# The $R_{\infty}$ property for nilpotent quotients of Generalized Solvable
Baumslag-Solitar groups

Wagner C. Sgobbi  Departamento de Matemática - Instituto de Matemática e
Estatística - Universidade de São Paulo, Caixa Postal 66.281 - CEP 05314-970,
São Paulo - SP, Brazil<EMAIL_ADDRESS>,  Dalton C. Silva
Instituto Federal de Educação, Ciência e Tecnologia de São Paulo, 11665-071,
Caraguatatuba, SP, Brazil<EMAIL_ADDRESS>and  Daniel Vendrúscolo
Departamento de Matemática, Universidade Federal de São Carlos, Rodovia
Washington Luiz, Km 235, São Carlos 13565-905, Brazil<EMAIL_ADDRESS>
(Date: May 2022)

###### Abstract.

We say a group $G$ has property $R_{\infty}$ if the number $R(\varphi)$ of
twisted conjugacy classes is infinite for every automorphism $\varphi$ of $G$.
For such groups, the $R_{\infty}$-nilpotency degree is the least integer $c$
such that $G/\gamma_{c+1}(G)$ has property $R_{\infty}$. In this work, we
compute the $R_{\infty}$-nilpotency degree of all Generalized Solvable
Baumslag-Solitar groups $\Gamma_{n}$. Moreover, we compute the lower central
series of $\Gamma_{n}$, write the nilpotent quotients
$\Gamma_{n,c}=\Gamma_{n}/\gamma_{c+1}(\Gamma_{n})$ as semidirect products of
finitely generated abelian groups and classify which integer invertible
matrices can be extended to automorphisms of $\Gamma_{n,c}$.

###### Key words and phrases:

$R_{\infty}$ Property, Generalized Solvable Baumslag-Solitar groups

###### 2000 Mathematics Subject Classification:

Primary: 20E36, 20E45 Secundary: 20F65

Wagner C. Sgobbi was supported by grants 2017/21208-0 and 2019/03150-0, São
Paulo Research Foundation (FAPESP)

Daniel Vendrúscolo was partially supported by grant 2016/24707-4 São Paulo
Research Foundation (FAPESP)

##  1\. Introduction

The main task of our paper is to show the following:

######  Theorem 1 (Theorem 4.4).

Let an integer $n\geq 1$ have a prime decomposition with at least two prime
numbers involved. Then the $R_{\infty}$-nilpotency degree of any Generalized
Solvable Baumslag-Solitar group $\Gamma_{n}$ is infinite.

Let us first define property $R_{\infty}$ and the $R_{\infty}$-nilpotency
degree and then problematize. An automorphism $\varphi$ of a group $G$ gives
rise to a “twisted conjugacy” equivalence relation on $G$ given by
$x\sim_{\varphi}y\ \iff\ \exists\ z\in G:\ zx\varphi(z)^{-1}=y$. The number of
equivalence classes (or Reidemeister classes, or twisted conjugacy classes) is
denoted by $R(\varphi)$, and we say $G$ has property $R_{\infty}$ if
$R(\varphi)$ is infinite for every automorphism $\varphi$ of $G$. This twisted
conjugacy relation first appeared in a work of K. Reidemeister [12] and has
many connections with other areas of mathematics, in particular to fixed point
theory (see [8]). We refer the introduction of the paper [3] to a good
exhibition of those connections and a discussion of the historical context and
development of $R_{\infty}$, including a list of families of groups with this
property. The search of $R_{\infty}$-groups is still very active. For example,
a first proof of $R_{\infty}$ for the pure Artin braid groups $P_{n}$, $n\geq
3$, was published in 2021 [5] and, even more recently, an alternative proof
was obtained in [1].

For groups $G$ with property $R_{\infty}$, the $R_{\infty}$-nilpotency degree
is the least integer $c$ such that the quotient $G/\gamma_{c+1}(G)$ has
property $R_{\infty}$, where $\gamma_{k}(G)$ are the terms of the lower
central series of $G$, that is, $\gamma_{1}(G)=G$ and
$\gamma_{c+1}(G)=[\gamma_{c}(G),G]$, for any $c\geq 1$. If none of the
quotients have $R_{\infty}$, we say the $R_{\infty}$-nilpotency degree of $G$
is infinite. This degree relates to the folowing problem: it is well known
that if $G$ has a characteristic subgroup $N$ (for example, $N=\gamma_{k}(G)$,
$k\geq 1$) such that $G/N$ has $R_{\infty}$, then $G$ has $R_{\infty}$. One
can ask then whether the converse is true, that is, are there groups $G$ whose
property $R_{\infty}$ is still valid for the quotients $G/\gamma_{k}(G)$? The
answer to this question is positive and gives us the more specific motivation
for our work, as we will discuss now.

Knowing that the free groups $F_{n}$ have $R_{\infty}$ (since they are
hyperbolic [9]), the authors K. Dekimpe and D. L. Gonçalves studied property
$R_{\infty}$ for infinitely generated free groups, free nilpotent groups and
free solvable groups [2]. They generalized a result of [13], showing that the
free nilpotent groups $F_{n,c}=F_{n}/\gamma_{c+1}(F_{n})$, that is, nilpotent
quotients of free groups $F_{n}$, have property $R_{\infty}$ if and only if
$c\geq 2n$. So, the $R_{\infty}$-nilpotency degree of free groups $F_{n}$ is
$2n$. This fact is actually what motivated the definition of
$R_{\infty}$-nilpotency degree above, which was first given in [3]. In that
work, also motivated by the fact that fundamental groups of hyperbolic
surfaces have $R_{\infty}$ (again by [9]), the same authors studied nilpotent
quotients of such surface groups, showing that the $R_{\infty}$-nilpotency
degree $c$ of an orientable surface $S_{g}$ of genus $g\geq 2$ is $c=4$ and
that, for $N_{g}$ a connected sum of $g\geq 3$ projective planes, $c=2(g-1)$.
It is worth noticing that a consequence of their research was the discovery of
new examples of nilmanifolds on which every self-homotopy equivalence can be
deformed into a fixed point free map.

On the non-hyperbolic side, an important example of $R_{\infty}$ groups are
the Baumslag-Solitar groups

$BS(m,n)=\left<a,t\ |\ ta^{m}t^{-1}=a^{n}\right>,$

for $m,n$ integers (see [6]). These are important examples of combinatorial
and geometric group theory. In [4], K. Dekimpe and D. L. Gonçalves determined
the $R_{\infty}$-nilpotency degree of $BS(m,n)$:

######  Theorem 2 (Theorem 5.4 in [4]).

Let $0<m\leq|n|$ with $m\neq n$ and take $d=gcd(m,n)$. Let $p$ denote the
largest integer such that $2^{p}|2\frac{m}{d}+2$. Then, the
$R_{\infty}$-nilpotency degree $r$ of $BS(m,n)$ is given by

  * •

In case $n<0$ and $n\neq-m$, then $r=2$.

  * •

In case $n=-m$ then $r=\infty$.

  * •

In case $n=m$ then $r=\infty$.

  * •

In case $n-m=d$, then $r=\infty$.

  * •

In case $n-m=2d$, then $2\leq r\leq p+2$.

  * •

In case $n-m\geq 3d$, then $r=2$.

In the particular case of the solvable Baumslag-Solitar groups $BS(1,n)$,
$n\geq 2$, we have $r=\infty$ for $n=2$, $r=4$ for $n=3$ and $r=2$ for $n\geq
4$. The investigation of the $R_{\infty}$-nilpotency degree for
generalizations of these groups is therefore natural. The authors comment
about the family of Generalized Baumslag-Solitar groups (or GBS groups), which
are $R_{\infty}$ groups. In this paper, we add one more important family to
that discussion, namely, the Generalized Solvable Baumslag-Solitar groups
$\Gamma_{n}$. Let $n\geq 2$ be an integer with prime decomposition
$n={p_{1}}^{y_{1}}...{p_{r}}^{y_{r}}$, the $p_{i}$ being pairwise distinct and
$y_{i}>0$. We consider the Generalized Solvable Baumslag-Solitar group

$\Gamma_{n}=\left<a,t_{1},...,t_{r}\ |\ t_{i}t_{j}=t_{j}t_{i},\ i\neq j,\
t_{i}a{t_{i}}^{-1}=a^{{p_{i}}^{y_{i}}},\ i=1,...,r\right>.$

In particular, if $n=p^{y}$ involves only one prime number, then
$\Gamma_{n}=BS(1,n)$ is a solvable Baumslag-Solitar group. We will focus,
therefore, in the case of two or more prime numbers in the decomposition of
$n$.

These groups arise mainly in a geometric generalization of the solvable
Baumslag-Solitar groups $BS(1,n)$. Consider the group
$PSL_{2}(\mathbb{Z}[1/n])$, which acts on the product of the hyperbolic space
$H_{2}$ with the product of the Bruhat-Tits trees for
$PSL_{2}(\mathbb{Q}_{p_{i}})$. The stabilizer of a point at infinity under
this action is the upper triangular subgroup of $PSL_{2}(\mathbb{Z}[1/n])$,
which we are denoting by $\Gamma_{n}$ (see [14]). In this paper, however,
dealing with the above presentation was enough to estabilish our results. The
$\Gamma_{n}$ are also known to be solvable and metabelian groups, fitting in a
short exact sequence of the form

$1\to\mathbb{Z}\left[\frac{1}{n}\right]\to\Gamma_{n}\to\mathbb{Z}^{r}\to 1.$

More details about $\Gamma_{n}$ can be seen in the paper [14], where J. Taback
and P. Wong show property for $\Gamma_{n}$ and for any group quasi-isometric
to it.

We started dealing with Baumslag-Solitar groups in the first author’s doctoral
dissertation. There, one realized that elementary techniques involving the BNS
invariant $\Sigma^{1}$ of the groups $BS(1,n)$ could also be applied to
$\Gamma_{n}$, leading to an elementary proof of $R_{\infty}$ to these groups.
One can then ask whether other techniques - such as the investigation of
nilpotent quotients of $BS(m,n)$ in [4] \- can also be adapted to
$\Gamma_{n}$. The main question was whether $\Gamma_{n}$ would behave more
like $BS(1,2)$ (with infinite $R_{\infty}$-nilpotency degree) or like
$BS(1,n)$, $n\geq 3$ (finite degree cases). As Theorem 1 shows, the groups
$\Gamma_{n}$ fit in the infinite degree case. We believe this happens because
of the large automorphism groups of its nilpotent quotients.

We will denote the nilpotent quotients of $\Gamma_{n}$ by

$\Gamma_{n,c}=\frac{\Gamma_{n}}{\gamma_{c+1}(\Gamma_{n})}$

for any $c\geq 1$, where $\gamma_{c+1}$ is the $(c+1)^{th}$ term of the lower
central series of $\Gamma_{n}$. We know that $\Gamma_{n,c}$ is a nilpotent
group with nilpotency class $\leq c$, for
$\gamma_{c+1}(\Gamma_{n,c})=\gamma_{c+1}(\frac{\Gamma_{n}}{\gamma_{c+1}(\Gamma_{n})})=\frac{\gamma_{c+1}(\Gamma_{n})}{\gamma_{c+1}(\Gamma_{n})}=1$.
In this work, the torsion subgroup of a nilpotent group $G$ will be denoted by
$\tau G=\\{g\in G\ |\ g^{n}=1\ \text{for some}\ n\geq 1\\}$.

This work is divided as follows: in section 2, we compute the torsion subgroup
and the lower central series of $\Gamma_{n,c}$. Then, in section 3, we use
these computations to create an useful isomorphism between $\Gamma_{n,c}$ and
a semidirect product of the form
$G_{n,c}=\mathbb{Z}_{m^{c}}\rtimes\mathbb{Z}^{r}$, for some $m\geq 1$.
Finally, in section 4, we classify which matrices in $GL_{r}(\mathbb{Z})$ can
be extended to automorphisms of $\Gamma_{n,c}$ and use the classification to
find specific automorphisms with finite Reidemeister numbers in
$\Gamma_{n,c}$.

##  2\. Torsion and lower central series of $\Gamma_{n,c}$

We begin by showing the following

######  Lemma 2.1.

Let $m=\gcd(p_{1}^{y_{1}}-1,...,p_{r}^{y_{r}}-1)$. Then
$a^{m^{k}}\in\gamma_{k+1}(\Gamma_{n})$ for all $k\geq 1$.

###### Proof.

Induction on $k$. First, $k=1$. By using the group relations, note that, for
any $1\leq i\leq r$,
$a^{p_{i}^{y_{i}}-1}=t_{i}at_{i}^{-1}a^{-1}=[t_{i},a]\in\gamma_{2}(\Gamma_{n})$.
Since this is true for any $i$ and $m$ is an integer combination of the
$p_{i}^{y_{i}}-1$, we have $a^{m}\in\gamma_{2}(\Gamma_{n})$. Now, suppose the
lemma is true for some $k\geq 1$. Then

$a^{(p_{i}^{y_{i}}-1)m^{k}}=a^{p_{i}^{y_{i}}m^{k}}a^{-m^{k}}=t_{i}a^{m^{k}}t_{i}^{-1}a^{-m^{k}}=[t_{i},a^{m^{k}}]\in\gamma_{k+2}(\Gamma_{n}).$

Again, since this is true for any $i$ and $m$ is an integer combination of the
$p_{i}^{y_{i}}-1$, we have
$a^{mm^{k}}=a^{m^{k+1}}\in\gamma_{k+2}(\Gamma_{n})$, as desired. This
completes the proof. ∎

To compute the torsion of the groups $\Gamma_{n,c}$, we need the following
standard results:

######  Proposition 2.2 (see [11]).

Let $k,m,n\geq 1$ and let $x,y,z\in G$ be elements of a group $G$ such that
$x\in\gamma_{k}(G)$, $y\in\gamma_{m}(G)$ and $z\in\gamma_{n}(G)$. Then:

  * a)

$xy=yx$ $\mod\gamma_{k+m}(G)$;

  * b)

$[x,yz]=[x,y][x,z]$ $\mod\gamma_{k+m+n}(G)$;

  * c)

$[xy,z]=[x,z][y,z]$ $\mod\gamma_{k+m+n}(G)$.

######  Proposition 2.3.

([10], Theorem 5.4) If $G$ is finitely generated by elements
$x_{1},\dots,x_{r}$ then, for any $k\geq 1$, $\gamma_{k}(G)/\gamma_{k+1}(G)$
is abelian and finitely generated by the cosets of the $k$-fold commutators
$[x_{i_{1}},\dots,x_{i_{k}}]$, where $1\leq i_{j}\leq r$.

By an easy recursive argument, one shows the following:

######  Lemma 2.4.

Let $G$ be a nilpotent group of class $\leq c$ and denote
$\gamma_{i}=\gamma_{i}(G)$. If the quotients
$\gamma_{2}/\gamma_{3},...,\gamma_{c}/\gamma_{c+1}$ are finite, then
$\gamma_{2}$ is a torsion subgroup of $G$. ∎

######  Proposition 2.5.

$\tau\Gamma_{n,c}=\left<\overline{a},\gamma_{2}(\Gamma_{n,c})\right>$, where
$\overline{a}=a\gamma_{c+1}=a\gamma_{c+1}(\Gamma_{n})\in\Gamma_{n,c}$.

###### Proof.

In the case $c=1$ we have $\Gamma_{n,1}$ is the abelianized group of
$\Gamma_{n}$, so

$\Gamma_{n,1}=\left<\overline{a},\overline{t_{1}},...,\overline{t_{r}}\ |\
\overline{t_{i}}\overline{t_{j}}=\overline{t_{j}}\overline{t_{i}},\
\overline{t_{i}}\overline{a}=\overline{a}\overline{t_{i}},\
\overline{a}^{p_{i}^{y_{i}}-1}=1\right>\simeq\mathbb{Z}_{m}\times\mathbb{Z}^{r},$

where $m=\gcd(p_{1}^{y_{1}}-1,...,p_{r}^{y_{r}}-1)$. So
$\tau\Gamma_{n,1}=\left<\overline{a}\right>=\left<\overline{a},\gamma_{2}(\Gamma_{n,1})\right>$,
since $\gamma_{2}(\Gamma_{n,1})=1$.

Now let us show the proposition in the case $c\geq 2$. For $(\subset)$, let
$x\gamma_{c+1}\in\tau\Gamma_{n,c}$. This means $x^{k}\in\gamma_{c+1}$ for some
$k\geq 1$. Since $c\geq 2$, we have $x^{k}\in\gamma_{c+1}\subset\gamma_{2}$,
so $x\gamma_{2}\in\tau\Gamma_{n,1}=\left<\overline{a}\right>$. Write then
$x=a^{l}g_{2}$ for $l\in\mathbb{Z}$ and
$g_{2}\in\gamma_{2}=\gamma_{2}(\Gamma_{n})$. This gives
$x\gamma_{c+1}=(a\gamma_{c+1})^{l}(g_{2}\gamma_{c+1})\in\left<\overline{a},\gamma_{2}(\Gamma_{n,c})\right>$,
as we wanted. To show $(\supset)$, we note that by Lemma 2.1 we get
$\overline{a}^{m^{c}}=1$ in $\Gamma_{n,c}$, so
$\overline{a}\in\tau\Gamma_{n,c}$. So, we just need to show that
$\gamma_{2}(\Gamma_{n,c})$ is a torsion subgroup of $\Gamma_{n,c}$. To do
this, we invoke Lemma 2.4, by which we know it is enough to show the quotients

$\frac{\gamma_{2}(\Gamma_{n,c})}{\gamma_{3}(\Gamma_{n,c})},...,\frac{\gamma_{c}(\Gamma_{n,c})}{\gamma_{c+1}(\Gamma_{n,c})}$

are all finite. But for every $2\leq i\leq c$, by the known Isomorphism
Theorem for quotients, we have

$\frac{\gamma_{i}(\Gamma_{n,c})}{\gamma_{i+1}(\Gamma_{n,c})}=\frac{\gamma_{i}(\Gamma_{n})/\gamma_{c+1}(\Gamma_{n})}{\gamma_{i+i}(\Gamma_{n})/\gamma_{c+1}(\Gamma_{n})}\simeq\frac{\gamma_{i}(\Gamma_{n})}{\gamma_{i+i}(\Gamma_{n})}=\gamma_{i}/\gamma_{i+1},$

so let us show that $\gamma_{2}/\gamma_{3},...,\gamma_{c}/\gamma_{c+1}$ are
finite by induction. By Proposition 2.3, we know they are abelian groups,
generated by their $i$-fold comutator cosets. The group
$\gamma_{2}/\gamma_{3}$ is generated by the elements $[t_{i},a]\gamma_{3}$,
$1\leq i\leq r$ and by $[t_{i},t_{j}]\gamma_{3}=1\gamma_{3}=\gamma_{3}$, which
are trivial. By Proposition 2.2, we get
$[t_{i},a]^{m}\gamma_{3}=[t_{i},a^{m}]\gamma_{3}=\gamma_{3}$, since
$a^{m}\in\gamma_{2}$ (Lemma 2.1). So all generators of $\gamma_{2}/\gamma_{3}$
have torsion. Since it is finitely generated and abelian, it must be a finite
group. Finally, suppose by induction that $\gamma_{i}/\gamma_{i+1}$ is finite
for some $i\geq 2$. By Proposition 2.3, $\gamma_{i+1}/\gamma_{i+2}$ is then
generated by the elements of the form $[x,y]\gamma_{i+2}$ with
$x\in\gamma_{i}$ and $y\in\Gamma_{n}$. Since $\gamma_{i}/\gamma_{i+1}$ is
finite, let $k=k(x,y)\geq 1$ such that $x^{k}\in\gamma_{i+1}$. Then
$[x,y]^{k}\gamma_{i+2}=[x^{k},y]\gamma_{i+2}=\gamma_{i+2}$. By the same
argument we just used, this implies $\gamma_{i+1}/\gamma_{i+2}$ is finite and
completes the proof. ∎

######  Proposition 2.6.

$\gamma_{k}(\Gamma_{n,c})=\left<\overline{a}^{m^{k-1}}\right>$ for all $k\geq
2$ and $c\geq 1$.

###### Proof.

First, we will show that

$\frac{\gamma_{k}(\Gamma_{n,c})}{\gamma_{k+1}(\Gamma_{n,c})}=\left<\overline{a}^{m^{k-1}}\gamma_{k+1}(\Gamma_{n,c})\right>.$
(1)

For $k=2$, by Proposition 2.3,
$\frac{\gamma_{2}(\Gamma_{n,c})}{\gamma_{3}(\Gamma_{n,c})}$ is generated by
the cosets $[\overline{t_{i}},\overline{a}]\gamma_{3}(\Gamma_{n,c})$. Since
$[\overline{t_{i}},\overline{a}]=\overline{a}^{p_{i}^{y_{i}}-1}$, we have

$\frac{\gamma_{2}(\Gamma_{n,c})}{\gamma_{3}(\Gamma_{n,c})}=\left<\overline{a}^{p_{1}^{y_{1}}-1}\gamma_{3}(\Gamma_{n,c}),...,\overline{a}^{p_{r}^{y_{r}}-1}\gamma_{3}(\Gamma_{n,c})\right>=\left<\overline{a}^{m}\gamma_{3}(\Gamma_{n,c})\right>$

(remember that $m=\gcd(p_{1}^{y_{1}}-1,...,p_{r}^{y_{r}}-1)$). Suppose now (1)
is true for some $k\geq 2$. We know
$\frac{\gamma_{k+1}(\Gamma_{n,c})}{\gamma_{k+2}(\Gamma_{n,c})}$ is generated
by the cosets $[x,z]\gamma_{k+2}(\Gamma_{n,c})$, where
$x\in\gamma_{k}(\Gamma_{n,c})$ and $z\in\Gamma_{n,c}$. By induction, we can
write $x=\overline{a}^{\alpha m^{k-1}}w_{k+1}$ for some
$w_{k+1}\in\gamma_{k+1}(\Gamma_{n,c})$ and $\alpha\in\mathbb{Z}$. Then, by
using Proposition 2.2 we get

$\displaystyle[x,z]\gamma_{k+2}(\Gamma_{n,c})$ $\displaystyle=$
$\displaystyle[\overline{a}^{\alpha
m^{k-1}}w_{k+1},z]\gamma_{k+2}(\Gamma_{n,c})$ $\displaystyle=$
$\displaystyle[\overline{a}^{m^{k-1}},z]^{\alpha}[w_{k+1},z]\gamma_{k+2}(\Gamma_{n,c})$
$\displaystyle=$
$\displaystyle[\overline{a}^{m^{k-1}},z]^{\alpha}\gamma_{k+2}(\Gamma_{n,c}),$

so the quotient
$\frac{\gamma_{k+1}(\Gamma_{n,c})}{\gamma_{k+2}(\Gamma_{n,c})}$ is actually
generated only by the cosets
$[\overline{a}^{m^{k-1}},z]\gamma_{k+2}(\Gamma_{n,c})$. Since
$[\overline{a}^{m^{k-1}},\overline{a}]$ is obviously trivial, the quotient
group is generated only by the generators
$[\overline{a}^{m^{k-1}},\overline{t_{i}}]\gamma_{k+2}(\Gamma_{n,c})$. Since
$[\overline{a}^{m^{k-1}},\overline{t_{i}}]=\overline{a}^{(p_{i}^{y_{i}}-1)m^{k-1}}$,
we obtain

$\frac{\gamma_{k+1}(\Gamma_{n,c})}{\gamma_{k+2}(\Gamma_{n,c})}=\left<\overline{a}^{(p_{1}^{y_{1}}-1)m^{k-1}}\gamma_{k+2}(\Gamma_{n,c}),...,\overline{a}^{(p_{r}^{y_{r}}-1)m^{k-1}}\gamma_{k+2}(\Gamma_{n,c})\right>=\left<\overline{a}^{\beta}\gamma_{k+2}(\Gamma_{n,c})\right>,$

where

$\beta=\gcd((p_{1}^{y_{1}}-1)m^{k-1},...,(p_{r}^{y_{r}}-1)m^{k-1})=m^{k-1}\gcd(p_{1}^{y_{1}}-1,...,p_{r}^{y_{r}}-1)=m^{k},$

and this shows (1). Now, let us show the proposition. The $(\supset)$ part is
a direct consequence of Lemma 2.1. Let us show $(\subset)$. In the case $c<k$,
we have
$\gamma_{k}(\Gamma_{n,c})=1\subset\left<\overline{a}^{m^{k-1}}\right>$.
Suppose then $c\geq k$ and let $x\in\gamma_{k}(\Gamma_{n,c})$. Since
$x\gamma_{k+1}(\Gamma_{n,c})\in\left<\overline{a}^{m^{k-1}}\gamma_{k+1}(\Gamma_{n,c})\right>$
(by (1)), write $x=\overline{a}^{j_{k}m^{k-1}}x_{k+1}$ for
$j_{k}\in\mathbb{Z}$ and $x_{k+1}\in\gamma_{k+1}(\Gamma_{n,c})$. By using (1)
again, we write $x_{k+1}=\overline{a}^{j_{k+1}m^{k}}x_{k+2}$ for
$j_{k+1}\in\mathbb{Z}$ and $x_{k+2}\in\gamma_{k+2}(\Gamma_{n,c})$. We can do
this recursively to obtain

$\displaystyle x$ $\displaystyle=$
$\displaystyle\overline{a}^{j_{k}m^{k-1}}\overline{a}^{j_{k+1}m^{k}}...\overline{a}^{j_{c}m^{c-1}}x_{c+1}$
$\displaystyle=$
$\displaystyle\overline{a}^{m^{k-1}(j_{k}+j_{k+1}m+...+j_{c}m^{c-k})}$
$\displaystyle\in$ $\displaystyle\left<\overline{a}^{m^{k-1}}\right>,$

and the proof is complete. ∎

By Lemma 2.1 and the two propositions above, we get

######  Corollary 2.7.

$\tau\Gamma_{n,c}=\left<\overline{a}\right>$ and
$\operatorname{card}(\tau\Gamma_{n,c})\leq m^{c}$. ∎

##  3\. An isomorphism for $\Gamma_{n,c}$

The next step is to find a presentation to $\Gamma_{n,c}$, so we will find an
isomorphism between $\Gamma_{n,c}$ and a more known group. We will use the
notations from the previous section and will also denote
$\mathbb{Z}_{m^{c}}=\left<x\ |x^{m^{c}}=1\right>$ and
$\mathbb{Z}^{r}=\left<s_{1},...,s_{r}\ |s_{i}s_{j}=s_{j}s_{i}\right>$. We
define the group

$G_{n,c}=\mathbb{Z}_{m^{c}}\rtimes\mathbb{Z}^{r},$

where the action of $\mathbb{Z}^{r}$ on $\mathbb{Z}_{m^{c}}$ is given by
$s_{i}xs_{i}^{-1}=x^{p_{i}^{y_{i}}},1\leq i\leq r$.

######  Observation 3.1.

Note first that the actions defined above are all automorphisms of
$\mathbb{Z}_{m^{c}}$, since $\gcd(p_{i}^{y_{i}},m)=1$ (and so
$\gcd(p_{i}^{y_{i}},m^{c})=1$ for any $c\geq 1$). Second, all such
automorphisms commute, for $\mathbb{Z}_{m^{c}}$ is cyclic. These facts show
that there is a well defined homomorphism $\mathbb{Z}^{r}\to
Aut(\mathbb{Z}_{m^{c}})$, so this semidirect product is well defined.

We will show that $\Gamma_{n,c}\simeq G_{n,c}$. To do this, we need:

######  Lemma 3.2.

$G_{n,c}$ is nilpotent of class $\leq c$.

###### Proof.

Since $[s_{i},x]=x^{p_{i}^{y_{i}}-1}\in\left<x^{m}\right>$ for every $i$, we
have $\gamma_{2}(G_{n,c})\subset\left<x^{m}\right>$. Similarly, since
$[s_{i},x^{m}]=x^{(p_{i}^{y_{i}}-1)m}\in\left<x^{m^{2}}\right>$ for every $i$,
in particular we have $[s_{i},z]\in\left<x^{m^{2}}\right>$ for every
$z\in\gamma_{2}(G_{n,c})$, so it is easy to see that
$\gamma_{3}(G_{n,c})\subset\left<x^{m^{2}}\right>$. Recursively, we can show
that $\gamma_{k}(G_{n,c})\subset\left<x^{m^{k-1}}\right>$ for every $k\geq 2$.
In particular, $\gamma_{c+1}(G_{n,c})\subset\left<x^{m^{c}}\right>=1$, since
$x^{m^{c}}=1$ in $\mathbb{Z}_{m^{c}}$. This shows the lemma. ∎

######  Corollary 3.3.

$\tau G_{n,c}$ is a subgroup of $G_{n,c}$. Moreover, $\tau
G_{n,c}=\mathbb{Z}_{m^{c}}=\left<x\right>$ and so $\operatorname{card}(\tau
G_{n,c})=m^{c}$.

######  Theorem 3.4.

$\Gamma_{n,c}\simeq G_{n,c}$.

###### Proof.

Let $f:\Gamma_{n}\to G_{n,c}$ be the map $f(a)=x$ and $f(t_{i})=s_{i}$. Since
$f(t_{i})f(a)f(t_{i})^{-1}=s_{i}xs_{i}^{-1}=x^{p_{i}^{y_{i}}}=f(a)^{p_{i}^{y_{i}}}$,
$f$ is a well defined group homomorphism. Since
$f(\gamma_{i}(\Gamma_{n}))\subset\gamma_{i}(G_{n,c})$, $f$ induces the
morphism

$f:\Gamma_{n,c}=\frac{\Gamma_{n}}{\gamma_{c+1}(\Gamma_{n})}\to\frac{G_{n,c}}{\gamma_{c+1}(G_{n,c})}=G_{n,c}$

given by $f(\overline{a})=x$ and $f(\overline{t_{i}})=s_{i}$. It is obviously
surjective. We are just left to show that $\ker(f)=1$, and to do that we will
make use of the torsion subgroups. Since $f(\tau\Gamma_{n,c})\subset\tau
G_{n,c}$ (this is true for any homomorphisms between nilpotent groups), there
is the restriction morphism $f_{\tau}:\tau\Gamma_{n,c}\to\tau G_{n,c}$. By
Corollaries 2.7 and 3.3, we can actually write
$f_{\tau}:\left<\overline{a}\right>\to\left<x\right>$. Since
$f_{\tau}(\overline{a})=x$, it is clearly surjective. Now, $f_{\tau}$ is a
surjective map from a finite set of $\leq m^{c}$ elements (Corollary 2.7) to a
finite set with exactly $m^{c}$ elements (Corollary 3.3), so we must have
$\operatorname{card}(\left<\overline{a}\right>)=m^{c}$ and $f_{\tau}$ an
isomorphism. In particular, $\ker(f_{\tau})=1$. We claim that
$\ker(f)\subset\tau\Gamma_{n,c}$. In fact, let $z\in\ker(f)$. By using the
relations in $\Gamma_{n}$, we can write

$z=\overline{t_{1}}^{k_{1}}...\overline{t_{r}}^{k_{r}}\overline{t_{1}}^{-\alpha_{1}}...\overline{t_{r}}^{-\alpha_{r}}\overline{a}^{l}{t_{r}}^{\alpha_{r}}...\overline{t_{1}}^{\alpha_{1}},$

for $k_{i},l\in\mathbb{Z}$ and $\alpha_{i}\geq 0$. So

$1=f(z)=s_{1}^{k_{1}}...s_{r}^{k_{r}}s_{1}^{-\alpha_{1}}...s_{r}^{-\alpha_{r}}x^{l}s_{r}^{\alpha_{r}}...s_{1}^{\alpha_{1}}.$

Since $x\in\tau G_{n,c}\lhd G_{n,c}$ we have
$s_{1}^{-\alpha_{1}}...s_{r}^{-\alpha_{r}}x^{l}s_{r}^{\alpha_{r}}...s_{1}^{\alpha_{1}}\in\tau
G_{n,c}=\left<x\right>$, so
$1=f(z)=s_{1}^{k_{1}}...s_{r}^{k_{r}}x^{l^{\prime}}$ for some
$l^{\prime}\in\mathbb{Z}$. By projecting this equality under the natural
homomorphism $G_{n,c}\to\mathbb{Z}^{r}$ we get
$1=s_{1}^{k_{1}}...s_{r}^{k_{r}}$, which implies $k_{i}=0$ for every $i$.
Therefore
$z=\overline{t_{1}}^{-\alpha_{1}}...\overline{t_{r}}^{-\alpha_{r}}\overline{a}^{l}\overline{t_{r}}^{\alpha_{r}}...\overline{t_{1}}^{\alpha_{1}}\in\tau
G_{n,c}$, since $\overline{a}\in\tau\Gamma_{n,c}\lhd\Gamma_{n,c}$, which shows
the claim. Finally, this gives
$\ker(f)=\ker(f)\cap\tau\Gamma_{n,c}=\ker(f_{\tau})=1$ and the theorem is
proved. ∎

######  Corollary 3.5.

For any $c\geq 1$, the nilpotent quotient $\Gamma_{n,c}$ has the following
presentation:

$\Gamma_{n,c}=\left<x,s_{1},...,s_{r}\ |\ x^{m^{c}}=1,\
s_{i}s_{j}=s_{j}s_{i},\ s_{i}xs_{i}^{-1}=x^{p_{i}^{y_{i}}}\right>.$

∎

##  4\. Reidemeister numbers

Because of the theorem above, from now on we will make the following
identifications

$\Gamma_{n,c}=G_{n,c}=\mathbb{Z}_{m^{c}}\rtimes\mathbb{Z}^{r}=\left<x\right>\rtimes\left<s_{1},...,s_{r}\right>.$

It’s also worth remembering that we will restrict us to investigate
Reidemeister numbers of $\Gamma_{n,c}$ only in the case $r\geq 2$, for, if
$r=1$, then $\Gamma_{n}$ is by definition a Baumslag-Solitar group $BS(1,n)$
and its Reidemeister numbers were studied in [4]. Let $\varphi\in
Aut(\Gamma_{n,c})$. Since $\varphi(\tau\Gamma_{n,c})\subset\tau\Gamma_{n,c}$,
we have an induced automorphism

$\overline{\varphi}:\frac{\Gamma_{n,c}}{\tau\Gamma_{n,c}}=\mathbb{Z}^{r}\to\mathbb{Z}^{r}=\frac{\Gamma_{n,c}}{\tau\Gamma_{n,c}}.$

From now on, we will use the usual identification
$Aut(\mathbb{Z}^{r})=GL_{r}(\mathbb{Z})$ which sees an automorphism of
$\mathbb{Z}^{r}$ as its (integer invertible) matrix with respect to the
coordinates $s_{i}$. So, if
$\overline{\varphi}(s_{i})=s_{1}^{a_{1i}}...s_{r}^{a_{ri}}$, we will identify

$\overline{\varphi}=(a_{ij})_{ij}=\begin{bmatrix}a_{11}&\cdots&a_{1r}\\\
\vdots&&\vdots\\\ a_{r1}&\cdots&a_{rr}\\\ \end{bmatrix}=\left[A_{1}\cdots
A_{r}\right],\ \ \text{where}\ A_{i}=\begin{bmatrix}a_{1i}\\\ \vdots\\\
a_{ri}\\\ \end{bmatrix}\in\mathbb{Z}^{r}.$

######  Proposition 4.1.

If $\varphi\in Aut(\Gamma_{n,c})$, the following are equivalent:

  * (1)

$R(\varphi)=\infty$;

  * (2)

$R(\overline{\varphi})=\infty$;

  * (3)

$\det(\overline{\varphi}-Id)=0$;

  * (4)

$\overline{\varphi}$ has $1$ as an eigenvalue.

###### Proof.

Items $(2),(3)$ and $(4)$ are well known to be all equivalent. Also, we have
an obvious commutative diagram involving the automorphisms $\varphi$,
$\overline{\varphi}$ and the projection $\pi:\Gamma_{n,c}\to\mathbb{Z}^{r}$.
Thus, by Lemma 1.1 of [7], it follows that $(2)$ implies $(1)$. So we only
have to prove that $(1)$ implies $(2)$.

To simplify the computation, let us use the following notation in this proof:
given $y=(y_{1},...,y_{r})\in\mathbb{Z}^{r}$ (either a row or a column
vector), we will denote the element
$s_{1}^{y_{1}}...s_{r}^{y_{r}}\in\Gamma_{n,c}$ by $S^{y}$, and the scalar
product of $k\in\mathbb{Z}$ by $y$ is denoted by $ky$. With this notation, it
turns out that any element of $\Gamma_{n,c}$ is of the form $S^{y}x^{\beta}$
for some $y\in\mathbb{Z}^{r}$ and $\beta\in\mathbb{Z}$. Suppose then that
$R(\overline{\varphi})=d<\infty$ and write
$\mathcal{R}(\overline{\varphi})=\\{[v_{1}]_{\overline{\varphi}},...,[v_{d}]_{\overline{\varphi}}\\}$
for $v_{i}\in\mathbb{Z}^{r}$ or, equivalently,
$\frac{\mathbb{Z}^{r}}{\operatorname{im}(\overline{\varphi}-Id)}=\\{\overline{v_{1}},...,\overline{v_{d}}\\}$
(where $\overline{v_{i}}=v_{i}+\operatorname{im}(\overline{\varphi}-Id)$).
Write $\varphi(x)=x^{\mu}$ (for some $\mu\in\mathbb{Z}$ with
$\gcd(\mu,m^{c})=1$) and $\varphi(s_{i})=S^{A_{i}}x^{\beta_{i}}$,
$\beta_{i}\in\mathbb{Z}$. Given that the $s_{i}$-coordinates behave well in
the $\Gamma_{n,c}$, for any $k=(k_{1},...,k_{r})\in\mathbb{Z}^{r}$ and
$l\in\mathbb{Z}$ we have

$\varphi(S^{k}x^{l})=S^{\overline{\varphi}(k)}x^{\theta},$

for some $\theta\in\mathbb{Z}$. This implies that, for any $j\in\mathbb{Z}$
and $y\in\mathbb{Z}^{r}$,

$(S^{k}x^{l})(S^{y}x^{j})\varphi(S^{k}x^{l})^{-1}=S^{y+(Id-\overline{\varphi})(k)}x^{\tilde{\theta}},$

for some $\tilde{\theta}\in\mathbb{Z}$. This means that, if two vectors
$y,y^{\prime}\in\mathbb{Z}^{r}$ are such that
$\overline{y}=\overline{y^{\prime}}\in\frac{\mathbb{Z}^{r}}{\operatorname{im}(\overline{\varphi}-Id)}$,
then every element $S^{y}x^{j}$ is $\varphi$-conjugated to some element
$S^{y^{\prime}}x^{\theta}$ for some $0\leq\theta<m^{c}$. Since
$\frac{\mathbb{Z}^{r}}{\operatorname{im}(\overline{\varphi}-Id)}=\\{\overline{v_{1}},...,\overline{v_{d}}\\}$,
every element $S^{y}x^{j}$ is $\varphi$-conjugated to some
$S^{v_{i}}x^{\theta}$, $1\leq i\leq d$, $0\leq\theta<m^{c}$, so
$R(\varphi)\leq dm^{c}<\infty$ and the proposition is proved.

∎

In the rest of the work we will use the following notation: we know that
$\gcd(p_{i}^{y_{i}},m^{c})=1$. This means that $p_{i}^{y_{i}}$ is an
invertible element in the commutative ring $\mathbb{Z}_{m^{c}}$ (now thought
in the abelian notation $\mathbb{Z}_{m^{c}}=\\{0,1,...,m^{c}-1\\}$). So, we
will naturally denote by $p_{i}^{-y_{i}}$ the inverse element
$(p_{i}^{y_{i}})^{-1}\in\mathbb{Z}_{m^{c}}$ and, similarly, we define
$p_{i}^{-ky_{i}}$ as $(p_{i}^{ky_{i}})^{-1}$ for any $k\geq 0$, so it makes
sense to write $p_{i}^{ky_{i}}$ for any $k\in\mathbb{Z}$, thinking of it as an
invertible element of the ring $\mathbb{Z}_{m^{c}}$. We are saying this to
avoid a possible misinterpretation of $p_{i}^{-y_{i}}$ as
$\frac{1}{p_{i}^{y_{i}}}\in\mathbb{Q}$, for example. With this notation, it is
clear that $s_{i}^{k}xs_{i}^{-k}=x^{p_{i}^{ky_{i}}}$ for any $k\in\mathbb{Z}$.

######  Proposition 4.2.

$\Gamma_{n,c}$ has not property $R_{\infty}$ if and only if there is
$M=(a_{ij})_{ij}\in Gl_{r}(\mathbb{Z})$ such that

  * •

$\det(M-Id)\neq 0$;

  * •

for any $1\leq i\leq r$,

$p_{1}^{a_{1i}y_{1}}p_{2}^{a_{2i}y_{2}}\dots p_{r}^{a_{ri}y_{r}}\equiv
p_{i}^{y_{i}}\ \ \text{mod}\ m^{c}.\ \ \ \ \ (M,c,i)$

###### Proof.

Suppose first that $\Gamma_{n,c}$ has not property $R_{\infty}$. Let
$\varphi\in Aut(\Gamma_{n,c})$ such that $R(\varphi)<\infty$. Let
$M=\overline{\varphi}\in Gl_{r}(\mathbb{Z})$, and write $M=(a_{ij})_{ij}$. By
Proposition 4.1, we have $\det(M-Id)\neq 0$. Since
$\varphi(\tau\Gamma_{n,c})\subset\tau\Gamma_{n,c}$, we have
$\varphi(x)=x^{\mu}$ for some $\mu\in\mathbb{Z}$ such that
$\gcd(\mu,m^{c})=1$. Let us show that for any $1\leq i\leq r$ the equation
$(M,c,i)$ holds. For any such $i$, since $\varphi$ is a homomorphism of
$\Gamma_{n,c}$ it must satisfy
$\varphi(s_{i})\varphi(x)\varphi(s_{i})^{-1}=\varphi(x)^{p_{i}^{y_{i}}}$, so

$s_{1}^{a_{1i}}\dots s_{r}^{a_{ri}}x^{\mu}s_{r}^{-a_{ri}}\dots
s_{1}^{-a_{1i}}=x^{\mu p_{i}^{y_{i}}}$

or, equivalently,

$x^{\mu p_{1}^{a_{1i}y_{1}}\dots p_{r}^{a_{ri}y_{r}}}=x^{\mu p_{i}^{y_{i}}}.$

Then $\mu p_{1}^{a_{1i}y_{1}}\dots p_{r}^{a_{ri}y_{r}}\equiv\mu p_{i}^{y_{i}}$
mod $m^{c}$, and since $\gcd(\mu,m^{c})=1$, we have $p_{1}^{a_{1i}y_{1}}\dots
p_{r}^{a_{ri}y_{r}}\equiv p_{i}^{y_{i}}$ mod $m^{c}$, which is exactly
$(M,c,i)$. This shows the “if” part. Suppose now that there is such a matrix
$M=(a_{ij})_{ij}$ and let us show $\Gamma_{n,c}$ has not $R_{\infty}$. Define
$\varphi:\Gamma_{n,c}\to\Gamma_{n,c}$ by $\varphi(x)=x$ and
$\varphi(s_{i})=s_{1}^{a_{1i}}s_{2}^{a_{2i}}...s_{r}^{a_{ri}}$. Let us check
that $\varphi$ is a well defined homomorphism:

$\varphi(s_{i})\varphi(x)\varphi(s_{i})^{-1}=s_{1}^{a_{1i}}s_{2}^{a_{2i}}...s_{r}^{a_{ri}}xs_{r}^{-a_{ri}}...s_{2}^{-a_{2i}}s_{1}^{-a_{1i}}=x^{p_{1}^{a_{1i}y_{1}}...p_{r}^{a_{ri}y_{r}}}=\varphi(x)^{p_{1}^{a_{1i}y_{1}}...p_{r}^{a_{ri}y_{r}}}=\varphi(x)^{p_{i}^{y_{i}}},$

the last equality being true by $(M,c,i)$. Also, since the $s_{i}$ commute, we
obviously have

$\varphi(s_{i})\varphi(s_{j})=s_{1}^{a_{1i}}...s_{r}^{a_{ri}}s_{1}^{a_{1j}}...s_{r}^{a_{rj}}=s_{1}^{a_{1j}}...s_{r}^{a_{rj}}s_{1}^{a_{1i}}...s_{r}^{a_{ri}}=\varphi(s_{j})\varphi(s_{i}).$

Finally,

$\varphi(x)^{m^{c}}=x^{m^{c}}=1,$

so $\varphi$ is in fact a homomorphism. Let us now construct an inverse
homomorphism. Let $N=M^{-1}\in GL_{r}(\mathbb{Z})$ and write
$N=(b_{ij})_{ij}$. Let us show that, for any $1\leq i\leq r$, $N$ satisfies
the equation $(N,c,i)$, that is
$p_{1}^{b_{1i}y_{1}}p_{2}^{b_{2i}y_{2}}..p_{r}^{b_{ri}y_{r}}=p_{i}^{y_{i}}$
mod $m^{c}$. Since $MN=Id$, for any $1\leq i,j\leq r$ we have

$\sum_{k=1}^{r}a_{ik}b_{kj}=(MN)_{ij}=Id_{ij}=\delta_{ij},$

where $\delta_{ij}$ is the Kronecker delta. Fix $i$. We do the following: for
each fixed $1\leq j\leq r$, we raise both sides of equation $(M,c,j)$ to the
power of $b_{ji}$ and obtain

$p_{1}^{a_{1j}b_{ji}y_{1}}p_{2}^{a_{2j}b_{ji}y_{2}}\dots
p_{r}^{a_{rj}b_{ji}y_{r}}\equiv p_{j}^{b_{ji}y_{j}}\ \ mod\ m^{c}$

Now, if we do the product of all the $r$ equations above (on both sides, of
course) and rearrange the left side according to the primes we get

$p_{1}^{(a_{11}b_{1i}+\dots+a_{1r}b_{ri})y_{1}}p_{2}^{(a_{21}b_{1i}+\dots+a_{2r}b_{ri})y_{2}}\dots
p_{r}^{(a_{r1}b_{1i}+\dots+a_{rr}b_{ri})y_{r}}\equiv
p_{1}^{b_{1i}y_{1}}p_{2}^{b_{2i}y_{2}}\dots p_{r}^{b_{ri}y_{r}}\ \ mod\
m^{c},$

or

$p_{1}^{(\sum_{k}a_{1k}b_{ki})y_{1}}p_{2}^{(\sum_{k}a_{2k}b_{ki})y_{2}}\dots
p_{r}^{(\sum_{k}a_{rk}b_{ki})y_{r}}\equiv
p_{1}^{b_{1i}y_{1}}p_{2}^{b_{2i}y_{2}}\dots p_{r}^{b_{ri}y_{r}}\ \ mod\
m^{c},$

or even

$p_{1}^{\delta_{1i}y_{1}}p_{2}^{\delta_{2i}y_{2}}\dots
p_{r}^{\delta_{ri}y_{r}}\equiv p_{1}^{b_{1i}y_{1}}p_{2}^{b_{2i}y_{2}}\dots
p_{r}^{b_{ri}y_{r}}\ \ mod\ m^{c},$

which results in

$p_{i}^{y_{i}}\equiv p_{1}^{b_{1i}y_{1}}p_{2}^{b_{2i}y_{2}}\dots
p_{r}^{b_{ri}y_{r}}\ \ mod\ m^{c},$

which is exactly $(N,c,i)$, as we wanted. Now define
$\psi:\Gamma_{n,c}\to\Gamma_{n,c}$ by $\psi(x)=x$ and
$\psi(s_{i})=s_{1}^{b_{1i}}s_{2}^{b_{2i}}...s_{r}^{b_{ri}}$. As we did with
$\varphi$, the fact that $N$ satisfies $(N,c,i)$ for all $i$ gives us that
$\psi$ is a group homomorphism. Of course we have $\varphi(\psi(x))=x$. Also,
by the fact that $MN=Id$, straightforward calculations show that
$\varphi(\psi(s_{i}))=s_{i}$. Similarly, we show that $\psi\varphi=Id$ by
using that $NM=Id$, so $\varphi\in Aut(\Gamma_{n,c})$. Since
$\overline{\varphi}=M$ we have $\det(\overline{\varphi}-Id)=\det(M-Id)\neq 0$
by hypothesis, so $R(\varphi)<\infty$ by Proposition 4.1. This completes the
proof. ∎

######  Observation 4.3.

Implicit in the proof of Proposition 4.2 above is the classification of all
matrices in $GL_{r}(\mathbb{Z})$ which can be extended to automorphisms of
$\Gamma_{n,c}$. In other words, given a matrix $M=(a_{ij})_{ij}\in
GL_{r}(\mathbb{Z})$, there is an automorphism $\varphi$ of $\Gamma_{n,c}$ such
that $\overline{\varphi}=M$ if and only if all equations $(M,c,i)$ are
satisfied.

To proceed, we need the following lemma, which can be easily shown by
elementary number theory and induction on $k$:

######  Lemma 4.4.

Let $x,m\geq 2$. If $x=1\ \text{mod}\ m$, then $x^{m^{k}}=1\ mod\ m^{k+1}$ for
any $k\geq 0$. ∎

######  Theorem 4.5.

Let $n\geq 2$ have prime decomposition $n={p_{1}}^{y_{1}}\dots p_{r}^{y_{r}}$,
the $p_{i}$ being pairwise distinct and $y_{i}>0$. Suppose $r\geq 2$, that is,
there are at least two primes involved. Then the nilpotent quotient group
$\Gamma_{n,c}=\Gamma_{n}/\gamma_{c+1}(\Gamma_{n})$ does not have property
$R_{\infty}$ for any $c\geq 1$. In other words, the $R_{\infty}$-nilpotency
degree of $\Gamma_{n}$ is infinite.

###### Proof.

Let $m=\gcd(p_{1}^{y_{1}}-1,\dots,p_{r}^{y_{r}}-1)$, as we have done in this
work. If $m=1$, then none of the groups $\Gamma_{n,c}$ have property
$R_{\infty}$. This is because $\Gamma_{n,c}\simeq\mathbb{Z}^{r}$ for any $c$
in this case (see Theorem 3.4), and we know $\mathbb{Z}^{r}$ has not
$R_{\infty}$. So, from now on, suppose $m\geq 2$. Of course $\Gamma_{n,1}$
does not have property $R_{\infty}$, for it is a finitely generated abelian
group. Now, for any fixed $c\geq 2$, we will use Proposition 4.2, that is, for
any $r\geq 2$, we will find a matrix $M=(a_{ij})_{ij}\in Gl_{r}(\mathbb{Z})$
with $\det(M-Id)\neq 0$ and satisfying equations $(M,c,i)$ for $1\leq i\leq
r$. We will look for a particular family of matrices $M$, that is,

$M=m^{k}N+Id.$

Here, $k$ will be some suitable positive number,
$N=(j_{\alpha\beta})_{\alpha\beta}$ will be some integer $r\times r$ matrix
with determinant $1$ and $m^{k}N=(m^{k}j_{\alpha\beta})_{\alpha\beta}$ is the
natural scalar product of a number by a matrix. The first thing to observe is
that any such matrix $M$ satisfies all the equations $(M,c,i)$ for some big
enough $k\geq 1$. Let us see that. It is easy to see that, for such $M$, the
equations $(M,c,i)$ become exactly

$(p_{1}^{j_{1i}y_{1}}p_{2}^{j_{2i}y_{2}}\dots
p_{r}^{j_{ri}y_{r}})^{m^{k}}\equiv 1\ \ \text{mod}\ m^{c}.\ \ \ \ \ (M,c,i)$

For us to use the previous lemma, the term inside the parenthesis in the above
equation must be congruent to $1$ modulo $m$, so we claim this is true. Since
$m$ divides each number $p_{s}^{y_{s}}-1$ ($1\leq s\leq r$) by definition, we
have $p_{s}^{y_{s}}=1\ \text{mod}\ m$, so by the multiplicative property of
integer congruence,it is clear that
$p_{1}^{j_{1i}y_{1}}p_{2}^{j_{2i}y_{2}}\dots p_{r}^{j_{ri}y_{r}}\equiv 1\ \
\text{mod}\ m$, which shows our claim. Now let $k=c-1$. By the above lemma we
have
$(p_{1}^{j_{1i}y_{1}}p_{2}^{j_{2i}y_{2}}...p_{r}^{j_{ri}y_{r}})^{m^{k}}=1\
\text{mod}\ m^{c}$, so for every $i$, equation $(M,c,i)$ is satisfied for such
$M$.

It is then enough for us to find, for any $r\geq 2$, an integer matrix $N$
which makes $det(M)=1$ and $det(M-Id)\neq 0$. Since $M=m^{k}N+Id$, we have

$det(M-Id)=det(m^{k}N)=m^{rk}det(N),$

so for $det(M-Id)$ to be non-zero it suffices us to have $det(N)\neq 0$. We
claim therefore that, for any $r\geq 2$, there is a matrix $N_{r}$ such that
$det(N_{r})=1$ and $det(M_{r})=det(m^{k}N_{r}+Id)=1$. For any $r\geq 2$, let

$N_{r}=\left[\begin{array}[]{ccccccc}1&-(m^{k}+2)&m^{k}+1&-(m^{k}+1)&\dots&(-1)^{r-4}(m^{k}+1)&(-1)^{r-3}(m^{k}+1)\\\
1&-(m^{k}+1)&m^{k}&-m^{k}&\dots&(-1)^{r-4}m^{k}&(-1)^{r-3}m^{k}\\\
0&1&0&0&\dots&0&0\\\ 0&0&1&0&\dots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&0&\dots&1&0\end{array}\right].$

By developing the determinant of $N_{r}$ using the last column, we get that
$\det(N_{r})=1$, since the two submatrices that appear are upper triangular
with diagonal entries equal 1. Now, our task is to prove that $\det(M_{r})=1$,
where

$M_{r}=\left[\begin{array}[]{ccccccc}d&-m^{k}(m^{k}+2)&m^{k}d&-m^{k}d&\dots&(-1)^{r-4}m^{k}d&(-1)^{r-3}m^{k}d\\\
m^{k}&-m^{k}d+1&m^{2k}&-m^{2k}&\dots&(-1)^{r-4}m^{2k}&(-1)^{r-3}m^{2k}\\\
0&m^{k}&1&0&\dots&0&0\\\ 0&0&m^{k}&1&\dots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&0&\dots&1&0\\\
0&0&0&0&\dots&m^{k}&1\end{array}\right]$

with $d=m^{k}+1$. We will prove this by induction. The case $r=2$ is verified
by the calculation of the determinant of

$M_{2}=\left[\begin{array}[]{cc}m^{k}+1&-m^{k}(m^{k}+2)\\\
m^{k}&-m^{k}(m^{k}+1)+1\end{array}\right].$

Now, for $r>2$, developing the determinant of $M_{r}$ by the last column gives
us:

$\begin{array}[]{rl}\det(M_{r})&=(-1)^{r+1}(-1)^{r-3}m^{k}dm^{k(r-1)}+(-1)^{r+2}(-1)^{r-3}m^{2k}dm^{k(r-2)}+(-1)^{2r}\det(M_{r-1})\\\
&=m^{kr}d-m^{kr}d+1\\\ &=1.\end{array}$

This completes the induction step and finishes our proof. ∎

## References

  * [1] M. Calvez and I. Soroko. _Property $R_{\infty}$ for some spherical and affine Artin-Tits groups_ Journal of Group Theory, vol. , no. , 2022. https://doi.org/10.1515/jgth-2022-0010
  * [2] K. Dekimpe and D. L. Gonçalves. _The $R_{\infty}$ property for free groups, free nilpotent groups and free solvable groups_, Bull. London Math. Soc., V. 46, N. 4, 2014, 737 – 746.
  * [3] K. Dekimpe and D. L. Gonçalves. _The $R_{\infty}$ property for nilpotent quotients of surface groups_, Transactions of the London Mathematical Society, Volume 3, Issue 1, 2016, pp. 28 – 46.
  * [4] K. Dekimpe and D. L. Gonçalves. _The $R_{\infty}$ property for nilpotent quotients of Baumslag-Solitar groups_, Journal of Group Theory, vol. 23, no. 3, 2020, pp. 545 – 562.
  * [5] K. Dekimpe, D. L. Gonçalves and O. Ocampo. _The $R_{\infty}$ property for pure Artin braid groups_, Monatsh Math 195, 15 – 33 (2021).
  * [6] A. L. Fel’shtyn and D. L. Gonçalves. _The Reidemeister number of any automorphism of a Baumslag-Solitar group is infinite_ , Geometry and dynamics of groups and spaces, 399 – 414, Progr. Math., 265, Birkhäuser, Basel, 2008.
  * [7] D. L. Gonçalves and P. Wong. _Twisted conjugacy classes in nilpotent groups_ , Journal fur die Reine und Angewandte Mathematik, V. 633, 2009, pp. 11 – 27.
  * [8] B. Jiang, _Lectures on nielsen fixed point theory_ , Contemporary Mathematics 14 (American Mathematical Society, Providence, RI, 1983) vii+100pp.
  * [9] G. Levitt and M. Lustig. _Most automorphisms of a hyperbolic group have very simple dynamics_ , Ann. Scient. Éc. Norm. Sup. 33, 507 – 517 (2000).
  * [10] W. Magnus, A. Karrass and D. Solitar. _Combinatorial group theory: Presentation of groups in terms of generators and relations_ , Dover publications, inc. New York, 1976.
  * [11] R. Mikhailov and I. B. S. Passi. _Lower Central and Dimension Series of Groups_ , Part of the book series: Lecture Notes in Mathematics (LNM, volume 1952), Springer, 2009.
  * [12] K. Reidemeister. _Automorphismen von Homotopiekettenringen_ , Math. Ann. 112 (1936) 586 – 593.
  * [13] V. Roman’kov. _Twisted conjugacy classes in nilpotent groups_ , J. Pure Appl. Algebra, V. 215, N. 4, 2011, 664 – 671.
  * [14] J. Taback and P. Wong. _Twisted conjugacy and quasi-isometry invariance for Generalized Solvable Baumslag-Solitar groups_ , J. London Math. Soc. (2) 75 (2007) 705 – 717, 2007.


# Effect of packing fraction on the jamming of granular flow through small
apertures

Rodolfo Omar Uñac1, Ana María Vidales1 and Luis A. Pugnaloni2 1 Departamento
de Física, Instituto de Física Aplicada (UNSL-CONICET), Universidad Nacional
de San Luis, Ejército de los Andes 950, D5700HHW San Luis, Argentina.
2 Instituto de Física de Líquidos y Sistemas Biológicos (CONICET La Plata,
UNLP), Calle 59 Nro 789, 1900 La Plata, Argentina<EMAIL_ADDRESS>(R O
Uñac<EMAIL_ADDRESS>(A M Vidales<EMAIL_ADDRESS>(L A
Pugnaloni)

###### Abstract

We investigate the flow and jamming through small apertures of a column of
granular disks via a pseudo-dynamic model. We focus on the effect that the
preparation of the granular assembly has on the size of the avalanches
obtained. Ensembles of packings with different mean packing fractions are
created by tapping the system at different intensities. Surprisingly, packing
fraction is not a good indicator of the ability of the deposit to jam a given
orifice. Different mean avalanche sizes are obtained for deposits with the
same mean packing fraction that were prepared with very different tap
intensities. It has been speculated that the number and size of arches in the
bulk of the granular column should be correlated with the ability of the
system to jam a small opening. We show that this correlation, if exists, is
rather poor. A comparison between bulk arches and jamming arches (i.e., arches
that block the opening) reveals that the aperture imposes a lower cut-off on
the horizontal span of the arches which is greater than the actual size of the
opening. This is related to the fact that blocking arches have to have the
appropriate orientation to fit the gap between two piles of grains resting at
each side of the aperture.

##  1 Introduction

The flow of granular materials through apertures is commonplace in a variety
of industrial applications. Studies in this respect can be separated into two
major areas: (1) continuous flow, and (2) jamming. Continuous granular flow is
observed for dry non-cohesive materials if the size of the aperture is large
(typically above five grain diameters for spherical particles). Jamming is
observed whenever the opening is smaller, which requires the input of external
perturbations in order to restart the flow. The cause of jamming is the
formation of a blocking arch. We distinguish two type of arches: blocking
arches (or jamming arches) and bulk arches. Blocking arches form at the
orifice during discharge and prevent the flow. Bulk arches form at any place
inside the packing during the dynamical process that leads the grains to reach
mechanical equilibrium. Arches are set of particles that are mutually stable.
The removal of any particle in the arch leads to the destabilization of the
others.

A number of studies have considered the jamming of an aperture during the
discharge of grains. These include experimental studies on two-dimensional
(2D) hoppers using circular grains [1, 2] and three-dimensional (3D) silos
using spherical and non-spherical particles [3, 4], numerical simulations
using discrete element methods in 2D [7, 8], experiments with quasi-2D silos
and spherical grains [5], 3D vibrated silos [6] and experiments with tilted
[9] and wedge-shaped hoppers [10]. Also, the properties of blocking arches and
bulk arches have been considered in the past [1, 14, 11, 12, 13, 15]. However,
none of these studies have considered the effect of the preparation of the
granular column prior to the discharge. It is known that the number and size
of arches inside a granular assembly are dependent on the packing fraction.
Therefore, it is expected that the jamming of columns prepared at different
packing fractions may occur with different probability. A related issue is the
question of to what extent the arches formed in the bulk of the system are
comparable with the arches that effectively block the aperture during
drainage.

In this paper we use a 2D pseudo-dynamic simulation scheme previously
developed by Manna and Khakhar [16, 17] to study the effect of the initial
packing fraction on the jamming probability and the correlation between bulk
arches and blocking arches. The jamming probability is directly connected with
the mean size of the avalanches [5]. An avalanche is the flow of grains that
occurs between the initiation of the discharge and the arrest of the flow due
to the formation of a blocking arch [3]. We will show that there is a strong
dependence of the size of the avalanches with the packing fraction. However,
there is not a monotonic relation between these two quantities. We also find
that there is a poor correlation between the size of arches in the bulk and
the size of the avalanches. A comparison between bulk arches and jamming
arches reveals that the aperture not only imposes a cut-off on arches of
horizontal span below the opening size, but also prevents the formation of
some blocking arches that, in principle, are wide enough to induce jamming.

##  2 The pseudo-dynamic algorithm

Our simulations are based on an algorithm for inelastic massless hard disks
designed by Manna and Khakhar [16, 17]. This is a pseudo-dynamics that
consists in small falls and rolls of the grains until they come to rest by
contacting other particles or the system boundaries. We use a container formed
by a flat base and two flat vertical walls. No periodic boundary conditions
are applied.

The deposition algorithm consists in choosing a disk in the system and
allowing a free fall of length $\delta$ if the disk has no supporting
contacts, or a roll of arc-length $\delta$ over its supporting disk if the
disk has one single supporting contact [16, 17, 18]. Disks with two supporting
contacts are considered stable and left in their positions. If in the course
of a fall of length $\delta$ a disk collides with another disk (or the base),
the falling disk is put just in contact and this contact is defined as its
first supporting contact. Analogously, if in the course of a roll of length
$\delta$ a disk collides with another disk (or a wall), the rolling disk is
put just in contact. If the first supporting contact and the second contact
are such that the disk is in a stable position, the second contact is defined
as the second supporting contact; otherwise, the lowest of the two contacting
particle is taken as the first supporting contact of the rolling disk and the
second supporting contact is left undefined. If, during a roll, a particle
reaches a lower position than the supporting particle over which it is
rolling, its first supporting contact is left undefined (in this way the
particle will fall vertically in the next step instead of rolling underneath
the first contact). A moving disk can change the stability state of other
disks supported by it, therefore, this information is updated after each move.
The deposition is over once each particle in the system has both supporting
contacts defined or is in contact with the base (particles at the base are
supported by a single contact). Then, the coordinates of the centers of the
disks and the corresponding labels of the two supporting particles, wall, or
base, are saved for analysis.

Figure 1: (Color online). Sample configurations of the granular column for the
steady state corresponding to $\Gamma=0.39$ before (a) and after (b)
discharging through and opening of width $D=2.75$. The red segments indicate
the arches in the system.

An important point in these simulations is the effect that the parameter
$\delta$ has in the results since particles do not move simultaneously but one
at a time. One might expect that in the limit $\delta\rightarrow 0$ we should
recover a fairly ”realistic” dynamics for fully inelastic non-slipping disk
dragged downwards at constant velocity. This should represent particles
deposited in a viscous medium or carried by a conveyor belt. Although this
dynamics contrasts with the dynamics of dry granulates, experiments on the
jamming of fully submerged grains in gels [19] have shown remarkable
similarities with the more widely available data on dry systems. We chose
$\delta=0.0062d$ (with $d$ the particle diameter) since we have observed that
for smaller values of $\delta$ results are indistinguishable from those
obtained here [18].

The pseudo-dynamics approach has been chosen in view of the low CPU time
demanded by this scheme. In this study, we need to prepare a large number of
samples (through tapping) with a given packing fraction and then trigger
discharges for different aperture sizes. More realistic simulations such as
granular dynamics (or discrete element method) would require a much higher
computational effort. In spite of the simplifications of the pseudo-dynamics,
it has been shown that results on tapping agree qualitatively with granular
dynamics simulations [21].

##  3 Initial packings

In order to study the effect of packing fraction on the jamming of the flow
through an aperture, we first need to prepare packings at reproducible packing
fractions. To achieve this, several techniques can be applied. For example,
the sequential deposition of grains submerged in a viscous liquid yield
reproducible packing fractions that can be tuned by changing the friction
coefficient of the particles or the density mismatch [22]. We have chosen
another well known technique to generate reproducible ensembles of packings:
tapping. Nowak et al. have shown that an appropriate tapping protocol can lead
to reproducible states in the sense that an ensemble of configurations with
well defined mean packing fraction is recovered if the same protocol is
followed irrespective of the initial state [23]. This has been more carefully
discussed by Ribière et al. [20]. Dijksman et al. showed how different states
can be obtained not only by changing the tap intensity but also by changing
the tap duration [24]. A similar effect was investigated by Pica Ciamarra et
al. in submerged samples where a fluid pulse is used to excite the granular
column [25]. Hence, we use a simulated tapping protocol (see below) to
generate sets of initial configurations that have well defined mean packing
fractions.

Figure 2: Mean packing fraction $\phi$ in the steady state of the tapping
protocol as a function of the tap intensity $\Gamma$.

The simulations are carried out in a rectangular box of width $24.78d$
containing $1500$ equal-sized disks of diameter $d$. Initially, disks are
placed at random in the simulation box (with no overlaps) and deposited using
the pseudo-dynamic algorithm. Once all the grains come to rest, the system is
expanded in the vertical direction and randomly shaken to simulate a vertical
tap. Then, a new deposition cycle begins. After many taps of given amplitude,
the system achieves a steady state where all characterizing parameters
fluctuate around equilibrium values independently of the previous history of
the granular bed. The existence of such “equilibrium” states has been
previously reported in experiments [20].

The tapping of the system is simulated by multiplying the vertical coordinate
of each particle by a factor $A$ (with $A>1$). Then, the particles are
subjected to several (about $20$) Monte Carlo loops where positions are
changed by displacing particles a random length $\Delta r$ uniformly
distributed in the range $0<\Delta r<A-1$. New configurations that correspond
to overlaps are rejected. This disordering phase is crucial to avoid particles
falling back again into the same positions. Moreover, the upper limit for
$\Delta r$ (i.e. $A-1$) is deliberately chosen so that a larger tap promotes
larger random changes in the particle positions. The expansion amplitude $A$
ranges from $1.03$ up to $3.0$. Following Refs. [26, 21] we quantify the tap
intensity by the parameter $\Gamma=\sqrt{A-1}$. For each value of $\Gamma$
studied, $10^{3}$ taps are carried out for equilibration followed by $5\times
10^{3}$ taps for production. $500$ deposited configurations are stored which
are obtained by saving every $10$ taps during the production run after
equilibration. These deposits will be used later as initial conditions for the
discharge and flow through an opening.

The deposited configurations are analyzed in search of bulk arches. We first
identify all mutually stable particles —which we define as directly connected—
and then we find the arches as chains of connected particles. Two disks A and
B are mutually stable if A is the left supporting particle of B and B is the
right supporting particle of A, or viceversa. We measure the total number of
arches, arch size distribution $n(k)$, and the horizontal span distribution of
the arches $n_{k}(x)$. The latter is the probability density of finding an
arch consisting of $k$ disks with horizontal span between $x$ and $x+dx$. The
horizontal span (or lateral extension) is defined as the projection onto the
horizontal axis of the segment that joins the centers of the right-end disk
and the left-end disk in the arch. In Fig. 1(a), we show an example of a
deposited configuration with arches indicated by segments (for a description
of Fig. 1(b) see next section). Notice that the pseudo-dynamics mimics the
behavior of disks that roll without slipping. This corresponds to a system
with infinite static friction which is expected to yield a large number and
variety of arches. The arch structure of frictionless systems may differ
significantly from the one seen in Fig. 1(a). However, simulations with
realistic discrete element methods with finite friction yield similar
structures [13].

In Fig. 2, we present the steady state mean packing fraction, $\phi$, of our
granular deposits as a function of $\Gamma$. There exists a rather sharp
decrease of $\phi$ as the tapping intensity is increased followed by a minimum
and a very smooth increase. The sharp drop of $\phi$ is associated to a
discontinuous order-disorder transition previously reported for this model
[18] and also observed in granular dynamics of polygonal grains [27]. The
appearance of the minimum packing fraction as a function of tap intensity has
been reported for several models (including a frustrated lattice gas model
[28], a Monte Carlo type deposition [21] and a realistic discrete element
method simulation [21]) and in experiments of tapping with a quasi-2D system
[30, 29]. For the model we use in this paper, the minimum $\phi$ has been
shown to exist even if bidisperse systems are considered [31].

Figure 3: (Color online). Number of arches per particle (red circles) and mean
size of the arches in terms of the number of grains involved in an arch (black
squares) as a function of $\Gamma$.

The minimum in $\phi$ is caused to a large extent by the formation of arches
[21]. Figure 3 shows the number and mean size of arches found in the system as
a function of $\Gamma$. When $\Gamma$ is increased considerably (above $0.7$),
every tap expands the assembly in such a way that particles get well apart
from each other. During deposition, particles will reach the free surface of
the bed almost sequentially (one at a time), reducing the chances of mutual
stabilization. Therefore, arches are less probable to form as $\Gamma$
increases and so $\phi$ must grow since fewer voids get trapped. Indeed, we
see in Fig. 3 that the number and size of arches decrease at large $\Gamma$
($\gtrsim 0.7$) for increasing tap intensities. Eventually, for very large
$\Gamma$, no arches are formed after each tap and $\phi$ will reach a limiting
value. At lower $\Gamma$ ($0.4\lesssim\Gamma\lesssim 0.7$), the free volume
injection due to a tap creates very narrow gaps between particles. For a given
arch to grow by the insertion of a new particle, it is necessary to create a
gap between two particles in the existing arch where the new particle can fit
in. This explains why increasing $\Gamma$ will promote the formation of larger
arches and reduce $\phi$ in this regime. For very low tapping intensities
($\Gamma\lesssim 0.3$), we find a rather constant $\phi$. However, the number
and size of the arches decrease with $\Gamma$ (see Fig. 3). This would imply
that a maximum in the packing fraction should be observed at such light
tapping. This maximum has been recently reported in other models [27, 32], but
is not present in our pseudo-dynamics. Notice that the number and size of the
arches give only a rough indication for the free volume in the sample since
the actual shape of the arches will also be important.

##  4 Flow and jamming

For each deposit generated as described in the previous section, we trigger a
discharge by opening an aperture of width $D$ relative to the diameter $d$ of
the disk in the center of the base of the containing box. Grains will flow out
of the box following the pseudo-dynamics until a blocking arch forms or until
the entire system is discharged (with the exception of two piles resting on
each side of the aperture). During the dynamics, disks that reach the bottom
and whose centers lie on the interval that defines the opening will fall
vertically (even if the surface of the disk touches the edge of the aperture).
This prevents the formation of arches with end disks sustained by the vertical
edge of the orifice. Although such arrangements happen in real experiments,
they are uncommon [14]. After each discharge, we record the size of the
avalanche (i.e. the number of grains flowed out) and the final arrangement of
the grains left in the box. Averages are taken over 500 discharges for each
value of $\Gamma$ used to prepare the initial packings.

One single discharge attempt is carried out for each initial deposit. This
allows us to assure that the initial preparation of the pack belongs to the
ensemble of deposits corresponding to the steady state of the particular tap
intensity chosen. In many experiments and most industrial applications,
discharges are triggered one after another from the same deposit without
preparing the system in the initial condition again [3, 4, 5, 6]. However,
some experiments do fill the container anew before each discharge [1, 2, 33].
We can see in Fig. 1(b) an image of the system after a discharge that resulted
in a jam. It is clear that the structure of the packing is greatly affected by
the partial discharge in our simulations. Therefore, these final structures
are not used for new discharges.

In Fig. 4 we plot the avalanche size distribution $p(s)$ for a few values of
$\Gamma$ and $D=2.25$. We obtain this by counting the number of grains $s$
that flow in each of the $500$ discharges corresponding to each initial
deposit generated for each $\Gamma$. An exponential tail in $p(s)$ has been
observed in several previous studies, both two-dimensional [5, 2, 8, 7] and
three-dimensional [3, 4]. Manna and Herrmann [34], using the same model, found
avalanches with a power law distribution. Notice however, that in Ref. [34]
the authors trigger one avalanche after the other by simply removing a grain
of the blocking arch. This minute perturbation to trigger avalanches may
induce strong correlations between successive discharges in contrast with the
strong rearrangements induced in most experiments. Based in our limited number
of discharges, we are unable to assert if an exponential or a power-law decay
is at play in our simulations (see log-lin and log-log plots in Fig. 4).
Although we report $p(s)$ up to $s=100$, larger avalanches of up to $1000$
disks are observed, whose statistics can be largely affected by the finite
size of the system ($N=1500$).

Figure 4: (Color online). Avalanche size distribution $p(s)$ for several
values of $\Gamma$ for $D=2.25$. (a) Semilog plot, (b) log-log plot.

The mean avalanche size, $\langle s\rangle$, as a function of $\Gamma$ is
shown in Fig. 5 for various aperture sizes. As it can be expected, $\langle
s\rangle$ increases if $D$ increases. As we can see, for small apertures,
$\langle s\rangle$ grows monotonically as $\Gamma$ increases. However, for
$D>2.0$, the mean avalanche size presents a local maximum and a local minimum
as a function of $\Gamma$.

It has been speculated [35, 11] that the size of the avalanches can be
connected with the arches inside the granular deposit. Although arches in the
initial configuration are not dragged to the aperture during flow since arches
actually break and form all the time in the process, it is believed that the
ability of the system to form arches in the initial deposit is connected with
the ability to form blocking arches during flow. Indeed, the features observed
in Fig. 5 are somewhat correlated with the number and size of the bulk arches.
As $\Gamma$ is increased from the lower values, the size of the arches remains
initially rather constant (whereas the number of arches decreases, Fig. 3).
This reduction in the number of arches leads to a smaller jamming probability
and a rapid increase in the size of the avalanches (see Fig. 5). This regime
ends when the sharp drop in $\phi$ ends ($\Gamma\approx 0.4$). For larger
$\Gamma$ and up to the packings with minimum packing fractions (i.e.
$0.4\lesssim\Gamma\lesssim 0.7$), the number and size of the arches increase.
As a consequence, $\langle s\rangle$ decreases due to the increased likelihood
of jamming. Finally, for $\Gamma\gtrsim 0.7$, the number and size of the
arches fall and a corresponding increase of the avalanche sizes is observed.
From these observations we can assert that the number and size of arches in a
given packing give an overall indication of the chances that the system will
jam if it is left to flow through a small aperture.

Figure 5: (Color online). Mean avalanche size $\langle s\rangle$ as a function
of $\Gamma$ for several sizes of the aperture $D$. Notice that $\langle
s\rangle$ is affected to a large extent by the rare large avalanches not
reported in Fig. 4.

Notwithstanding the previous analysis, the size of the avalanches is not only
dependent on the preparation protocol —defined in our case by $\Gamma$— but
also on the actual size of the outlet imposed. For example, as we mentioned,
for $D=2.0$ the mean avalanche size does not present the maximum and minimum
suggested by the number and size of arches. In order to take into account the
effect of the size of the aperture we measure the probability
$P_{\mathrm{arch}}(D)$ of finding an arch in the bulk of the deposits wide
enough to block a given aperture $D$. We measure the horizontal span of each
arch as the projection on the horizontal axis of the segment that joins the
centers of the end particles of the arch. An arch of span $x$ can jam an
opening of width $x+d$ (with $d$ the diameter of a grain). In this analysis we
include the grains that do not form arches, which can jam any orifice with
$D\leq 1$. In Fig. 6 we plot $P_{\mathrm{arch}}(D)$ as a function of $\Gamma$
and compare with the corresponding $\langle s\rangle$. Overall, the
probability of finding an arch wide enough to block an aperture of size $D$
decreases with $D$ in correspondence with the overall increase of $\langle
s\rangle$. However, for a given $D$, the dependence with $\Gamma$ does not
show a clear anti-correlation between the probability and the mean avalanche
size. This implies that the bulk arches can give only a rough indication of
the eventual size of the avalanches that would discharge if an aperture is
opened.

Figure 6: (Color online). Mean avalanche size $\langle s\rangle$ as a function
of $\Gamma$ and probability $P_{\mathrm{arch}}(D)$ of finding an arch of
horizontal span $D-1$ for several apertures: (a) $D=2.0$, (b) $D=2.25$, (c)
$D=2.5$, (d) $D=2.75$.

It is important to mention that we have always opened the aperture at the
center of the bottom of the container. The ordering observed for $\Gamma<0.5$
(see Fig. 1) suggests that an effect related to the relative position of the
aperture and the first layer of grains may be expected. The main effect would
be related to the fact that particles just at the edges of the orifice do not
flow in the pseudo-dynamics and these will produce a reduced effective
aperture. If all discharges start from an initial packing so ordered that
grains at the first layer always sit on the same horizontal positions, the
mean avalanche size would depend on the horizontal position of the aperture.
When this happens, one observes oscillations in the avalanche size as a
function of the aperture size [33]. However, this does not happen in our
simulations as we can see in Fig. 5. This effect is observed only for highly
ordered structures obtained with frictionless particles [33]. Our particles
model non-slipping grains and the small deviations in position of the disks of
the first layer with respect to a truly crystalline structure are sufficiently
large to mask any systematic effect due to ordering that may require a
detailed study on the position chosen for the aperture.

##  5 Effect of packing fraction

It is generally believed that packing fraction is a good parameter to
characterize many properties of a granular bed [36, 37]. Results obtained for
deposits prepared at a given $\phi$ are not necessarily general and must be
repeated for different packing fractions. However, this does not mean that
packing fraction is the only or the main factor that can affect the results.
In Table 1 we present part of the data of Fig. 5 but ordered by mean packing
fractions $\phi$ corresponding to the steady state obtained for each given
$\Gamma$ (see Fig. 2). As we can see, the mean avalanche size depends on
$\phi$ in a non trivial way. At the highest $\phi$, obtained by light tapping,
the mean avalanche size can range from a few grains to a hundred grains
depending on the value of $\Gamma$ used to create the packings. The larger the
aperture $D$, the wider the range in $\langle s\rangle$ within this regime
where the system is rather ordered but the number and size of the arches fall
with increasing $\Gamma$ (see Fig. 3). For low $\phi$, deposits with similar
packing fractions but created with low and high tap intensities display
different values of $\langle s\rangle$ for any given $D$.

As we can see, deposits with the same $\phi$ can present different values of
$\langle s\rangle$. Therefore, the steady state ensembles of packings with
equal $\phi$ obtained by tapping may behave differently. This has been pointed
out in Refs. [29, 30] where steady states with the same $\phi$ but bearing
different stresses were obtained through tapping. In our simulations, forces
are not calculated and therefore the stress tensor cannot be obtained.
However, a clear difference in the response of the granular columns with same
$\phi$ is observed in the sense that avalanches are, in average, of different
size.

|  |  |  | $\langle s\rangle$ |
---|---|---|---|---|---
$\phi$ | $\Gamma$ | $D=2.00$ | $D=2.25$ | $D=2.50$ | $D=2.75$
0.8425 | 0.173 | 7.2 | 7 | 7.1 | 8
0.8425 | 0.224 | 7 | 9 | 8 | 9
0.8424 | 0.274 | 9 | 11 | 12 | 33
0.8424 | 0.316 | 7 | 29 | 74 | 107
0.7393 | 0.548 | 26 | 27 | 41 | 85
0.7354 | 1.000 | 47 | 59 | 78 | 104
0.7540 | 0.447 | 10 | 36 | 80 | 130
0.7603 | 1.414 | 50 | 60 | 71 | 110
Table 1: Mean avalanche size $\langle s\rangle$ for different tap intensities
$\Gamma$ that yield similar packing fractions $\phi$.

##  6 Connection between bulk arches and jamming arches

Although we have shown in the previous section that arches found in the bulk
of the granular packing give a rough indication as to whether the system would
be more or less likely to jam, arches actually formed at the aperture during
discharge are different. Detailed studies of such blocking arches have been
reported for two-dimensional experimental setups [14, 38]. Since we have
access to both bulk and blocking arches in our simulations, we compare a few
properties and discuss on the implications for the jamming probability.

In Fig. 7 we compare the arch size distribution $n(k)$ for the bulk arches and
for the jamming arches for different values of $D$. $n(k)$ is the probability
of finding an arch of $k$ grains ($k\geq 2$). For bulk arches, $n(k)$ is
calculated as the number of arches of $k$ disks found in all initial packings
divided by the total number of arches (i.e., summing for all $k\geq 2$). For
jamming arches, $n(k)$ is calculated as the number of discharges that led to a
blocking arch of $k$ disks divided by the total number of discharges
(discharges that ended without producing a jam were not considered). The plot
is presented as a function of $k-D$ since this produces a collapse of the
curves by subtracting a quantity ($D$) proportional to the lower cut-off
imposed by the orifice on the arch sizes (see Ref. [14]). The fact that all
the normalized histograms fall on the same curve indicates that the nature of
the arches that jam the aperture is the same for small and big orifices. In
general, an exponential tail is observed and a cut off for small arches is
imposed by the orifice. The large arches generally form upstream supported by
two stationary piles resting on both sides of the opening. It is clear from
Fig. 7 that blocking arches tend to be larger than bulk arches (even after the
correction due to the cut-off imposed by $D$). We believe this is due to the
fact that many small arches that are stable in the bulk thanks to the many
neighbors cannot accommodate in the conical shaped funnel created by the two
stationary piles. This is best demonstrated in the next paragraph.

Figure 7: (Color online). Distribution $n(k)$ of arch sizes for
$\Gamma=0.387$. The black symbols correspond to arches found in the bulk prior
to the discharge, the color data correspond to the blocking arches for
different values of $D$ as indicated in the legend. Notice that the horizontal
axis is shifted by the size of the aperture $D$ for each set of data.

The horizontal span of the arches for a given number of grains $k$ is shown in
Fig. 8. We include data for the bulk arches found in the initial deposits and
for the jamming arches found for an aperture of width $D=2.5$. As we can see,
for a given $k$, arches are more likely to be wider in the case of jamming
arches as compared to bulk arches. This is to be expected for small $k$ since
small arches with small span might not be able to jam the aperture. However,
even arches with bigger $k$ are biased in the distribution of blocking arches.
This is due to the two piles formed at each side of the orifice. Blocking
arches that are wider than the aperture $D$ must span this funnel. Arches of
$k$ disks with horizontal span sufficient to jam the orifice might still be
unable to span the funnel (see the inset in Fig. 8(c)). This results in
blocking arches generally wider, for a given $k$, than the corresponding bulk
arches. Notice that arches of horizontal span $x<D-1$ may also jam the orifice
(see Fig. 8(a)) due to the grains sitting at the edges of the aperture that
reduce the effective size of the opening. This effect has been studied in more
detail by Pournin et al. [33].

Figure 8: (Color online). Horizontal span distribution $n_{k}(x)$ for arches
formed by different number of grains $k$ at $\Gamma=0.707$. (a) $k=2$, (b)
$k=3$, (c) $k=4$, (d) $k=5$. The red dashed lines correspond to blocking
arches for an aperture $D=2.5$, whereas the black solid lines correspond to
arches found in the bulk of the initial deposits. The inset of panel (c) shows
two schematic arches of $k=4$ which are wide enough to jam the orifice but
only one of them fits in the gap left by the two piles at rest on each side of
the aperture.

##  7 Conclusions

We have considered granular avalanches discharged though small apertures at
the bottom of a container by using a 2D pseudo-dynamic model. We have focused
on the effect of the packing fraction of the granular deposit prior to the
avalanche discharge. The results indicate that the initial packing fraction
has an important effect on the mean avalanche size for a given opening size.
However, similar $\langle s\rangle$ can be obtained for packings with very
different $\phi$. Most importantly, very different values of $\langle
s\rangle$ can correspond to initial packings with the same packing fractions
that were prepared by using different tap intensities. It is important to note
that these results are obtained not for single packings but for ensembles of
deposits representative of steady states corresponding to a particular tap
intensity.

Our main conclusion is that packing fraction is not a good macroscopic
parameter to predict the size of the avalanches that would flow through a
given aperture. It seems that further information is necessary. Although this
information is expected to reside in the size and number of arches, we have
seen that the correlation of these with $\langle s\rangle$ is not consistent
for all openings $D$. It seems that is not possible to predict the jamming
probability of a granular column as it flows through a small aperture based on
a few global properties of the initial deposits.

A side result from our study is that blocking arches are generally wider than
the arches found in bulk. This is not only due to the fact that the aperture
imposes a lower cut-off for the possible jamming arches, but also to the fact
that even arches formed in the bulk which are wide enough to block the outlet
have shapes not compatible with the conical boundary effectively created by
the two piles of stationary grains at the sides of the aperture.

LAP acknowledges valuable discussion with Angel Garcimartín. This work has
been partially supported by CONICET (Argentina).

## References

## References

  * [1] K. To, P-Y Lai, and H. K. Pak, Phys. Rev. Lett. 86, 71 (2001).
  * [2] K. To, Phys. Rev. E 71, 060301(R) (2005).
  * [3] I. Zuriguel, L. A. Pugnaloni, A. Garcimartín, and D. Maza, Phys. Rev. E 68, 030301(R) (2003).
  * [4] I. Zuriguel, A. Garcimartín, D. Maza, L. A. Pugnaloni, and J. M. Pastor, Phys. Rev. E 71, 051303 (2005).
  * [5] A. Janda, I. Zuriguel, A. Garcimartín, L. A. Pugnaloni and D. Maza, Europhys. Lett. 84, 44002 (2008).
  * [6] C. Mankoc, A. Garcimartín, I. Zuriguel, D. Maza and L. A. Pugnaloni, Phys. Rev. E 80, 011309 (2009).
  * [7] C. F. M. Magalhães, J. G. Moreira and A. P. F. Atman, Phys. Rev. E 82, 051303 (2010).
  * [8] G. Pérez, Pramana 70, 989 (2008).
  * [9] H. G. Sheldon and D. J. Durian, Granular Matter 12, 579 (2010).
  * [10] S. Saraf and S. V. Franklin, Phys. Rev. E 83, 030301(R) (2011).
  * [11] L. A. Pugnaloni, G. C. Barker and A. Mehta, Adv. Complex Syst. 4, 289 (2001).
  * [12] L. A. Pugnaloni and G. C. Barker, Physica A 337, 428 (2004).
  * [13] R. Arévalo, D. Maza and L. A. Pugnaloni, Phys. Rev. E 74, 021303 (2006).
  * [14] A. Garcimartín, I. Zuriguel, L. A. Pugnaloni and A. Janda, Phys.Rev. E 82, 031306 (2010).
  * [15] A. Mehta, G. C. Barker and J. M. Luck, J. Stat. Mech: Theo. Experim. P10014 (2004).
  * [16] S. S. Manna and D. V. Khakhar, Phys. Rev. E 58, R6935 (1998).
  * [17] S. S. Manna, Phase Transition: A Multinational Journal 75, 529 (2002).
  * [18] L. A. Pugnaloni, M. G. Valluzi and L. G. Valluzzi, Phys. Rev. E 73, 051302 (2006).
  * [19] N. Roussel, Thi Lien Huong Nguyen and P. Coussot, Phys. Rev. Lett. 98, 114502 (2007).
  * [20] Ph. Ribière, P. Richard, P. Philippe, D. Bideau and R. Delannay, Eur. Phys. J. E 22, 249 (2007).
  * [21] L. A. Pugnaloni, M. Mizrahi, C. M. Carlevaro and F. Vericat, Phys. Rev. E 78, 051305 (2008).
  * [22] G. R. Farrell, K. M. Martini and N. Menon, Soft Matter 6, 2925 (2010).
  * [23] E. R. Nowak, J. B. Knight, M. L. Povinelli, H. M. Jaeger and S .R. Nagel, Powder Tech. 94, 79 (1997).
  * [24] J. A. Dijksman and M. van Hecke, Europhysics Lett 88, 44001 (2009).
  * [25] M. Pica Ciamarra, A. Coniglio and M. Nicodemi, Phys. Rev. Lett. 97, 158001 (2006).
  * [26] P. Philippe and D. Bideau, Phys. Rev. E 63, 051304 (2001).
  * [27] C. M. Carlevaro and L. A. Pugnaloni, J. Stat. Mech: Theo. Expt. P01007 (2011).
  * [28] P. A. Gago, N. E. Bueno and L. A. Pugnaloni, Gran. Matt. 11, 365 (2009).
  * [29] L. A. Pugnaloni, J. Damas, I. Zuriguel and D. Maza, Papers in Physics 3, 030004 (2011).
  * [30] L. A. Pugnaloni, I. Sánchez, P. A. Gago, J. Damas, I. Zuriguel and D. Maza, Phys Rev E 82, 050301(R) (2010).
  * [31] A. M. Vidales, L. A. Pugnaloni and I. Ippolito, Phys. Rev. E 77, 051305 (2008).
  * [32] A. D. Rosato, O. Dybenko, D. J. Horntrop, V. Ratnaswamy and L. Kondic, Phys. Rev. E 81, 061301 (2010).
  * [33] L. Pournin, M. Ramaioli, P. Folly and Th.M. Liebling, Eur. Phys. J. E 23, 229 (2007).
  * [34] S. S. Manna and H.J. Herrmann, Eur. Phys. J. E 1, 341 (2000).
  * [35] A. Mehta, Soft Matter 6, 2875 (2010).
  * [36] I. Bratberg, F. Radjai and A. Hansen, Phys. Rev. E 66, 031303 (2002).
  * [37] N. Gravish, P. B. Umbanhowar and D. I. Goldman Phys. Rev. Lett. 105, 128301 (2010).
  * [38] K. To and P.-Y. Lai, Phys. Rev. E 66, 011308 (2002).