)$
$\displaystyle\cong$
$\displaystyle\text{Ind}_{G_{1}}^{G}\,\text{Spec}\left(B_{1}[[t]]\right),$
$\displaystyle V\,\times_{T}\,\text{Spec}\left(k[u^{-1}][[t]]\right)$
$\displaystyle\cong$
$\displaystyle\text{Ind}_{G_{2}}^{G}\,\text{Spec}\left(B_{2}[[t]]\right),\,\text{and}$
$\displaystyle
V\,\times_{T}\,\text{Spec}\left(\widehat{\mathcal{O}_{T,\sigma}}\right)$
$\displaystyle\cong$
$\displaystyle\text{Ind}_{I}^{G}\,\text{Spec}\left(\widehat{Z}\right)$

as Galois covers of $\text{Spec}\left(A[[t]]\right)$,
$\text{Spec}\left(k[u^{-1}][[t]]\right)$ and
$S=\text{Spec}\left(\widehat{\mathcal{O}}_{T,\sigma}\right)$, respectively.
Let $h^{0}\,\colon\,V^{0}\longrightarrow X_{k((t))}$ be the generic fibre of
$h$. Then there exists a set $B^{\prime}=\\{x_{1},\ldots,x_{r}\\}\subset X$
disjoint from $B$ such that $h^{0}$ is étale away from $\\{b_{k((t))}\,|\,b\in
B\sqcup B^{\prime}\\}$, $I$ occurs as an inertia group above $x_{k((t))}$, for
each point $y\neq x$ in $B$, $I_{y}$ occurs as an inertia group above
$y_{k((t))}$, for each $1\leq i\leq r$, $J_{i}$ occurs as an inertia group
above $x_{i,k((t))}$. As $h$ is a cover of smooth irreducible $k((t))$-curves
and the closed fibre of $h$ is generically smooth, the result follows from [9,
Corollary 2.7]. ∎

As a consequence of the above, we have the following characteristic-free
general result on the realization of certain pairs (Definition 2.1). This can
be applied to potentially increase the Galois group without changing the
$\mathbb{Z}/p$ inertia.

######  Proposition 3.2.

Let $p$ be a prime. Let $G$ be a finite group, $G_{1}\times G_{2}\subset G$,
where $G_{1}$ and $G_{2}$ are quasi $p$-groups. Suppose that for $i=1,\,2$,
$\lambda_{i}$ is an element of order $p$ in $G_{i}$ such that the pair
$(G_{1}\times G_{2},\,\langle\,(\lambda_{1},\lambda_{2})\,\rangle)$ is
realizable. Let $c\in G$ be an element of order prime-to-$p$ that interchanges
$\lambda_{1}$ and $\lambda_{2}$ via conjugation, i.e.
$c^{-1}\,(\lambda_{1},\lambda_{2})\,c=(\lambda_{2},\lambda_{1})$. Consider the
subgroup $H\coloneqq\langle\,G_{1}\times G_{2},\,\langle\,c\,\rangle\,\rangle$
of $G$. Let $T$ be the maximal common quotient of $H$ and
$\langle\,c\,\rangle$. Then the pair
$(H\times_{T}\langle\,c\,\rangle,\,\langle\,(\lambda_{1},\lambda_{2})\,\rangle)$
is realizable.

###### Proof.

Let $Y_{1}\longrightarrow\mathbb{P}^{1}$ be a connected $G_{1}\times
G_{2}$-Galois cover, étale away from $\infty$, such that
$\langle\,(\lambda_{1},\lambda_{2})\,\rangle$ occurs as an inertia group above
$\infty$. Let $Y_{2}\longrightarrow\mathbb{P}^{1}$ be a
$\langle\,(\lambda_{1},\lambda_{2})\,\rangle\rtimes\langle\,c\,\rangle$-Galois
Harbater-Katz-Gabber cover that is totally ramified over $\infty$,
$\langle\,c\,\rangle$ occurs as an inertia group above $0$, and is étale
everywhere else. By Lemma 3.1, we obtain a connected $H$-Galois cover
$Y\longrightarrow\mathbb{P}^{1}$, étale away from $\\{0,\infty\\}$, such that
$\langle\,(\lambda_{1},\lambda_{2})\,\rangle\rtimes\langle\,c\,\rangle$ occurs
as an inertia group above $\infty$, and $\langle\,c\,\rangle$ occurs as an
inertia group above $0$. Consider the connected $\langle\,c\,\rangle$-Galois
Kummer cover $\psi\,\colon\,Z\cong\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1}$
that is étale away from $\\{0,\infty\\}$, over which the cover is totally
ramified. Let $W$ be a dominant component in the normalization of
$Y\times_{\mathbb{P}^{1}}Z$, and consider the cover $f\colon W\longrightarrow
Z\cong\mathbb{P}^{1}$. Since the function field $k(W)$ is the compositum
$k(Y)\cdot k(Z)$, we see that $f$ is a Galois cover with group
$H\times_{T}\langle\,c\,\rangle$. By [14, Proposition 3.5], the Galois cover
$f$ is étale away from $\infty$, and
$\langle\,(\lambda_{1},\lambda_{2})\,\rangle$ occurs as an inertia group above
$\infty$. ∎

Now we are ready to prove the PWIC (Conjecture 1.2) for $A_{rp}$.

######  Theorem 3.3.

Let $p$ be an odd prime, $r\geq 1$. Then the PWIC holds for $A_{rp}$.

###### Proof.

By [3, Theorem 1.2], the IC – Conjecture 1.1 (and hence the PWIC) holds for
$A_{p}$ with $p\geq 5$. The PWIC holds for $A_{3}$ since $A_{3}$ is the
$3$-cyclic group. We suppose that $r\geq 2$.

For $1\leq i\leq r$, consider the $p$-cycle

$\tau_{i}\coloneqq((i-1)p+1,\ldots,ip)\in A_{rp}.$

Since $A_{rp}$ is a simple quasi $p$-group, it is generated by the conjugates
of any cyclic subgroup of order $p$. As any $p$-subgroup of $A_{rp}$ contains
an element of order $p$, and the inertia groups above a point in any connected
$A_{rp}$-Galois cover are conjugates in $A_{rp}$, in view of [8, Theorem 2],
it is enough to prove the following.

_For each $1\leq u\leq r$, the pair
$(A_{rp},\,\langle\,\tau_{1}\cdots\tau_{u}\,\rangle)$ is realizable._

When $u<r$, this is [4, Corollary 5.6]. Let $u=r$, and set
$\tau=\tau_{1}\cdots\tau_{r}$. We proceed via induction on $r$ and apply
Proposition 3.2 to show that the pair $(A_{rp},\,\langle\,\tau\,\rangle)$ is
realizable. By the induction hypothesis, the pair
$(A_{(r-1)p},\,\langle\,\tau_{1}\cdots\tau_{r-1}\,\rangle)$ is realizable. By
[4, Theorem 5.2], the pair $(A_{(r-1)p}\times
A_{p},\,\langle\,(\tau_{1}\cdots\tau_{r-1},\tau_{r})\,\rangle)$ is realizable,
as well. Now take
$G_{1}=A_{(r-1)p},\,G_{2}=A_{p}=\text{Alt}\\{(r-1)p+1,\ldots,rp\\},\,G=S_{rp},\,\lambda_{1}=\tau_{1}\cdots\tau_{r-1}$
and $\lambda_{2}=\tau_{r}$ in Proposition 3.2. Consider the odd permutation

$c\coloneqq(1,(r-1)p+1)\,(2,(r-1)p+2)\,\cdots\,(p,rp)\in S_{rp}$

of order $2$. Under the natural embedding $A_{(r-1)p}\times
A_{p}\hookrightarrow S_{rp}$, we identify the element
$(\tau_{1}\cdots\tau_{r-1},\tau_{r})$ with $\tau=\tau_{1}\cdots\tau_{r}$. The
element $c$ acts on $\tau$ via conjugation. Then we have
$H=\langle\,G_{1}\times G_{2},\,\langle\,c\,\rangle\,\rangle=S_{rp}$. As the
maximal common quotient of $H$ and $\langle\,c\,\rangle\cong\mathbb{Z}/2$ is
$\langle\,c\,\rangle$, by Proposition 3.2, we obtain a connected
$A_{rp}$-Galois cover of $\mathbb{P}^{1}$, branched only at $\infty$, and
$\langle\,\tau\,\rangle$ occurs as an inertia group above $\infty$. ∎

A similar argument as in [5, Theorem 7.4] establishes the GWPIC for product of
Alternating groups.

######  Corollary 3.4.

Let $p$ be an odd prime, $n\geq 1$. For $1\leq i\leq n$, let
$d_{i}\geq\text{max}\\{p,5\\}$. The GPWIC holds for the product of Alternating
groups $A_{d_{1}}\times\cdots\times A_{d_{n}}$.

We note the following cases where Proposition 3.2 can be applied.

######  Remark 3.5.

The pair $(G_{1}\times G_{2},\,\langle\,(\tau_{1},\tau_{2})\,\rangle)$ is
realizable in the following cases.

  1. (1)

([4, Theorem 5.2]) When the PWIC holds for two perfect (groups whose derived
subgroup is the whole group) quasi $p$-group $G_{1}$ and $G_{2}$, and for any
$1\leq a\leq p-1$, there is an automorphism of $G_{1}\times G_{2}$ taking
$(\tau_{1}^{a},\tau_{2})$ to $(\tau_{1},\tau_{2})$, the pair $(G_{1}\times
G_{2},\,\langle\,(\tau_{1},\tau_{2})\,\rangle)$ is realizable.

  2. (2)

([5, Theorem 7.5]) Suppose that for $i=1,\,2$, the pair
$(G_{i},\,\langle\,\tau_{i}\,\rangle)$ is realizable. Let
$\pi_{i}\,\colon\,G_{1}\times G_{2}\longrightarrow G_{i}$ be the projections.
If $G_{1}$ and $G_{2}$ do not have a common quotient, and $Q\subset
G_{1}\times G_{2}$ is a $p$-subgroup such that
$\pi_{i}(Q)\cong\langle\,\tau_{i}\,\rangle$, then the pair $(G_{1}\times
G_{2},\,Q)$ is realizable. In particular, this holds for
$Q=\langle\,(\tau_{1},\tau_{2})\,\rangle$.

  3. (3)

We note that the above result (2) can be generalized to:

_for perfect quasi $p$-groups $G_{1}$ and $G_{2}$ for which the PWIC hold, the
PWIC also holds for $G_{1}\times G_{2}$._

To see this, consider a $p$-subgroup $P\subset G\coloneqq G_{1}\times G_{2}$
with $\langle\,P^{G}\,\rangle=G$. Let $\pi_{i}$ be the obvious projections as
before. By Goursat’s Lemma, $P=\pi_{1}(P)\times_{Q}\pi_{2}(P)$ for a common
quotient $Q$ of $\pi_{1}(P)$ and $\pi_{2}(P)$. By [13, Corollary 4.6], the
pairs $(G_{1}\times\pi_{2}(P),P)$ and $(\pi_{1}\times G_{2},P)$ are
realizable. Applying [5, Lemma 4.6] to the Galois covers realizing these
pairs, we obtain respective covers with the following additional property: the
local $P$-Galois extensions over $K_{\mathbb{P}^{1},\infty}$ are isomorphic.
This allows us to use a standard formal patching technique to obtain a
connected $G_{1}\times G_{2}$-Galois cover of $\mathbb{P}^{1}$, étale away
from $\infty$, such that $P$ occurs as an inertia group above $\infty$.

Before moving on to the even characteristic case, we see another consequence
of the patching result Lemma 3.1 and the above corollary. This allows one to
check the full Inertia Conjecture for a fewer candidates of potential inertia
groups.

######  Proposition 3.6.

Let $p$ be a prime number, $G$ be a finite group, and $\mathbb{Z}/p\cong P$ be
a subgroup of $G$ such that the pair $(G,\,P)$ is realizable. Let $\beta\in
N_{G}(P)$ be an element of order prime-to-$p$. Suppose that $H$ is a subgroup
of $G$ containing $I\coloneqq P\rtimes\langle\,\beta\,\rangle$, and the pair
$(H,\,I)$ is realizable. Then for any $p$-subgroup $P^{\prime}$ of $G$
containing $P$ that is normalized by $\beta$, the pair
$(G,\,P^{\prime}\rtimes\langle\,\beta\,\rangle)$ is realizable.

In particular, let $p$ be an odd prime, $d^{\prime}\geq d\geq p$. Suppose that
for some element $\beta\in N_{A_{d}}(\langle\,(1,\ldots,p)\,\rangle)$ of order
prime-to-$p$, the pair
$(A_{d},\,\langle\,(1,\ldots,p)\,\rangle\rtimes\langle\,\beta\,\rangle)$ is
realizable. If $P\subset A_{d^{\prime}}$ is a $p$-subgroup containing the
$p$-cycle $(1,\ldots,p)$ that is normalized by $\beta$, the pair
$(A_{d^{\prime}},\,P\rtimes\langle\,\beta\,\rangle)$ is realizable.

###### Proof.

By our assumption, there are connected Galois covers $\psi_{1}$ and $\psi_{2}$
of $\mathbb{P}^{1}$, étale away from $\infty$, with Galois groups $G$ and $H$,
respectively, and such that the inertia groups above $\infty$ are the
conjugates of $P$ and $I$, respectively, in the corresponding Galois groups.
By Lemma 3.1, the pair $(G,\,I)$ is realizable. By [8, Theorem 2], the pair
$(G,\,P^{\prime}\rtimes\langle\,\beta\,\rangle)$ is realizable as well.

By Corollary 3.4, for any $d^{\prime}\geq p$, the pair
$(A_{d^{\prime}},\,\langle\,(1,\ldots,p)\,\rangle)$ is realizable. So, the
first statement applies with $G=A_{d^{\prime}}$ and $H=A_{d}\subset
A_{d^{\prime}}$. ∎

Using the above, we prove the full Inertia Conjecture (Conjecture 1.1) for
$A_{p+1}$ for $p\geq 5$. This was proved in [5, Theorem 5.3] under the
assumption that $p\equiv 2\pmod{3}$.

######  Corollary 3.7.

For any prime number $p\geq 5$, the Inertia Conjecture is true for $A_{p+1}$.

###### Proof.

This follows from Proposition 3.6 since

$N_{A_{p}}(\langle\,(1,\cdots,p)\,\rangle)=N_{A_{p+1}}(\langle\,(1,\cdots,p)\,\rangle),$

and the Inertia Conjecture (Conjecture 1.1) holds for $A_{p}$, $p\geq 5$ by
[3, Theorem 1.2]. ∎

###  3.2. Even characteristic case

Now we show that the Generalized Purely Wild Inertia Conjecture (GPWIC,
Conjecture 1.3) is true for any product of certain Symmetric and Alternating
groups in characteristic $2$. This is the first ever non-trivial result
towards the PWIC and the GPWIC in characteristic $2$. One of the main
ingredients of the proofs will be Raynaud’s patching result [16, Theorem
2.2.3]. We start with the study of the PWIC.

Let $P$ be a $2$-subgroup of a finite quasi $2$-group $G$ such that
$\langle\,P^{G}\,\rangle\,=\,G$, i.e. the conjugates of $P$ in $G$ generate
$G$. The PWIC asserts that the pair $(G,\,P)$ is realizable (Definition 2.1).
By [8, Theorem 2], if $P$ is a $2$-subgroup as above for which the pair
$(G,\,P)$ is realizable, and $P^{\prime}\subset G$ is a $2$-subgroup
containing $P$, then the pair $(G,\,P^{\prime})$ is also realizable. Note that
a Symmetric group $S_{d}$ is a quasi $2$-group for all $d\geq 2$ ($S_{3}$ has
order strictly divisible by $2$, and the PWIC holds for it by [16, Corollary
2.2.2]). In fact, for $d\geq 2$, $S_{d}$ is a quasi $p$-group only for the
prime $p\,=\,2$. On the other hand, $A_{d}$ is a quasi $2$-group for all
$d\geq 5$ ($A_{3}$ and $A_{4}$ are quasi $p$-groups only for the prime
$p\,=\,3$). In these cases, we characterize the potential purely wild inertia
groups ($2$-subgroups) $P$.

######  Lemma 3.8.

Let $d\geq 5$. We have the following.

  1. (1)

Let $P\subset S_{d}$ be a $2$-subgroup such that
$\big{\langle}\,P^{\,S_{d}}\,\big{\rangle}\,=\,S_{d}$. Then $P$ contains an
odd permutation $\tau$ such that
$\big{\langle}\,\langle\,\tau\,\rangle^{\,S_{d}}\,\big{\rangle}\,=\,S_{d}$.

  2. (2)

Let $P\subset A_{d}$ be a $2$-subgroup such that
$\big{\langle}\,P^{\,A_{d}}\,\big{\rangle}\,=\,A_{d}$. Then $P$ contains an
even permutation $\tau$ of order $2$ such that
$\big{\langle}\,\langle\,\tau\,\rangle^{\,A_{d}}\,\big{\rangle}\,=\,A_{d}$.

If $P\subset S_{4}$ is a $2$-subgroup such that
$\big{\langle}\,P^{\,S_{4}}\,\big{\rangle}\,=\,S_{4}$, then $P$ contains
either a transposition or a $4$-cycle $\tau$ such that
$\big{\langle}\,\langle\,\tau\,\rangle^{\,S_{4}}\,\big{\rangle}\,=\,S_{4}$.

###### Proof.

  1. (1)

If $P\subset A_{d}$, every conjugate of $P$ is also contained in $A_{d}$.
Since $\big{\langle}\,P^{\,S_{d}}\,\big{\rangle}\,=\,S_{d}$, we may choose an
odd permutation $\tau\in P$. Since
$\big{\langle}\,\langle\,\tau\,\rangle^{\,S_{d}}\,\big{\rangle}$ is a non-
trivial normal subgroup of $S_{d}$ containing an odd permutation $\tau$,
$\big{\langle}\,\langle\,\tau\,\rangle^{\,S_{d}}\,\big{\rangle}\,=\,S_{d}$.

  2. (2)

Since $A_{d}$ is a simple group, for every non-identity element $\tau\in
A_{d}$, we have
$\big{\langle}\,\langle\,\tau\,\rangle^{\,A_{d}}\,\big{\rangle}\,=\,A_{d}$. As
$P$ always contains an element $\tau$ of order $2$, the result follows.

As in (1), $P$ contains an odd permutation $\tau\in S_{4}$ such that
$\big{\langle}\,\langle\,\tau\,\rangle^{\,S_{4}}\,\big{\rangle}\,=\,S_{4}$. In
$S_{4}$, such an element must be either a transposition or a $4$-cycle. ∎

We first consider connected $A_{d}$-Galois and $S_{d}$-Galois covers ($d\geq
5$) of $\mathbb{P}^{1}$ over an algebraically closed field of characteristic
$2$, étale away from $\\{\infty\\}$, such that the inertia groups above
$\infty$ are cyclic of order $2$; cf. Lemma 3.8.

######  Proposition 3.9.

Let $d\geq 5$, and $1\leq r<\lfloor d/2\rfloor$ be an integer. For $1\leq
u\leq r$, let $\tau_{u}$ be the transposition $(2u-1,2u)$, and consider the
element $\tau\coloneqq\tau_{1}\cdots\tau_{r}\in S_{d}$ of order $2$. Over any
algebraically field of characteristic $2$, we have the following.

  1. (1)

If $r$ is an even integer, the pair $(A_{d},\,\langle\,\tau\,\rangle)$ is
realizable.

  2. (2)

If $r$ is an odd integer, the pair $(S_{d},\,\langle\,\tau\,\rangle)$ is
realizable.

###### Proof.

For each $1\leq u\leq r$ and each $b\in\\{2r+1,\ldots,d\\}$ (as $2r<d$ by out
assumption, this set is non-empty), consider the $3$-cycle

$\sigma_{u,b}\,\coloneqq\,(2u-1,2u,b).$

Then for any $u$ and $b$, we have
$\tau_{u}^{-1}\sigma_{u,b}\tau_{u}=\sigma_{u,b}^{2}$. As $\tau_{u^{\prime}}$
has support disjoint from the set $\\{2u-1,\,2u,\,b\\}$ for $u^{\prime}\neq
u$, $\tau^{-1}\sigma_{u,b}\tau=\sigma_{u,b}^{2}$. So for each $u$ and $b$,
$\tau\in N_{S_{d}}(\langle\,\sigma_{u,b}\,\rangle)$, and the subgroup
$H_{u,b}\coloneqq\langle\,\sigma_{i,b}\,\rangle\rtimes\langle\,\tau\,\rangle$
is a quasi $2$-group of order $6$. As $\langle\,\tau\,\rangle$ is a Sylow
$2$-subgroup of $H_{u,b}$, by Raynaud’s proof of the Abhyankar’s Conjecture on
the affine line ([16, Corollary 2.2.2]), the pair
$(H_{u,b},\,\langle\,\tau\,\rangle)$ is realizable when the base field has
characteristic $2$. Consider the quasi $2$-subgroup

$H\,\coloneqq\,\langle\,H_{u,b}\,|\,1\leq u\leq r,\,2r+1\leq b\leq d\,\rangle$

of $S_{d}$. From the construction, $H$ is a primitive subgroup of $S_{d}$. By
the patching result [16, Theorem 2.2.3], the pair
$(H,\,\langle\,\tau\,\rangle)$ is realizable.

When $d=5$, by our assumption, $r=1\text{ or }2$. If $r=1$, $H$ contains the
transposition $(1,2)$. By Jordan’s Theorem [12, Theorem 1.1], $A_{5}\subset
H$. If $r=2$, $H$ contains the $3$-cycle $(1,2,5)$ that fixes $2$ points in
$\\{1,\ldots,5\\}$. By [12, Theorem 1.2(3)], $A_{5}\subset H$ (note that
$PGL_{2}(4)\cong A_{5}$). For $d\geq 6$, $H$ is a primitive permutation group
containing a $3$-cycle $(1,2,d)$; so again by Jordan’s Theorem, $A_{d}\subset
H$. Since for all $d\geq 5$, $H$ is a quasi $2$-subgroup of $S_{d}$ containing
$A_{d}$, we have $H\in\\{A_{d},S_{d}\\}$.

If $r$ is an even integer, $\tau$ is an even permutation. So
$\left\langle\,\langle\,\tau\,\rangle^{\,S_{d}}\,\right\rangle=A_{d}$. Since
$\left\langle\,\langle\,\tau\,\rangle^{\,H}\,\right\rangle=H$, we have
$H=A_{d}$. If $r$ is an odd integer, $\tau$ is an odd permutation and we have
$H=S_{d}$. ∎

######  Remark 3.10.

A similar argument as above shows that for any transposition $\tau\in S_{4}$,
the pair $(S_{4},\,\langle\,\tau\,\rangle)$ is realizable. The subgroup $H$ as
in the proof becomes a primitive quasi $2$-subgroup of $S_{4}$ containing the
transposition $\tau$ and a $3$-cycle. As $PGL_{2}(3)\cong S_{4}$, by [12,
Theorem 1.2(3)], $H$ contains $A_{4}$. Since $H$ is a quasi $2$-subgroup of
$S_{4}$, we conclude that $H=S_{4}$.

######  Theorem 3.11.

Let $d\geq 5$ be an integer that is not a multiple of $4$. The PWIC is true
for $A_{d}$ in characteristic $2$.

###### Proof.

Let $P\subset A_{d}$ be a $2$-subgroup such that
$\left\langle\,P^{\,A_{d}}\,\right\rangle=A_{d}$. By Lemma 3.8 (2), there is
an even permutation $\tau\in P$ of order $2$ such that
$\left\langle\,\langle\,\tau\,\rangle^{\,A_{d}}\,\right\rangle=A_{d}$.
Consider a disjoint cycle decomposition $\tau=\tau_{1}\cdots\tau_{r}$ of
$\tau$ in $A_{d}$ where each $\tau_{u}$ is a transposition, and $r$ is
necessarily an even integer, $2r\leq d$. Since $4\nmid d$ by our assumption,
$r<\lfloor d/2\rfloor$. As $r\geq 2$, $\tau$ is conjugate to the permutation
$(1,2)\cdots(2r-1,2r)$. By Proposition 3.9 (1), the pair
$(A_{d},\,\langle\,\tau\,\rangle)$ is realizable. Now applying [8, Theorem 2],
the pair $(A_{d},\,P)$ is realizable. ∎

Remark 3.5(3) and the above result have the following immediate consequence.

######  Corollary 3.12.

In characteristic $2$, the GPWIC (Conjecture 1.3) is true for any product
$A_{d_{1}}\times\cdots\times A_{d_{u}}$, $u\geq 1$, where each $d_{i}\geq 5$
is an integer not divisible by $4$.

######  Remark 3.13.

The smallest case the above theorem does not include is $A_{8}$. When
$\tau=(1,2)(3,4)(5,6)(7,8)\in A_{8}$, the realization of the potential purely
wild inertia group $\langle\,\tau\,\rangle$ remains unknown. Note that when
$4|d$, $d\geq 5$, $r=d/2$, and $\tau=(1,2)\cdots(2r-1,r)$, proving that the
pair $(A_{d},\,\langle\,\tau\,\rangle)$ is realizable would imply that the
PWIC holds for $A_{d}$ (with any $d\geq 5$) in characteristic $2$.

######  Theorem 3.14.

In characteristic $2$, the PWIC is true for $S_{2}$, $S_{3}$, and $S_{d}$ for
all odd integer $d\geq 5$.

###### Proof.

By [16, Corollary 2.2.2], the PWIC in characteristic $2$ is true for the
groups $S_{2}$ and $S_{3}$. Let $d\geq 2$ be an odd integer. Let $P\subset
S_{d}$ be a $2$-subgroup such that
$\left\langle\,P^{\,S_{d}}\,\right\rangle=S_{d}$. Using Lemma 3.8 (1), choose
an odd permutation $\tau\in P$ such that
$\left\langle\,\langle\,\tau\,\rangle^{\,S_{d}}\,\right\rangle=S_{d}$. By [8,
Theorem 2], it is enough to show that the pair
$(S_{d},\,\langle\,\tau\,\rangle)$ is realizable. Suppose that
$\text{ord}(\tau)=2^{f}$, $f\geq 1$.

If $\tau$ has order $2$, then $\tau=\tau_{1}\cdots\tau_{r}$ for disjoint
transpositions $\tau_{u}$’s, for some odd integer $r\leq\lfloor d/2\rfloor$.
As $d$ is an odd integer, $2r<d$. Moreover, $\tau$ is conjugate to the element
$(1,2)\cdots(2r-1,2r)$. By Proposition 3.9 (1), the pair
$(S_{d},\,\langle\,\tau\,\rangle)$ is realizable.

Let $\text{ord}(\tau)\geq 4$. Then $\tau^{2}$ is a non-trivial even
permutation. By Theorem 3.11, the pair $(A_{d},\,\langle\,\tau^{2}\,\rangle)$
is realizable. Also the pair
$(\langle\,\tau\,\rangle,\,\langle\,\tau\,\rangle)$ is realizable as the PWIC
holds for any $2$-group. By [16, Theorem 2.2.3], the pair
$(S_{d}=\langle\,A_{d},\langle\,\tau\,\rangle\,\rangle,\,\langle\,\tau\,\rangle)$
is realizable. ∎

######  Example 3.15 (Failure of Abhyankar-style equation to the PWIC for
$S_{4}$ in characteristic $2$).

The smallest case not covered in the above result is $d=4$. One can construct
the connected $A_{d}$-Galois and $S_{d}$-Galois covers using covers
$\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1}$ given by explicit affine
equations, following Abhyankar’s work or [5, Section 3]. It is not clear
whether we can use this method to obtain a connected $S_{4}$-Galois cover
étale cover of $\mathbb{A}^{1}$ such that $\langle\,(1,2,3,4)\,\rangle$ occurs
as an inertia group above $\infty$. We illustrate one such example in the
following.

Let $\psi\,\colon\,Y=\mathbb{P}^{1}_{y}\longrightarrow\mathbb{P}^{1}_{x}$ be a
degree-$4$ cover given by the affine equation $f(x,y)=0$, where

$f(x,y)=y^{4}-y^{3}+x.$

Since $(0,0)$ is the only common zero of $f$ and its $y$-derivative
$\frac{\partial f}{\partial y}$, the cover $\psi$ is étale away from
$\\{0,\infty\\}$. From the equation, it follows that there are two points
$(y=0)$ and $(y=1)$ above $x=0$ in $\psi$ having ramification indices $3$ and
$1$, respectively, and $\psi^{-1}(\infty)$ consists of the unique point
$(y=\infty)$ having ramification index $4$. Let
$\phi\,\colon\,Z\longrightarrow\mathbb{P}^{1}_{x}$ be the Galois closure of
$\psi$, and let $G$ be the Galois group. Since $f(x,y)$ is an irreducible
polynomial in $k(x)[y]$, it follows that $G$ is a transitive subgroup of
$S_{4}$. It can also be shown that $\phi$ is étale away from $\\{0,\infty\\}$,
the inertia groups above $0$ are the cyclic groups generated by the $3$-cycles
in $G$; moreover, if $I_{\infty}\subset G$ occurs as an inertia groups above
$\infty$, a $4$-cycle $\tau$ is contained in $I_{\infty}$. From this it
follows that $G=S_{4}$. Note that
$N_{S_{4}}(\langle\,\tau\,\rangle)\cong\langle\,(1,2,3,4),(1,3)\,\rangle$ is
the Dihedral group $D_{8}$ of order $8$ (which is also a Sylow $2$-subgroup of
$S_{4}$). So $I_{\infty}=p(I_{\infty})$ is either a cyclic group of order $4$
or is a Sylow $2$-group. If $I_{\infty}\cong\mathbb{Z}/4$ (generated by
$\tau$), the induced cover $Z\longrightarrow Y=\mathbb{P}^{1}_{y}$ is a
connected $S_{4}$-Galois cover, tamely ramified over $(y=1)$, étale everywhere
else. By the Riemann Hurwitz formula, such a cover cannot exist. So the cover
$\phi$ has the Sylow $2$-group as the inertia groups above $\infty$.

Now we establish the GPWIC (Conjecture 1.3) in characteristic $2$ for any
product of the Alternating and the Symmetric groups we encountered above.
Since $S_{d}$ is not a perfect group, Remark 3.5 (3) do not apply; instead, we
apply patching results by Harbater and Raynaud together with Theorem 3.11. The
proof is broken down into three steps. After a simple group theoretic
observation in Step 1 that the potential purely wild inertia groups contain
certain elements $g^{(i)}$’s, we proceed to construct simpler realizable pairs
in Step 2 with inertia groups $\langle\,g^{(i)}\,\rangle$. We ‘patch’ together
these covers in Step 3 to obtain the result.

######  Theorem 3.16.

The GPWIC in characteristic $2$ is true for any product
$G=S_{d_{1}}\times\cdots\times S_{d_{n}}$, $n\geq 1$, where each $d_{i}\geq 5$
is an odd integer.

###### Proof.

We write any element of $G$ as $g=(g_{1},\cdots,g_{n})$. For $g\in G$, set
$\text{Supp}(g)\coloneqq\\{1\leq i\leq n\,|\,g_{i}\neq 1\\}$. Let $r\geq 1$ be
an integer, $B=\\{x_{1},\ldots,x_{r}\\}\subset\mathbb{P}^{1}$ be a set of
closed points, and $P_{1},\ldots,P_{r}\subset G$ be $2$-groups such that
$G=\left\langle\,P_{1}^{G},\ldots,P_{r}^{G}\,\right\rangle$ where $P_{j}^{G}$
denote the set of conjugates of $P_{j}$ in $G$. We assert that there is a
connected $G$-Galois cover of $\mathbb{P}^{1}$ that is étale away from $B$,
and $P_{j}$ occurs as an inertia group above $x_{j}$ for $1\leq j\leq r$.
Since the inertia groups above a point in a connected $G$-Galois cover are
conjugate, we assume that $P_{1},\ldots,P_{r}$ are contained in a single Sylow
$2$-subgroup of $G$.

Step 1: We start with a group theoretic observation that each $2$-group
$P_{j}$ contains a product $P_{j}^{\prime}$ of cyclic groups, and the
conjugates of all these $P_{j}^{\prime}$’s generate $G$.

There is a natural projection map $\pi\,\colon\,G\longrightarrow
Q\coloneqq\prod_{1\leq i\leq n}\mathbb{Z}/2$. Since $G$ is generated by the
conjugates of $P_{j}$’s, $Q$ is generated by the conjugates of $\pi(P_{j})$’s.
As $Q$ is abelian, we have
$\left\langle\,\pi(P_{1}),\ldots,\pi(P_{r})\,\right\rangle=Q$. If
$\pi(P_{j_{0}})=\\{1\\}$ for some $j_{0}$, we have
$\left\langle\,P_{j}^{G}\,|\,1\leq j\leq r,\,j\neq j_{0}\,\right\rangle=G$. In
view of [5, Theorem 4.7], it is enough to consider that
$\pi(P_{j})\neq\\{1\\}$ for any $1\leq j\leq r$. Since each $\pi(P_{j})\subset
Q$ is a non-trivial elementary abelian $2$-group, there are integers
$0=t_{0}<t_{1}<\cdots<t_{r-1}<t_{r}$ such that the following hold.

  1. (1)

For each $1\leq j\leq r$, there is a subgroup
$P^{\prime}_{j}\coloneqq\left\langle\,g^{(t_{j-1}+1)}\,\right\rangle\times\cdots\times\left\langle\,g^{(t_{j})}\,\right\rangle\subset
P_{j}$ such that $\pi(P^{\prime}_{j})=\pi(P_{j})$;

  2. (2)

for each $1\leq i\leq t_{r}$, there exists an integer $\lambda(i)$ such that
$g^{(i)}_{\lambda(i)}\in S_{\lambda(i)}$ is an odd permutation;

  3. (3)

$\cup\\{\lambda(i)\,|\,1\leq i\leq t_{r}\\}=\\{1,\cdots,n\\}$, i.e.,
conjugates of $\left\langle\,P^{\prime}_{j}\,|\,1\leq j\leq r\,\right\rangle$
generate $G$.

Step 2: Let $1\leq j\leq r$, and $t_{j-1}+1\leq i\leq t_{j}$. We use formal
patching techniques to realize $\left\langle\,g^{(i)}\,\right\rangle$ as the
inertia groups for certain quasi $2$-subgroups of $G$.

First suppose that $g^{(i)}_{\lambda(i)}$ has order $\geq 4$. Set
$h^{(i)}\coloneqq(g^{(i)})^{2}$. Then for any
$v\in\text{Supp}\left(h^{(i)}\right)$, $h^{(i)}_{v}\in S_{d_{v}}$ is a non-
trivial even permutation. Set
$K_{i}\coloneqq\prod_{v\in\text{Supp}\left(h^{(i)}\right)}A_{d_{v}}\subset G$.
By Theorem 3.11, the pair
$\left(K_{i},\left\langle\,h^{(i)}\,\right\rangle\right)$ is realizable. As
the pair
$\left(\left\langle\,g^{(i)}\,\right\rangle,\,\left\langle\,g^{(i)}\,\right\rangle\right)$
is also realizable, by [16, Theorem 2.2.3], we conclude that the pair

(3.1)
$\left(H_{i}\coloneqq\left\langle\,K_{i},\left\langle\,g^{(i)}\,\right\rangle\,\right\rangle,\,\left\langle\,g^{(i)}\,\right\rangle\right)\,\text{
is realizable}.$

Since $g^{(i)}_{\lambda(i)}$ is an odd permutation, the image of $H_{i}$ under
the $\lambda(i)^{\text{th}}$ projection $G\twoheadrightarrow
S_{d_{\lambda(i)}}$ is $S_{d_{\lambda(i)}}$.

Now let $\text{ord}\left(g^{(i)}_{\lambda(i)}\right)=2$. Then
$g^{(i)}_{\lambda(i)}$ is conjugate to the element
$\tau\coloneqq(1,2)\cdots(2u-1,2u)$ in $S_{d_{\lambda(i)}}$ for some odd
integer $1\leq u<d/2$, and let $w\in S_{d_{\lambda(i)}}$ be such that
$g^{(i)}_{\lambda(i)}=w^{-1}\tau w$. For each $1\leq t\leq u$ and $2u+1\leq
b\leq d_{\lambda(i)}$, consider the $3$-cycle
$\sigma_{t,b}\coloneqq(2t-1,2t,b)\in S_{d_{\lambda(i)}}$, and let
$\gamma_{t,b}\in H_{i}$ be such that
$(\gamma_{t,b})_{\lambda(i)}=\sigma_{t,b}$, $(\gamma_{t,b})_{s}=1$ for $s\neq
t$. As in the proof of Proposition 3.9, we have
$\tau^{-1}\gamma_{t,b}\tau=\gamma_{t,b}^{2}$. Hence for each $t$ and $b$,
setting $\gamma_{t,b}^{\prime}\coloneqq w^{-1}\gamma_{t,b}w$, we have

$\left(g^{(i)}\right)^{-1}\gamma_{t,b}^{\prime}g^{(i)}=\left(\gamma_{t,b}^{\prime}\right)^{2}.$

By [16, Corollary 2.2.2], the pairs
$\left(\left\langle\,\gamma_{t,b}^{\prime}\,\right\rangle\rtimes\left\langle\,g^{(i)}\,\right\rangle,\,\left\langle\,g^{(i)}\,\right\rangle\right)$
are realizable for each $1\leq t\leq u$ and $2u+1\leq b\leq d_{\lambda(i)}$.
By [16, Theorem 2.2.3], we conclude that the pair

(3.2)
$\left(B_{i}\coloneqq\left\langle\,\left\langle\,\gamma_{t,b}^{\prime}\,\right\rangle\rtimes\left\langle\,g^{(i)}\,\right\rangle\,|\,1\leq
t\leq u,\,2u+1\leq b\leq
d_{\lambda(i)}\,\right\rangle,\,\left\langle\,g^{(i)}\,\right\rangle\right)\,\text{
is realizable}.$

The image of $B_{i}$ under the projection $G\twoheadrightarrow
S_{d_{\lambda(i)}}$ is $S_{d_{\lambda(i)}}$.

Step 3: For $1\leq j\leq r$, let $G_{j}$ be the subgroup of $G$ generated by
$H_{i}$’s ($\text{ord}\left(g^{(i)}_{\lambda(i)}\right)\geq 4$) and $B_{l}$’s
($\text{ord}\left(g^{(l)}_{\lambda(l)}\right)=2$), $t_{j-1}+1\leq i,l\leq
t_{j}$. Since $G_{j}\twoheadrightarrow\prod_{t_{j-1}+1\leq i\leq
t_{j}}S_{d_{\lambda(i)}}$ by construction, $\left\langle\,G_{j}\,|\,1\leq
j\leq r\right\rangle=G$.

Applying [16, Theorem 2.2.3] to the realizable pairs in (3.1) and (3.2), for
each $1\leq j\leq r$, the pair $(G_{j},\,P^{\prime}_{j})$ is realizable;
further applying [8, Theorem 2], the pair $(G_{j},\,P_{j})$ is realizable.
Finally, we apply [5, Theorem 4.7] inductively to obtain our required
$G$-Galois cover. ∎

######  Corollary 3.17.

The GPWIC (Conjecture 1.3) is true in characteristic $2$ for any product
$G=G_{1}\times\cdots\times G_{n}$, $n\geq 1$, where $G_{i}=A_{d_{i}}$ for some
$d_{i}\geq 5$ with $4\nmid d_{i}$ or $G_{i}=S_{d_{i}}$ for an odd integer
$d_{i}\geq 5$.

###### Proof.

This follows immediately from Corollary 3.12, Theorem 3.14 and [5, Theorem
7.5]. ∎

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