# Group actions and power maps for groups over non-Archimedean local fields

Arunava Mandal and C.R.E. Raja

###### Abstract

We consider linear groups and Lie groups over a non-Archimedean local field
$\mathbb{F}$ for which the power map $x\mapsto x^{k}$ has a dense image or it
is surjective. We prove that the group of $\mathbb{F}$-points of such
algebraic groups is a compact extension of unipotent groups with the order of
the compact group being relatively prime to $k$. This in particular shows that
the power map is surjective for all $k$ is possible only when the group is
unipotent or trivial depending on whether the characteristic of $\mathbb{F}$
is zero or positive. Similar results are proved for Lie groups via the adjoint
representation. To a large extent, these results are extended to linear groups
over local fields and global fields.

##  1 Introduction

We will be considering existence of the solution to the equation

$x^{k}=g,~{}~{}k>1$ $None$

in a group $G$ for every $g\in G$. Our aim is to find structural conditions
that are equivalent to equation (1) having a solution in $G$ for every $g\in
G$. We first note that equation (1) having a solution in $G$ for every $g\in
G$ is equivalent to the corresponding power map $P_{k}\colon G\to G$ defined
by $P_{k}(g)=g^{k}$ being surjective. Thus, we pay attention surjectivity of
the power maps. For a (topological) group $G$ and $k\geq 1$, we say that
$P_{k}$ is surjective on $G$ (resp. dense in $G$) if $P_{k}\colon G\to G$
defined by $P_{k}(x)=x^{k}$ for all $x\in G$ is surjective (resp. has dense
image).

Surjectivity of power maps have been studied for algebraic groups over local
fields and algebraically closed fields of characteristic zero in [3]-[5] and
for semisimple algebraic groups over algebraically closed fields in [20]: see
also the references cited therein [5]. Certain class of solvable (not
necessarily algebraic) group is considered in [9].

Density of the power map has also been considered for connected Lie groups in
[1], [11], [13] and for real algebraic groups in [12].

We consider groups that have linear representation over non-Archimedean local
fields and our approach involves linear dynamics and tidy subgroups: see [8]
for linear dynamics and [21] for tidy subgroups.

Let $\mathbb{F}$ be a field, and let $G$ be a smooth affine
$\mathbb{F}$-group. Then the $\mathbb{F}$-unipotent radical
$R_{u,\mathbb{F}}(G)$ of $G$ is defined to be the largest smooth connected
normal unipotent $\mathbb{F}$-subgroup of $G$. We say that $G$ is pseudo-
reductive $\mathbb{F}$-group if $R_{u,\mathbb{F}}(G)=\\{e\\}$ where $e$ is the
identity element of $G$. An $\mathbb{F}$-split unipotent radical
$R_{us,\mathbb{F}}(G)$ is a maximal smooth connected $\mathbb{F}$-split
unipotent normal subgroup of $G$ and $G$ is said to be quasi-reductive if
$R_{us,\mathbb{F}}(G)=\\{e\\}$. Note that $R_{us,\mathbb{F}}(G)\subset
R_{u,\mathbb{F}}(G)$. We equip $G(\mathbb{F})$ with topology inherited from
$\mathbb{F}$ when $\mathbb{F}$ is a local field. If $\mathbb{F}$ is a non-
Archimedean local field, then $G(\mathbb{F})$ is totally disconnected locally
compact group.

It is well-known that if $G$ is a smooth connected affine $\mathbb{F}$-group
then $G/R_{u,\mathbb{F}}(G)$ is a pseudo-reductive $\mathbb{F}$-group and
$G/R_{us,F}(G)$ is a quasi-reductive $\mathbb{F}$\- group (see [7] Corollary
B.3.5).

We first prove surjectivity and density are equivalent for $G(\mathbb{F})$ and
characterize the surjectivity/density of the $k$-th power map $P_{k}$ for
$G(\mathbb{F})$ in terms of the $\mathbb{F}$-points of the quasi-reductive
quotient group $G/R_{us,\mathbb{F}}(G)$. We recall that the canonical map from
$G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$ into
$(G/R_{us,\mathbb{F}}(G))(\mathbb{F})$ is an isomorphism as
$H^{1}(\mathbb{F},R_{us,\mathbb{F}}(G))=0$.

######  Theorem 1.1.

Let $\mathbb{F}$ be a non-Archimedean local field. Let $G$ be an
$\mathbb{F}$-group, and $R_{us,\mathbb{F}}(G)$ be the $\mathbb{F}$-split
unipotent radical of $G$. Suppose that the characteristic of $\mathbb{F}$ does
not divide $k$. Then the following are equivalent:

  1. $(a)$

$P_{k}$ is dense in $G(\mathbb{F})$;

  2. $(b)$

$G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$ is compact and $P_{k}$ is
surjective on $G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$;

  3. $(c)$

$P_{k}$ is surjective in $G(\mathbb{F})$.

Suppose the residual characteristic of $\mathbb{F}$ divides $k$. Then density
of $P_{k}$ on $G(\mathbb{F})$ implies that $G(\mathbb{F})$ is a finite
extension of a split unipotent group. In addition if characteristic of
$\mathbb{F}$ is positive, then $G(\mathbb{F})$ is finite.

If the field $\mathbb{F}$ is perfect then for an smooth connected
$\mathbb{F}$-group $G$, $R_{us,\mathbb{F}}(G)=R_{u,\mathbb{F}}(G)=R_{u}(G)$,
the unipotent radical of $G$, and $G/R_{u}(G)$ is a reductive
$\mathbb{F}$-group. The following Corollary is an immediate consequence of
Theorem 1.1 generalizes Theorem 1.2 of [4] \- characteristic zero case: refer
Section 2.2 for details on order of compact groups.

######  Corollary 1.2.

Let $\mathbb{F}$ be non-Archimedean perfect field and $G$ be an algebraic
group over $\mathbb{F}$. Suppose that the characteristic of $\mathbb{F}$ does
not divide $k$. Then $P_{k}$ is dense in $G(\mathbb{F})$ if and only if
$P_{k}$ is dense in $(G/R_{u}(G))(\mathbb{F})$ if and only if
$(G/R_{u}(G))(\mathbb{F})$ is compact and $k$ is co-prime to the order of
$(G/R_{u}(G))(\mathbb{F})$.

Next we consider inheritance of surjectivity of $P_{k}$ for algebraic groups:
recall that not all (even closed) subgroups inherit surjectivity or density of
$P_{k}$, e.g., $\mathbb{Z}_{p}$ (resp. $\mathbb{Z}$) is a closed subgroup of
$\mathbb{Q}_{p}$ (resp. $\mathbb{R}$) and $P_{k}$ is surjective on
$\mathbb{Q}_{p}$ (resp. $\mathbb{R}$) for all $k\geq 1$ but $P_{k}$ is not
even dense in $\mathbb{Z}_{p}$ (resp. $\mathbb{Z}$) for any $k$ divisible by
$p$ (resp. for any $k>1$): compare with Corollary 1.3 of [4].

######  Theorem 1.3.

Let $\mathbb{F}$ be a non-Archimedean local field and $G$ be an algebraic
group defined over $\mathbb{F}$. Then we have the following:

  1. (1)

If $P_{k}$ is surjective on $G(\mathbb{F})$ and $H$ is an algebraic subgroup
of $G$ defined over $\mathbb{F}$, then $P_{k}$ is surjective on
$H(\mathbb{F})$.

  2. (2)

If $H$ is a closed (not necessarily algebraic) normal subgroup in
$G(\mathbb{F})$ and $P_{k}$ is dense in $H$ as well as in $G(\mathbb{F})/H$,
then $P_{k}$ is surjective on $G(\mathbb{F})$.

In particular, for any algebraic normal subgroup $H$ of $G$ defined over
$\mathbb{F}$, $P_{k}$ is surjective on $G(\mathbb{F})$ if and only if $P_{k}$
is surjective on both $H(\mathbb{F})$ and $G(\mathbb{F})/H(\mathbb{F})$.

We have the following corollary regarding infinitely divisible algebraic
groups over non-Archimedean local fields.

######  Corollary 1.4.

Let $G$ be an algebraic group over a non-Archimedean local field. Suppose
$P_{k}$ is surjective on $G(\mathbb{F})$ for all $k\in\mathbb{N}$. Then
$G(\mathbb{F})$ is unipotent. In addition if characteristic of $\mathbb{F}$ is
positive, then $G(\mathbb{F})=\\{e\\}$.

The above results are proved using canonical realization of algebraic groups
as subgroups of matrix groups. We note that apart from algebraic groups, there
are some other groups that have interesting and enough linear representations.
Therefore, we consider linear representation of groups and prove the results
for general groups in terms of their linear representations.

Recall that a linear representation of a group $G$ over a local field
$\mathbb{F}$ is a continuous homomorphism $\rho\colon G\to{\rm
GL}(n,\mathbb{F})$.

Apart from algebraic groups, Lie groups over local fields is an interesting
class that admit a linear (not necessarily injective) representation, namely
the adjoint representation: recall that ${\rm Ad}$ is the adjoint
representation of $G$ defined on the Lie algebra of $G$ (see [2] and [16] for
more information on Lie groups).

Various classes of $p$-adic Lie groups were introduced using the adjoint
representation and interesting results were obtained (ref. [15] and [14]).
Motivated by these studies we now introduce the following: let $\rho\colon
G\to{\rm GL}(n,\mathbb{F})$ be a linear representation of $G$.

We say that $G$ is called $\rho$-type $R$ if eigenvalues of $\rho(g)$ are of
absolute value $1$ for all $g\in G$ and $G$ is called $\rho$-unipotent (resp.,
$\rho$-compact) if ${\rm\rho}(G)$ is contained in an unipotent (resp.,
compact) subgroup of ${\rm GL}(n,\mathbb{F})$.

We now give the results for Lie groups.

######  Theorem 1.5.

Let $G$ be a Lie group over a non-Archimedean local field $\mathbb{F}$ and
$P_{k}$ is dense in $G$ for $k>1$. Then we have the following:

  1. (1)

$G$ is type $R$.

  2. (2)

${\rm Ad}(G)$ is contained in a compact extension of an unipotent normal
subgroup.

  3. (3)

If $G$ is compactly generated, then $G$ is Ad-compact.

  4. (4)

If the residue characteristic divides $k$ and the characteristic of of
$\mathbb{F}$ is zero, then $\overline{{\rm Ad}(G)}$ is a finite extension of
an unipotent group.

  5. (5)

If the characteristic $p>0$ divides $k$, ${\rm Ad}(G)$ is finite.

  6. (6)

If $P_{k}$ is dense in $G$ for all $k\geq 1$, then ${\rm Ad}~{}(G)$ is a
$\mathbb{F}$-split unipotent group, in particular, $G$ is ${\rm
Ad}$-unipotent. In addition if the characteristic of $\mathbb{F}$ is positive,
then Ad is trivial.

##  2 Preliminaries

###  2.1 Semi-direct product

Let $H$ and $N$ be locally compact groups. We say that $H$ acts on $N$ by
automorphisms if there is a homomorphism $\phi\colon H\to{\rm Aut}~{}(N)$ such
that the map $(g,x)\mapsto\phi(g)(x)$ is continuous. In this case we define
the semi-direct product $H\ltimes N$ of $H$ and $N$ as the product space
$H\times N$ with binary operation:

$(g,x)(h,y)=(gh,xg(y))$

for all $g,h\in G$ and $x,y\in X$.

Then $H\ltimes N$ is a locally compact group. Identifying $g\in H$ with
$(g,e)\in H\ltimes N$ and $x\in N$ with $(e,x)\in H\ltimes N$, we may view $H$
and $N$ as closed subgroups of $H\ltimes N$. This in particular implies that
$N$ is a normal subgroup of $H\ltimes N$. Semi-direct product is a useful
technique, particularly helps us to prove Lemma 3.1.

######  Example 2.1.

  1. 1.

Any closed subgroup of ${\rm GL}(n,\mathbb{F})$ acts on $\mathbb{F}^{n}$ by
linear transformations for a local field $\mathbb{F}$.

  2. 2.

Generally, any closed subgroup of ${\rm Aut}~{}(G)$ acts canonically on $G$ by
automorphisms.

  3. 3.

Given two closed subgroups $H$ and $N$ of a locally compact group $G$ such
that $N$ is normalized by $H$. Then taking $\phi(g)$ to be inner automorphism
restricted to $N$, defines an action of $H$ on $N$ by automorphisms. In this
case, if $HN=G$, then $G$ is called semidirect product of $H$ and $N$.

###  2.2 Profinite groups

A topological group $G$ is said to be profinite if $G$ is a inverse limit of
finite groups. It is easy to see that any profinite group is a totally
disconnected compact group and the converse is also true (see Corollary 1.2.4
of [24]).

A Steinitz number or supernatural number is a formal infinite product
$\Pi_{p}p^{n(p)}$, over all primes $p$, where $n(p)$ is a non-negative integer
or infinity. One may define the product and l.c.m of super-natural numbers in
the natural way: see 2.1 of [24].

For a finite set $X$, ${\rm Ord}(X)$ denotes the order of $X$ which is the
number of elements in $X$.

For a pro-finite group $G$, the order of $G$ (possibly a supernatural number)
denoted by ${\rm Ord}(G)$ is defined by

${\rm Ord}(G):={\rm l.c.m}\\{{\rm Ord}(G/U):{\rm
for\;any\;open\;subgroup\;}U\subset G\\}.$

Since $G$ has arbitrarily small compact open normal subgroups, we may replace
open subgroups in the above definition of order, by open normal subgroups.

We recall the following result from [17] and include a simple proof.

######  Proposition 2.2.

Let $G$ be a profinite group. Then $P_{k}:G\to G$ is surjective if and only if
$k$ is coprime to ${\rm Ord}(G)$, that is for any compact open normal subgroup
$U$ of $G$, $k$ is coprime to the order of the finite group $G/U$.

###### Proof.

It is sufficient to prove the if direction. Since $G$ is a profinite group,
$G$ has a basis $(U_{i})$ of compact open normal subgroups at $e$. If $k$ is
co-prime to the order of $G/U_{i}$ for all $i$, then $P_{k}$ is surjective
$G/U_{i}$ for all $i$, hence for $g\in G$ there exist $x_{i}\in G$ and
$u_{i}\in U_{i}$ such that $g=x_{i}^{k}u_{i}$. Since $(U_{i})$ is a basis at
$e$, $(gU_{i})$ is a basis at $g$, hence any neighborhood of $g$ contains
$x_{i}^{k}$ for some $i$. Thus, $P_{k}$ is dense in $G$. Since $G$ is compact,
$P_{k}$ is surjective on $G$. ∎

The following lemma is easy to see and known and will be used often.

######  Lemma 2.3.

Let $G$ be a profinite group and $H$ be a closed subgroup of $G$. If $P_{k}$
is surjective on $G$ then $P_{k}$ is surjective on $H$. Conversely, if $H$ is
normal in $G$ and $P_{k}$ is surjective on $H$ as well as on $G/H$, then
$P_{k}$ is surjective on $G$.

###### Proof.

As $G$ is profinite, $H$ is also profinite. Note that, Lagrange’s Theorem
holds for profinite group (see Proposition 2.1.2 of [24]), so ${\rm Ord}(H)$
divides ${\rm Ord}(G)$. Now, $P_{k}:G\to G$ is surjective if and only if
$(k,{\rm Ord}(G))=1$. This implies that $(k,{\rm Ord}(H))=1$, and hence
$P_{k}:H\to H$ is surjective.

Conversely, if $H$ is normal in $G$ and $P_{k}$ is surjective on both $H$ and
$G/H$, then by Proposition 2.2 we get that $k$ is co-prime to both ${\rm
Ord}(H)$ and ${\rm Ord}(G/H)$. By Lagrange’s Theorem on profinite groups, we
get that $k$ is co-prime to ${\rm Ord}(G)$ (ref. Proposition 2.1.2 of [24]).
Another application of Proposition 2.2 proves that $P_{k}$ is surjective on
$G$. ∎

Next lemma relates order of a profinite group and its open subgroups.

######  Lemma 2.4.

Let $K$ be a profinite group and $L$ be an open subgroup of $K$. Then ${\rm
Ord}(K)={\rm l.c.m}\\{|K/U|:U\,{\rm is\,open\,normal\,in\,}L\\}=[K:L]{\rm
Ord}(L).$

###### Proof.

Note that $L$ is a finite index subgroup in $K$. For any open normal subgroup
$U$ of $L$, $|K/U|=|L/U|[K:L]$ (see Proposition 2.1.2 of [24]). This proves
the second equality.

Let $V$ be any open normal subgroup of $K$. Then $U=V\cap L$ is an open normal
subgroup of $L$. Also, $K/V\simeq(K/U)/(V/U)$ and hence $|K/V||V/U|=|K/U|$,
that is $K/V$ divides $K/U$. This proves the first equality. ∎

###  2.3 Algebraic groups

Let $\mathbb{F}$ be a local field and $G$ be defined over $\mathbb{F}$. The
$\mathbb{F}$-unipotent (resp., $\mathbb{F}$-split unipotent) radical of $G$
denoted by $R_{u,\mathbb{F}}(G)$ (resp., $R_{us,\mathbb{F}}(G)$) is defined to
be the maximal connected unipotent normal subgroup of $G$ that is defined over
$\mathbb{F}$ (resp., $\mathbb{F}$-split). These subgroups are contained in the
usual unipotent radical of $G$. Every connected linear algebraic
$\mathbb{F}$-group $G$ has $R_{us,\mathbb{F}}(G)$. We say that $G$ is pseudo-
reductive over $\mathbb{F}$ if $R_{u,\mathbb{F}}(G)=1$ and $G$ is quasi-
reductive over $\mathbb{F}$ if $R_{us,\mathbb{F}}(G)=1$. Since
$R_{us,\mathbb{F}}(G)\subset R_{u,\mathbb{F}}(G)$, every pseudo-reductive
$\mathbb{F}$-group is also quasi-reductive. If $\mathbb{F}$ is perfect, then
any connected unipotent group defined over $\mathbb{F}$ and is
$\mathbb{F}$-split, hence quasi-reductive $\mathbb{F}$-group and pseudo-
reductive are equivalent and in fact reductive, that is unipotent radical is
trivial. When $\mathbb{F}$ is not perfect, there do exist quasi-reductive
$\mathbb{F}$-groups that are not pseudo-reductive (ref. [19]).

A connected linear algebraic group G over a field of characteristic $0$ admits
a Levi decomposition, that is, it can be written as the semidirect product of
its unipotent radical and a reductive subgroup known as Levi factor. In the
case the field $\mathbb{F}$ is of positive characteristic, Levi factors need
not exist, even if $\mathbb{F}$ is algebraically closed (cf. [7, A.6]) the
unipotent radical need not be defined over $\mathbb{F}$.

However, we can get following two short exact sequences.

$1\to R_{u,\mathbb{F}}(G)\to G\to G/R_{u,\mathbb{F}}(G)\to 1.$

and

$1\to R_{us,\mathbb{F}}(G)\to G\to G/R_{us,\mathbb{F}}(G)\to 1.$

Let $N=R_{us,\mathbb{F}}(G)$ and $M=R_{u,\mathbb{F}}(G)$. Let
$P=G/R_{u,\mathbb{F}}(G)$ and $Q=G/R_{us,\mathbb{F}}(G)$.

The map $G(\mathbb{F})\to(G/N)(\mathbb{F})$ is a surjective submersion of
$\mathbb{F}$-analytic manifolds. In particular, this map induces an
isomorphism between $G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$ and
$Q(\mathbb{F})$ as $H^{1}(\mathbb{F},R_{us,\mathbb{F}}(G))=0$.

###  2.4 $\mathbb{F}$-nilpotent groups

By an $\mathbb{F}$-nilpotent group, we mean a nilpotent group $N$ such that if
$N=N_{0}\supset N_{1}\supset\cdots\supset N_{m}=\\{e\\}$ is a central series
of $N$, then each $N_{j}/N_{j+1}$ is a finite-dimensional $\mathbb{F}$-vector
space. Let $N$ be an $\mathbb{F}$ -nilpotent group and $N_{j}$ be a central
series. Let $G$ be a group acting on $N$ as a group of automorphisms of $N$ .
The $G$-action on $N$ is said to be $\mathbb{F}$ -linear if the induced action
of $G$ on $N_{j}/N_{j+1}$ is $\mathbb{F}$ -linear for all $j$ .

Fact 1 (see [9]): Let $G$ be a group and $N$ a normal subgroup of $G$. Suppose
that $N$ is $\mathbb{F}$ -nilpotent with respect to a field $\mathbb{F}$, and
that the conjugation action of $G$ on $N$ is $\mathbb{F}$-linear. Let
$N=N_{0}\supset N_{1}\supset\cdots\supset N_{r}=\\{e\\}$ be the central series
of $N$. Let $A=G/N$, $x\in G$ and $a=xN\in A$. Let $k\in\mathbb{N}$ be co-
prime to the characteristic of $\mathbb{F}$. Let $B=\\{b\in A|b^{k}=a\\}$ and
let $B^{*}$ be the subset consisting of all $b$ in $B$ such that for any $j$,
any element of $N_{j}/N_{j+1}$ which is fixed under the action of $a$ is also
fixed under the action of $b$. Then for any $b\in B^{*}$, $u\in N$, there
exists $y\in G$ such that $yN=b$ and $y^{k}=xu$. But when $G/N$ is a profinite
group, we have the following:

######  Lemma 2.5.

Let $\mathbb{F}$ be a non-archimedean local field and $G$ be a locally compact
group containing a closed normal subgroup $N$ such that $N$ is
$\mathbb{F}$-nilpotent and $G/N$ is a profinite group. Assume that the
conjugation action of $G$ on $N$ is $\mathbb{F}$-linear. Suppose $k$ is co-
prime to the characteristic of $\mathbb{F}$ and $P_{k}$ is surjective on
$G/N$. Then $P_{k}$ is surjective on $G$.

###### Proof.

Let $x\in G$ and $a=xN$. Since $P_{k}$ is surjective on the compact group
$G/N$, Lemma 2.3 implies that $P_{k}$ is surjective on $\overline{\left\langle
a\right\rangle}$. Let $N=N_{0}\supset N_{1}\supset\cdots\supset N_{r}={e}$ be
the central series of $N$. We observe that, for each $j$, any element of
$N_{j}/N_{j+1}$ fixed by $a$ is also fixed by the group
$\overline{\left\langle a\right\rangle}$. In view of Fact 1, we conclude that
$x\in P_{k}(G)$, that is $P_{k}$ is surjective on $G$. ∎

###  2.5 Scale function

Let $G$ be a locally compact totally disconnected group. Then $G$ has
arbitrarily small compact open subgroups $U$ of $G$. Let ${\rm Cos}(G)$ be the
set of all compact open subgroup of $G$.

Let ${\rm Aut}(G)$ be the collection of all (continuous) automorphisms of $G$.
Then the scale function $s:{\rm Aut}(G)\to\mathbb{N}$ is defined as follows:

$s(\alpha):={\rm min}\\{[\alpha(U):U\cap\alpha(U)]|U\in{\rm Cos}(G)\\}$

for any $\alpha\in{\rm Aut}(G)$ and the compact open subgroup for which the
minimum is attained is called tidy subgroup of $\alpha$ (see [22]). The scale
function was introduced by G. Willis [21] and it has proved to be useful. A
property of scale function that we often uses is the following:
$s(\alpha)=1=s(\alpha^{-1})$ if and only if $G$ contains a $\alpha$-invariant
compact open subgroup (ref. Proposition 4.3 of [21]).

For each $x\in G$, let $\alpha_{x}\colon G\to G$ be the inner-automorphism
defined by $x$, that is $\alpha_{x}(y)=xyx^{-1}$ for all $y\in G$. Now define
$s(x)=s(\alpha_{x})$ and the tidy subgroup of $\alpha_{x}$ is defined to be
the tidy subgroup of $x$.

It is known that $s(\alpha^{n})=s(\alpha)^{n}$: see [22].

##  3 Representations and Lie groups

We now prove a dynamic consequence of density of the power map using
semidirect product technique and scale function: recall that $P_{k}(x)=x^{k}$
is the $k$-th power map for $k\geq 1$.

######  Lemma 3.1.

Let $G$ be a locally compact totally disconnected group and $K$ be a compact
normal subgroup of $G$. Suppose $G$ acts on a totally disconnected locally
compact group $X$ by automorphisms. Then we have the following:

  1. 1.

If $g\in\overline{P_{k}(G)K}$ for infinitely many $k$, then the $g$ action on
$X$ fixes a compact open subgroup $K_{g}$ of $X$.

  2. 2.

If $P_{k}$ is dense in $G/K$ for some $k>1$, then the $g$ action on $X$ fixes
a compact open subgroup $K_{g}$ of $X$.

###### Proof.

Let $Y=G\ltimes X$ be the semidirect product of $G$ and $X$ for the given
action of $G$ on $X$. Then $Y$ is a totally disconnected locally compact group
containing $G$ as a closed subgroup and $X$ as a closed normal subgroup. Let
$s$ be the scale function on $Y$. Then $s$ is continuous on $Y$ (ref.
Corollary 4 of [21]).

Let $V$ be a compact open subgroup of $G$ containing $K$ and $W$ be a compact
open subgroup of $X$ fixed by $V$. Then $V\times W$ is a compact open subgroup
of $Y$ invariant under conjugation by elements of $K$.

Let $x\in G$. Then by Lemma 4.2 of [18], the group generated by $x$ and $K$
has a common tidy subgroup in $Y$. This implies by Corollary 2.7 of [18] that
$s(ab)\leq s(a)s(b)$ for all $a$ and $b$ in the group generated by $x$ and $K$
(see also Proposition 7.2 of [10]). Since $s(h)=1$ for all $h\in K$, we have
$s(xh)\leq s(x)\leq s(xh)s(h^{-1})=s(xh).$ Therefore, $s(xh)=s(x)$ for all
$h\in K$.

For $g\in G$, let $V_{g}=\\{x\in G\mid s(x)=s(g)\\}$. Then since $s$ is
continuous on $Y$, $V_{g}$ is an open neighborhood of $g$ in $G$ such that
$V_{g}K=V_{g}$.

Suppose $g\in\overline{P_{k}(G)K}$ for infinitely many $k$. Then there are
infinitely many $k$ such that $x_{k}^{k}\in V_{g}$ for some $x_{k}\in G$. This
implies that $s(g)=s(x_{k})^{k}$ for infinitely many $k$. Thus,
$s(g)\in\mathbb{N}$ has infinitely many roots, hence $s(g)=1$. Similarly
$s(g^{-1})=1$. Now the first part follows from Proposition 4.3 of [21].

Suppose $P_{k}$ is dense in $G/K$ for some $k>1$. Let $g\in G$. Then there is
a $x_{1}\in G$ such that $x_{1}^{k}\in V_{g}$, hence $s(x_{1})^{k}=s(g)$. Now
by considering $V_{x_{1}}$, there is a $x_{2}\in G$ such that $x_{2}^{k}\in
V_{x_{1}}$. This implies that $s(x_{2})^{k^{2}}=s(x_{1})^{k}=s(g)$.
Inductively, we get a sequence $(x_{n})$ in $G$ such that
$s(x_{n})^{k^{n}}=s(g)$. Thus, $s(g)\in\mathbb{N}$ has infinitely many roots,
hence $s(g)=1$. Similarly $s(g^{-1})=1$. Now the second part follows from
Proposition 4.3 of [21]. ∎

In case, the dynamics in the above Lemma 3.1 is linear, then we can proceed
further.

######  Lemma 3.2.

Let $G$ be a locally compact totally disconnected group and $V$ be a finite-
dimensional vector space over a non-Archimedean field $\mathbb{F}$. Suppose
$\rho\colon G\to{\rm GL}(V)$ is the map defining an action of $G$ on $V$ and
$s(x)=1$ for all $x\in G$ where $s$ is the scale function on $V$. Then $G$ is
$\rho$-type $R$ and there exists a compact group $K\subset{\rm GL}(V)$ and a
$\mathbb{F}$-split unipotent algebraic group $U\subset{\rm GL}(V)$ normalized
by $K$ such that $K\cap U$ is trivial, $\rho(G)\subset KU$ and $\rho(G)U$ is
dense in $KU$.

###### Proof.

Let $L=\rho(G)$. Then $s(g)=1$ for all $g\in L$, hence each $g\in L$ fixes a
compact open subgroup of $V$. Therefore, it follows that all eigenvalues of
$g\in L\subset{\rm GL}(V)$ are absolute value $1$. Thus $G$ is $\rho$-type
$R$.

Now it follows from Theorem 1 of [8] that there exists a flag
$\\{0\\}=V_{0}\subset V_{1}\subset\dots\subset V_{m-1}\subset V_{m}=V$ such
that $L$ on $V_{i}/V_{i-1}$ has only bounded orbits for any $i\geq 1$. Let
$U=\\{\alpha\in{\rm GL}(V)\mid\alpha(v+V_{i-1})=v+V_{i-1}~{}~{}{\rm
for~{}~{}all}~{}~{}v\in V_{i}~{}~{}{\rm and}~{}~{}i\geq 1\\}$ and $K$ be the
direct product of closure of the image of $L$ in ${\rm GL}(V_{i}/V_{i-1})$.
Then $U$ is a split unipotent algebraic group and $K$ is a compact group that
normalizes $U$ such that $K\cap U$ is trivial and $L\subset KU$ \- note that
$KU$ is the semidirect product $K\ltimes U$ of $K$. This implies that
$KU/U\simeq K$, hence replacing $K\simeq KU/U$ by the closure of $LU/U\subset
KU/U$, we may assume that $LU$ is dense in $KU$. ∎

We next obtain results in terms of the representation of the groups.

######  Proposition 3.3.

Let $\mathbb{F}$ be a non-Archimedean local field and $H$ be a group with a
linear representation $\rho\colon H\to{\rm GL}(d,\mathbb{F})$. Suppose that
$P_{k}$ is dense in $H$ for some $k>1$. Then we have the following:

  1. (1)

$H$ is $\rho$-type $R$.

  2. (2)

There exists a compact group $K\subset{\rm GL}(d,\mathbb{F})$ and a split
unipotent algebraic group $U\subset{\rm GL}(d,\mathbb{F})$ normalized by $K$
such that $K\cap U$ is trivial, $\rho(H)\subset KU$ and $\rho(H)U$ is dense in
$KU$. Moreover, $P_{k}$ is surjective on $KU/U\simeq K$.

  3. (3)

If $k$ is co-prime to the characteristic of $\mathbb{F}$, then $P_{k}$ is
surjective on $KU$.

  4. (4)

If the residual characteristic $p$ of $\mathbb{F}$ divides $k$, then $K$ is
finite, that is $\rho(H)$ is contained in a finite extension of a split
unipotent algebraic group $U$ and $P_{k}$ is dense in $\rho(H)\cap U$.

  5. (5)

If the residual characteristic $p$ of $\mathbb{F}$ divides $k$ and the
characteristic of $\mathbb{F}$ is zero, then $\overline{\rho(H)}$ is a finite
extension of a split unipotent algebraic group.

  6. (6)

If the characteristic $p$ of $\mathbb{F}$ divides $k$, then $\rho(H)$ is
finite.

  7. (7)

If $G$ is an $\mathbb{F}$-group and $\rho(H)=G(\mathbb{F})$, then $G$ has no
$\mathbb{F}$-split torus.

######  Remark 3.4.

The above results (5) and (6) generalize Corollary 1.7 of [5] to any linear
group (not necessarily algebraic) over any non-Archimedean local field.

Proof of Proposition 3.3: Let $L=\rho(H)$. Then $P_{k}$ is dense in $L$. By
Lemma 3.1, each $g\in L$ fixes a compact open subgroup of $\mathbb{F}^{n}$.
Therefore, it follows that all eigenvalues of $g\in L\subset{\rm
GL}(d,\mathbb{F})$ are absolute value $1$. This proves (1).

It follows from Lemma 3.1 that the scale function is trivial on $L$. Therefore
by Lemma 3.2, there exists a compact group $K\subset{\rm GL}(d,\mathbb{F})$
and a split unipotent algebraic group $U\subset{\rm GL}(d,\mathbb{F})$
normalized by $K$ such that $K\cap U$ is trivial, $\rho(H)\subset KU$ and
$\rho(H)U$ is dense in $KU$. Since $P_{k}$ is dense in $L$, $P_{k}$ is dense
in $LU/U$. Since $LU$ is dense in $KU$ and $KU/U$ is compact, we get that
$P_{k}$ is surjective on $KU/U\simeq K$. This proves (2).

Suppose $k$ is co-prime to the characteristic of $\mathbb{F}$. Then since
$P_{k}$ is surjective on the compact group, $KU/U$, Lemma 2.5 implies that
$P_{k}$ is surjective on $KU$. This proves (3).

Suppose the residual characteristic $p$ of $\mathbb{F}$ divides $k$. Since $K$
is a compact linear group over $\mathbb{F}$, we get that $K$ contains an open
subgroup $K_{0}$ such that $K_{0}$ is pro-p group. By Lemma 2.3, $P_{k}$ is
surjective on $K_{0}$. Since $p$ divides $k$, $K_{0}$ is trivial. This implies
that $K$ is a finite group and hence $U$ is an open subgroup of $KU$.
Therefore, $L\cap U$ is open in $L$. Since $P_{k}$ is dense in $L$,
$P_{k}(L)\cap U$ is dense in $L\cap U$. Let $g\in P_{k}(L)\cap U$. Then there
exist $x\in L$ such that $x^{k}=g\in L\cap U$. Since $L\subset KU$, there
exist $a\in K$ and $v\in U$ such that $x=av$, hence
$x^{k}=(av)^{k}=a^{k}v_{k}$ where $v_{k}=\prod_{j=1}^{k}a^{-k+j}va^{k-j}\in
U$. This implies that $a^{k}\in K\cap U$. Since $K\cap U$ is trivial,
$a^{k}=e$, hence $k$ divides the order of $K$ or $a=e$. Since $P_{k}$ is
surjective on $K$, by Proposition 2.2, we get that $a=e$. Thus, $x=v\in L\cap
U$. Therefore, $g\in P_{k}(L\cap U)$. Thus, $P_{k}$ is dense in $L\cap U$.
This proves (4).

Suppose the characteristic of $\mathbb{F}$ is zero and the residual
characteristic $p$ of $\mathbb{F}$ divides $k$. Since $P_{k}$ is dense in $L$,
$P_{k}$ is dense in $\overline{L}$. Therefore, replacing $L$ by
$\overline{L}$, we may assume that $L$ is closed. Since $L/L\cap U\simeq LU/U$
is finite, it is sufficient to claim that $L\cap U$ is a unipotent group. By
(4), $P_{k}$ is dense in $L\cap U$, we may assume that $L$ is a closed
subgroup of $U$ and $U$ is the smallest unipotent group containing $L$. If $U$
is abelian, then $U$ is the vector space spanned by $L$. Let $V$ the maximal
vector space contained in $L$. Then $L/V$ is compact. Since $P_{k}$ is dense
in $L$, $P_{k}$ is surjective on $L/V$ but $p$ divides $k$, hence $L/V$ is
finite. Since $L/V$ is a subgroup of the unipotent group $U/V$ which has no
elements of finite order, we get that $L=V$, hence $L=U$. If $U$ is a general
unipotent group, let $Z$ be the center of $U$. Then since $P_{k}$ is dense in
$L$, $P_{k}$ dense in $LZ/Z\subset U/Z$, hence by induction $\overline{LZ}=U$.
This proves that $[U,U]\subset L$. Using the commutative case for $U/[U,U]$,
we may conclude that $L=U$. This proves (5).

Suppose characteristic of $\mathbb{F}$ divides $k$. Since $U$ is a split
unipotent group and $p$ divides $k$, we get that $P_{k}(U)=\\{e\\}$. Since
$P_{k}$ is dense in $L\cap U$, $L\cap U$ is trivial. Since $L$ is contained in
a finite extension of $U$, $L$ is finite. This proves (6).

Suppose $H$ is the group of $\mathbb{F}$-points of an algebraic group $G$
defined over $\mathbb{F}$. Then the set of eigenvalues of elements of any non-
trivial split torus in $H$ is $\mathbb{F}^{*}$. Thus, (1) implies that $G$ has
no $\mathbb{F}$-split tours. ∎

It is known that $P_{k}$ is not dense in any finitely generated infinite
abelian groups: any such group is isomorphic to $F\times\mathbb{Z}^{d}$ for
$d\geq 1$ for some finite group $F$, hence have $\mathbb{Z}$ as a quotient. We
extend this to compactly generated groups and its linear representations over
non-Archimedean local fields.

######  Corollary 3.5.

Let $\mathbb{F}$ be a non-Archimedean local field and $H$ be a group with a
linear representation $\rho\colon H\to{\rm GL}(d,\mathbb{F})$. If $P_{k}$ is
dense in $H$ for some $k>1$ and $H$ is compactly generated then $H$ is
$\rho$-compact.

###### Proof.

Let $L=\rho(H)$. Then $P_{k}$ is dense in $L$. By (2) of Proposition 3.3,
there is a unipotent group $U$ and a compact linear group $K$ normalizing $U$
such that $L\subset KU$. Let $C$ be a compact generating subset of $L$. Then
$C\subset KM$ where $M$ is a compact subgroup of $U$. Since $K$ normalizes
$U$, any compact subset of $U$ is a contained in a compact $K$-invariant
subset of $U$. Thus, we may assume that $M$ is a $K$-invariant compact
subgroup. This implies that $KM$ is a compact subgroup. Since $C\subset KM$
and $L$ is generated by $C$, we get that $\overline{L}$ is compact. ∎

The next result shows that unipotent groups are the only infinite divisible
linear groups.

######  Corollary 3.6.

Let $\mathbb{F}$ be a non-Archimedean local field and $H$ be a group with a
linear representation $\rho\colon H\to{\rm GL}(d,\mathbb{F})$. Suppose $P_{k}$
is dense in $H$ for all $k$. Then $\overline{\rho(H)}$ is a split unipotent
algebraic group. In addition if $\mathbb{F}$ has positive characteristic,
$\rho$ is trivial.

###### Proof.

If the characteristic of $\mathbb{F}$ is zero, then by (5) of Proposition 3.3,
$\overline{\rho(H)}$ is a finite extension of an unipotent algebraic group
$U$. This implies that $\overline{\rho(H)}/U=(\rho(H)U)/U$. Let
$k=|\overline{\rho(H)}/U|$. Then since $P_{k}$ is dense in $H$, $P_{k}$ is
surjective on $\rho(H)U/U=\overline{\rho(H)}/U$ which has order $k$, hence
$\overline{\rho(H)}/U$ is trivial. Thus, $\overline{\rho(H)}=U$.

If $\mathbb{F}$ has positive characteristic, then by (6) of Proposition 3.3,
$\rho(H)$ is a finite group, hence $\rho(H)=\overline{\rho(H)}$. This implies
that $U$ is finite, hence trivial. ∎

For subgroups of linear groups over global fields: compare with Section 6 of
[4].

######  Corollary 3.7.

Let $\mathbb{E}$ be a global field and $H$ be a subgroup of ${\rm
GL}(d,\mathbb{E})$. Assume that $P_{k}$ is surjective on $H$ for some $k>1$.

  1. (1)

If the characteristic of $\mathbb{E}$ is $0$, then $H$ contains an unipotent
normal subgroup of finite index.

  2. (2)

If the characteristic of $\mathbb{E}$ is $p>0$, then $H$ is locally finite,
that is any finitely generated subgroup of $H$ is finite.

  3. (3)

If the characteristic $p$ of $\mathbb{E}$ divides $k$, then $H$ is finite.

  4. (4)

If $P_{k}$ is surjective on $H$ for all $k\geq 1$, then either $H$ is a
unipotent group or $H$ is trivial depending on characteristic of $\mathbb{E}$
is $0$ or positive.

###### Proof.

Suppose the characteristic of $\mathbb{E}$ is $0$. Let $E_{p}$ be the $p$-adic
completion for $p$ dividing $k$. Then by (5) of Proposition 3.3, $H$ is
contains an unipotent normal subgroup of finite index.

Suppose the characteristic of $\mathbb{E}$ is $p>0$. Then any completion
$E_{v}$ of $\mathbb{E}$ is non-Archimedean of characteristic $p>0$. By $(2)$
of Proposition 3.3, $H$ is contained in a compact extension of a split
unipotent group in ${\rm GL}(d,\mathbb{E}_{v})$. If $N$ is a finitely
generated subgroup of $H$, then $N$ is contained in a compact subgroup of
${\rm GL}(d,\mathbb{E}_{v})$. This implies that $N$ is finite.

Suppose the characteristic $p$ of $\mathbb{E}$ divides $k$. Then by $(6)$ of
Proposition 3.3, in any completion of $\mathbb{E}$, $H$ is finite. Thus, $H$
is finite.

Last part follows from 1, 3 and Proposition 2.2. ∎

Proof of Theorem 1.5: Let $d$ be the dimension of $G$. Then ${\rm Ad}(G)$ is a
subgroup of ${\rm GL}(d,\mathbb{F})$. Thus, Proposition 3.3 and its
Corollaries 3.5, 3.6.

##  4 Proof of Theorems 1.1, 1.3 and Corollary 1.4

We first deal the following:

######  Lemma 4.1.

Let $\mathbb{F}$ be a non-Archimedean local field and $H$ be a closed subgroup
of ${\rm GL}(d,\mathbb{F})$. Suppose the smallest algebraic group over
$\mathbb{F}$ containing $H$ has no $\mathbb{F}$-split unipotent normal
subgroup. Then $P_{k}$ is dense in $H$ if and only if $H$ is compact and
$(k,{\rm Ord}(H))=1$ (if and only if $P_{k}$ is surjective on $H$).

###### Proof.

Let $s$ be the scale function on $\mathbb{F}^{d}$. Applying Lemma 3.1 to the
canonical action of $H$ on $\mathbb{F}^{d}$, we get that $s(x)=1$ for all
$x\in H$. Now by Lemma 3.2, there exists a compact group $K\subset{\rm
GL}(d,\mathbb{F})$ and a split unipotent algebraic group $U\subset{\rm
GL}(d,\mathbb{F})$ normalized by $K$ such that $K\cap U$ is trivial, $H\subset
KU$ and $HU$ is dense in $KU$. Let $G$ be the normalizer of $U$. Then $G$ is
defined over $\mathbb{F}$ and contains $H$. Thus, the smallest algebraic
group, say $G_{1}$, defined over $\mathbb{F}$ containing $H$ normalizes $U$.
Since $G_{1}$ and $U$ are algebraic groups, $G_{1}(\mathbb{F})U$ is closed.
This implies that $KU\subset G_{1}(\mathbb{F})U$. Since $G_{1}$ has no
$\mathbb{F}$-split unipotent normal subgroup, $G_{1}(\mathbb{F})\cap U$ is
trivial, hence the map $f\colon G_{1}(\mathbb{F})\to G_{1}(\mathbb{F})U/U$
given by $f(g)=gU$ is an isomorphism of topological groups. In particular,
$f(H)$ is a closed subgroup. Since $HU\subset KU$, $f(H)$ is a closed subgroup
of $KU/U$. Therefore $f(H)$ is compact. Since $f$ is an isomorphisms, $H$ is
compact. ∎

Proof of Theorem 1.1: Let $N=R_{us,\mathbb{F}}(G)$ and
$Q=G/R_{us,\mathbb{F}}(G)$.

$(a)\Rightarrow(b):$ Recall that
$G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$ is isomorphic to
$Q(\mathbb{F})$ as $H^{1}(\mathbb{F},R_{us,\mathbb{F}}(G))=0$. Since $P_{k}$
is dense in $G(\mathbb{F})$, $P_{k}$ is dense in
$G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$ and hence by Lemma 4.1, we
get that $G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$ is compact.
Compactness of $G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$ implies that
$P_{k}$ is surjective on $G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$.

We next observe that $(b)\Rightarrow(c)$ follows from Lemma 2.5 and that
$(c)\Rightarrow(a)$ is trivial.

∎

Proof of Theorem 1.3: Suppose that $P_{k}:G(\mathbb{F})\to G(\mathbb{F})$ is
surjective. Let $N=R_{us,\mathbb{F}}(G)$, and $N_{H}=R_{us,\mathbb{F}}(H)$.

Assume that the characteristic of $\mathbb{F}$ does not divide $k$. By Theorem
1.1, we have $G(\mathbb{F})/N(\mathbb{F})$ is compact and $P_{k}$ is
surjective on $G(\mathbb{F})/N(\mathbb{F})$. Let $\phi\colon G(\mathbb{F})\to
G(\mathbb{F})/N(\mathbb{F})$ be the canonical quotient. Then
$\phi(H(\mathbb{F}))$ is a closed subgroup of $G(\mathbb{F})/N(\mathbb{F})$.
By Lemma 2.3, we have $P_{k}$ is surjective on $\phi(H(\mathbb{F}))$. Since
${\rm Ker\phi}\cap H(\mathbb{F})=N(\mathbb{F})\cap H(\mathbb{F})\subset
N_{H}(\mathbb{F})$, we get that $H(\mathbb{F})/N_{H}(\mathbb{F})$ is a
quotient of $\phi(H(\mathbb{F}))$, hence $H(\mathbb{F})/N_{H}(\mathbb{F})$ is
compact and $P_{k}$ is surjective on $H(\mathbb{F})/N_{H}(\mathbb{F})$. Now
the result follows from Theorem 1.1.

Suppose the characteristic of $\mathbb{F}$ divides $k$. Then $G(\mathbb{F})$
is finite, hence the result follows from Lemma 2.3.

Now let $H$ be a closed normal subgroup of $G(\mathbb{F})$ and $P_{k}$ is
dense in $H$ as well as in $G(\mathbb{F})/H$.

If characteristic of $\mathbb{F}$ divides $k$, then by (6) of Proposition 3.3,
$H$ is finite and $({\rm Ord}(H),k)=1$. Since $H$ is finite, $G(\mathbb{F})/H$
is also linear group, hence (6) of Proposition 3.3 implies that
$G(\mathbb{F})/H$ is also finite and $k$ is co-prime to the order of
$G(\mathbb{F})/H$. Thus, $G(\mathbb{F})$ is finite and
$|G(\mathbb{F})|=|H||G(\mathbb{F})/H|$, hence $k$ is co-prime to the order of
$G(\mathbb{F})$. Thus, $P_{k}$ is surjective on $G(\mathbb{F})$.

We may now assume that the characteristic of $\mathbb{F}$ does not divide $k$.

Let $Q=G/R_{us,\mathbb{F}}(G)$ and
$M=G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$. Then $M$ is isomorphic to
$Q(\mathbb{F})$. So we may assume that $M=Q(\mathbb{F})$. Let $M_{1}$ be the
closure of
$HR_{us,\mathbb{F}}(G)(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$. Then
$M_{1}$ is a closed normal subgroup of $M$. Since $P_{k}$ is dense in $H$,
$P_{k}$ is dense in $M_{1}$. Since $Q$ has no $\mathbb{F}$-split unipotent
normal subgroup and $M_{1}$ is a closed subgroup of $M=Q(\mathbb{F})$, Lemma
4.1 implies that $M_{1}$ is compact.

Let $d\geq 1$ be such that $M(=Q(\mathbb{F}))$ is a closed subgroup of
$GL(d,\mathbb{F})$ and $s$ be the scale function on $\mathbb{F}^{d}$. Since
$P_{k}$ is dense in $G(\mathbb{F})/H$, $P_{k}$ is dense on $M/M_{1}$.
Therefore by Lemma 3.1, $s$ is trivial on $M$. By Lemma 3.2 implies that there
is a compact subgroup $K\subset{\rm GL}(d,\mathbb{F})$ and a split unipotent
algebraic group $U\subset{\rm GL}(d,\mathbb{F})$ normalized by $K$ such that
$K\cap U$ is trivial, $M\subset KU$ and $MU$ is dense in $KU$. Since $Q$ is an
algebraic group and $M=Q(\mathbb{F})$, $MU$ is closed, hence $MU=KU$ and since
$Q$ has no $\mathbb{F}$-split unipotent normal subgroup, $M\cap U$ is trivial.
Therefore, $M\simeq MU/U=KU/U\simeq K$. Thus, $M$ is compact. Therefore,
$P_{k}$ is surjective on the compact groups $M_{1}$ and $M/M_{1}$, hence by
Lemma 2.3 we get that $P_{k}$ is surjective on the compact group
$M=G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$. Now applying Lemma 2.5, we
get that $P_{k}$ is surjective on $G(\mathbb{F})$.

∎

Proof of Corollary 1.4 Since $G(\mathbb{F})$ is a closed subgroup of ${\rm
GL}(d,\mathbb{F})$ for some $d\geq 1$, Corollary 3.6 implies that
$G(\mathbb{F})$ is split unipotent.

If $p>0$ is the characteristic of $\mathbb{F}$, then $P_{p}$ is surjective on
$R_{us,\mathbb{F}}(G)$ implies that $G(\mathbb{F})=R_{us,\mathbb{F}}(G)$ is
trivial. ∎

##  5 Lattices

We now consider the situation when the group $G$ has a finite co-volume
subgroup or a co-compact subgroup on which $P_{k}$ is dense.

######  Proposition 5.1.

Let $G$ be a totally disconnected locally compact group acting on a totally
disconnected locally compact group $X$ by automorphisms. Suppose $G$ has a
finite co-volume or cocompact subgroup $H$ and $P_{k}$ is dense in $H$. Then
every element of $G$ fixes a compact open subgroup of $X$.

###### Proof.

Since $P_{k}$ is dense in $H$, by Lemma 3.1 applied to the conjugate action of
$H$ of $G$, we get that every element of $H$ normalizes a compact open
subgroup of $G$. This implies that the modular function of $G$ is trivial on
$H$ and $H$ is unimodular. If $G/H$ is compact, we get that $G$ is unimodular.
Thus, $G/H$ has an invariant measure. Since $G/H$ is compact, $G/H$ has finite
volume. Thus, we may assume that $H$ has finite co-volume.

Since every element of $H$ normalizes a compact open subgroup of $G$, by
Theorem 2.5 of [23], every element of $G$ also normalizes a compact open
subgroup of $G$.

Let $g\in G$ and $V$ be a compact open subgroup of $G$ normalized by $g$. Then
there exists a $n\geq 1$ such that $g^{n}V\cap H\not=\emptyset$. Let $h\in H$
be such that $h\in g^{n}V$.

Let $s$ be the scale function on $X$. Then by Lemma 4.2 of [18], the group
generated by $g$ and $V$ has a common tidy subgroup in $X$. This implies by
Corollary 2.7 of [18] that $s(ab)\leq s(a)s(b)$ for all $a$ and $b$ in the
group generated by $g$ and $V$ (see also Proposition 7.2 of [10]). Since
$s(a)=1$ for all $a\in V$, we have

$s(g^{n}a)\leq s(g^{n})\leq s(g^{n}a)s(a^{-1})=s(g^{n}a).$

Therefore, $s(g^{n}a)=s(g^{n})$ for all $a\in V$. Since $h\in g^{n}V$,
$s(h)=s(g^{n})$. It now follows from Lemma 3.1 that
$1=s(h)=s(g^{n})=s(g)^{n}$, hence $s(g)=1$.

∎

It can easily be seen that we can’t expect $P_{k}$ to be dense in $G$ if
$P_{k}$ is dense in a co-compact or a finite co-volume subgroup $H$. For
example, take $H$ to be the trivial subgroup of a compact group which forces
us to ask is this the only obstruction. In case $G$ is a group of
$\mathbb{F}$-points of an algebraic group defined over $\mathbb{F}$, we have
the following affirmative answer.

######  Proposition 5.2.

Let $G$ be an algebraic group defined over a non-Archimedean local field
$\mathbb{F}$ and $H$ be a closed subgroup of $G(\mathbb{F})$ with finite co-
volume or co-compact. Suppose $P_{k}$ is dense in $H$. Then we have the
following:

  1. (1)

$G(\mathbb{F})$ is a compact extension of $R_{us,\mathbb{F}}(G)(\mathbb{F})$.

  2. (2)

If the residual characteristic of $\mathbb{F}$ does not divide $k$, then
$G(\mathbb{F})$ contains an open subgroup $G_{0}$ of finite index such that
$P_{k}$ is surjective on $G_{0}$ and $G_{0}$ contains $H$.

  3. (3)

If the residual characteristic of $\mathbb{F}$ divides $k$, then the
characteristic of $\mathbb{F}$ is zero implies that $H$ is a finite extension
of $R_{us,\mathbb{F}}(G)(\mathbb{F})$ and the characteristic of $\mathbb{F}$
is positive implies that $G(\mathbb{F})$ is compact.

###### Proof.

Let $M=G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$ and
$Q=G/R_{us,\mathbb{F}}(G)$. Then $M\simeq Q(\mathbb{F})$, hence we may assume
that $M=Q(\mathbb{F})$. Then $M$ is a closed subgroup of ${\rm
GL}(d,\mathbb{F})$ for some $d\geq 1$. Let $s$ be the scale function on
$\mathbb{F}^{d}$. Then by Proposition 5.1 $s$ is trivial on $M$. Now applying
Lemma 3.2 to $M$, we get that there is a compact group $K\subset{\rm
GL}(d,\mathbb{F})$ and a split unipotent group $U\subset{\rm
GL}(d,\mathbb{F})$ normalized by $K$ such that $M\subset KU$ and $MU$ is dense
in $KU$. Since both $Q$ and $U$ are algebraic, $MU$ is closed, hence $MU=KU$.

Let $f\colon KU\to KU/U$ be the canonical quotient map. Since $Q$ has no
$\mathbb{F}$-split unipotent normal subgroup, $M\cap U$ is trivial. Therefore
$f$ restricted to $M$ is an isomorphism onto $KU/U\simeq K$. Thus, $M$ is
compact.

Suppose the residual characteristic $p$ of $\mathbb{F}$ does not divide $k$.
Let $M_{0}$ be an open normal pro-$p$ subgroup of $M$. Then by Proposition
2.2, we get that $P_{k}$ is surjective on $M_{0}$. Let $M_{1}$ be the closure
of
$(HR_{us,\mathbb{F}}(G)(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F}))M_{0}$.
Then $M_{1}$ is an open subgroup of finite index in $M$ and by Lemma 2.3,
$P_{k}$ is surjective on $M_{1}$. Let $G_{0}$ be the subgroup of
$G(\mathbb{F})$ containing $R_{us,\mathbb{F}}(G)(\mathbb{F})$ such that
$G_{0}/R_{us,\mathbb{F}}(G)(\mathbb{F})=M_{1}$. Then $G_{0}$ is an open
subgroup of finite index in $G(\mathbb{F})$. Since the residual characteristic
$p$ of $\mathbb{F}$ does not divide $k$ and
$G_{0}/R_{us,\mathbb{F}}(G)(\mathbb{F})$ is compact on which $P_{k}$ is
surjective, Lemma 2.5 implies that $P_{k}$ is surjective on $G_{0}$.

If the residual characteristic $p$ of $\mathbb{F}$ divides $k$ and the
characteristic of $\mathbb{F}$ is zero, then by (5) of Proposition 3.3 we get
that $H$ is a finite extension of a split unipotent algebraic group $V$. Since
$H$ is a finite co-volume or co-compact subgroup of $G(\mathbb{F})$, $V$ is
also finite co-volume or co-compact subgroup of $G(\mathbb{F})$. Since
$G(\mathbb{F})/R_{us,\mathbb{F}}(G)(\mathbb{F})$ is compact, $V\subset
R_{us,\mathbb{F}}(G)(\mathbb{F})$. In particular, $V$ is a finite co-volume
subgroup of $R_{us,\mathbb{F}}(G)(\mathbb{F})$. Now, it can be easily shown
that $V=R_{us,\mathbb{F}}(G)(\mathbb{F})$. Thus, $H$ is a finite extension of
$R_{us,\mathbb{F}}(G)(\mathbb{F})$.

If the residual characteristic $p$ of $\mathbb{F}$ divides $k$ and the
characteristic of $\mathbb{F}$ is positive, then by (6) of Proposition 3.3,
$H$ is finite. This implies that $G(\mathbb{F})$ itself has finite volume,
hence $G(\mathbb{F})$ is compact.

∎

Acknowledgements We would like to thank Prof. B. Conrad for some helpful
suggestions.

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Arunava Mandal and C.R.E. Raja

Theoretical Statistics and Mathematics Unit,

Indian Statistical Institute, Bangalore Centre,

Bengaluru 560059, India.

E-mail<EMAIL_ADDRESS>and<EMAIL_ADDRESS>


# Accessing a broader range of energy states in metallic glasses by variable-
amplitude oscillatory shear

Nikolai V. Priezjev1,2 1Department of Mechanical and Materials Engineering,
Wright State University, Dayton, OH 45435  2National Research University
Higher School of Economics, Moscow 101000, Russia

###### Abstract

The influence of variable-amplitude loading on the potential energy and
mechanical properties of amorphous materials is investigated using molecular
dynamics simulations. We study a binary mixture that is either rapidly or
slowly cooled across the glass transition temperature and then subjected to a
sequence of shear cycles with strain amplitudes above and below the yielding
strain. It was found that well annealed glasses can be rejuvenated by small-
amplitude loading if the strain amplitude is occasionally increased above the
critical value. By contrast, poorly annealed glasses are relocated to
progressively lower energy states when subyield cycles are alternated with
large-amplitude cycles that facilitate exploration of the potential energy
landscape. The analysis of nonaffine displacements revealed that in both
cases, the typical size of plastic rearrangements varies depending on the
strain amplitude and number of cycles, but remains smaller than the system
size, thus preserving structural integrity of amorphous samples.

Keywords: metallic glasses, thermo-mechanical processing, yielding transition,
oscillatory shear deformation, molecular dynamics simulations

##  I Introduction

The fundamental understanding of the interrelationship between the amorphous
structure, mechanical and physical properties of bulk metallic glasses is
important for numerous biomedical and structural applications Wang2012 ;
Trexler10 ; ZhengBio16 . In contrast to crystalline materials, where
irreversible deformation is controlled by topological line defects, it was
realized that glasses deform plastically via a sequence of collective
rearrangements of groups of atoms, often referred to as shear transformations
Spaepen77 ; Argon79 . The advantageous properties of metallic glasses include
relatively high strength, large elastic strain limit, high resistance to
corrosion, and biocompatibility, among others, but they typically suffer from
catastrophic failure upon external deformation, i.e., they are brittle when in
a well annealed state Wang2012 . To remediate the latter issue, a number of
processing methods are employed in order to rejuvenate metallic glasses; for
example, cold rolling, high pressure torsion, irradiation, elastostatic
loading, and surface treatments like shot peening Greer16 . More recently, it
was found that a particularly elegant and minimally invasive approach to
enhance potential energy is to thermally cycle glasses between the room and
cryogenic temperatures Ketov15 ; Guo19 ; Priez18tcyc ; Priez19T2000 ;
Priez19T5000 ; Mirdamadi19 ; Priez19one ; Jittisa20 ; Meylan20 ; Du20 ; Guan20
. Despite considerable efforts, however, the development of efficient
processing methods to access a broader range of energy states in metallic
glasses and, at the same time, maintain their structural integrity remains a
challenging problem.

In the last decades, atomistic simulations have played an important role in
understanding relaxation, rejuvenation and yielding phenomena in disordered
materials subjected to periodic deformation Lacks04 ; Lo10 ; Priezjev13 ;
Sastry13 ; Reichhardt13 ; IdoNature15 ; Shi15 ; Yang16 ; Priezjev16 ;
Kawasaki16 ; Priezjev16a ; Sastry17 ; Priezjev17 ; OHern17 ; Priezjev18 ;
Priezjev18a ; Heuer18 ; NVP18strload ; Sastry19band ; PriezSHALT19 ; ShuoLi19
; Peng19 ; Priez20ba ; Jana20 ; KawBer20 ; NVP20altY ; Kawasaki20 ; Priez20del
; Priez20heal . Most notably, it was discovered that after a certain number of
cycles at zero temperature, disordered solids become locked into the so-called
‘limit cycles’ where the trajectory of each atom is exactly reversible during
one or more periods Reichhardt13 ; IdoNature15 . At finite temperatures, the
structural relaxation, sometimes termed as mechanical annealing, proceeds via
collective irreversible rearrangements during hundreds of loading cycles, and
the potential energy gradually approaches a constant value Priezjev18 ;
Priezjev18a ; NVP18strload ; PriezSHALT19 . Depending on the initial energy
state, it typically takes a number of transient cycles to yield when the
strain amplitude is above the critical value, and the number of cycles is
reduced upon periodically alternating the loading direction Priezjev17 ;
Priezjev18a ; Sastry19band ; Priez20ba ; KawBer20 ; NVP20altY ; Priez20del .
What remains unknown, however, is how to apply cyclic loading and rejuvenate
glasses without the formation of shear bands, and, on the other hand, how to
access low-energy states in poorly annealed glasses by mechanical agitation.

In this paper, the effect of variable-amplitude oscillatory shear deformation
of binary glasses on their energy states and mechanical properties is
investigated using molecular dynamics (MD) simulations. We consider a model
glass former initially cooled well below the glass transition temperature and
then periodically deformed with strain amplitudes alternating between values
below and above the yielding amplitude. It will be shown that well annealed
glasses become rejuvenated by cyclic loading when the strain amplitude is once
in a while increased above the critical amplitude, followed by a sequence of
low-amplitude cycles. Remarkably, the same deformation protocol drives poorly
annealed glasses to lower energy states as it allows for a more efficient
exploration of the potential energy landscape.

The rest of this paper is organized as follows. The molecular dynamics
simulations, parameter values, and the deformation protocol are described in
the next section. The results for the potential energy series under variable-
amplitude cycle loading, mechanical properties, and the spatiotemporal
analysis of irreversible displacements are presented in section III. The brief
summary is given in the last section.

##  II Molecular dynamics simulations

In our study, we use the binary (80:20) Lennard-Jones (LJ) mixture model to
represent an amorphous alloy in three dimensions. This popular model of a
glass former was first introduced by Kob and Andersen (KA), who investigated
its structural and dynamical properties near the glass transition temperature
KobAnd95 . The mixture consists of two types of atoms with strongly non-
additive interaction between different types, which suppresses crystallization
upon cooling across the glass transition. Specifically, any two atoms of types
$\alpha,\beta=A,B$ interact via the truncated LJ potential, as follows:

$V_{\alpha\beta}(r)=4\,\varepsilon_{\alpha\beta}\,\Big{[}\Big{(}\frac{\sigma_{\alpha\beta}}{r}\Big{)}^{12}\\!-\Big{(}\frac{\sigma_{\alpha\beta}}{r}\Big{)}^{6}\,\Big{]},$
(1)

with the parameters: $\varepsilon_{AA}=1.0$, $\varepsilon_{AB}=1.5$,
$\varepsilon_{BB}=0.5$, $\sigma_{AA}=1.0$, $\sigma_{AB}=0.8$,
$\sigma_{BB}=0.88$, and $m_{A}=m_{B}$ KobAnd95 . A similar parametrization of
the pairwise interaction was used by Weber and Stillinger to study the
amorphous metal-metalloid alloy $\text{Ni}_{80}\text{P}_{20}$ Weber85 . In our
simulations, the cutoff radius of the LJ potential was set to
$r_{c,\,\alpha\beta}=2.5\,\sigma_{\alpha\beta}$. The system consists of
$N=60\,000$ atoms. As usual, the simulation results are reported in terms of
the reduced units of length, mass, and energy $\sigma=\sigma_{AA}$, $m=m_{A}$,
and $\varepsilon=\varepsilon_{AA}$. The equations of motion were solved
numerically using the velocity Verlet algorithm with the time step $\triangle
t_{MD}=0.005\,\tau$, where $\tau=\sigma\sqrt{m/\varepsilon}$ is the LJ time
Allen87 ; Lammps .

The sample preparation protocol involved a thorough equilibration of the
liquid phase at a constant density $\rho=\rho_{A}+\rho_{B}=1.2\,\sigma^{-3}$
in a periodic box of linear size $L=36.84\,\sigma$. The system temperature was
regulated via the Nosé-Hoover thermostat Allen87 ; Lammps . For reference, the
computer glass transition temperature at $\rho=1.2\,\sigma^{-3}$ is
$T_{c}=0.435\,\varepsilon/k_{B}$, where $k_{B}$ is the Boltzmann constant
KobAnd95 . Following the equilibration period, the binary mixture was cooled
to the low temperature $T_{LJ}=0.01\,\varepsilon/k_{B}$ with a slow
($10^{-5}\varepsilon/k_{B}\tau$) and fast ($10^{-2}\varepsilon/k_{B}\tau$)
cooling rates in order to obtain well (low energy) and poorly (high energy)
annealed samples at $\rho=1.2\,\sigma^{-3}$.

After cooling to $T_{LJ}=0.01\,\varepsilon/k_{B}$, both samples were subjected
to periodic shear deformation along the $xz$ plane at constant volume, as
follows:

$\gamma_{xz}(t)=\gamma_{0}\,\text{sin}(2\pi t/T),$ (2)

where the oscillation period is $T=5000\,\tau$, and, correspondingly,
oscillation frequency is $\omega=2\pi/T=1.26\times 10^{-3}\,\tau^{-1}$. Unless
otherwise noted, the strain amplitude during $n-1$ cycles is set to
$\gamma_{0}=0.06$, while the amplitude during every $n$-th cycle is changed to
$\gamma_{0}=0.08$. An example of the time dependent shear strain for $n=10$ is
shown in Fig. 1. In the present study, the following values of the periodicity
$n$ were considered $n=2$, $5$, $10$, $20$, $50$, and $100$. The typical
simulation run during 3600 cycles takes about 95 days using 40 processors.
During production runs, the potential energy, atomic positions, and shear
strain were stored for post-processing. The simulations were performed only
for one well annealed and one poorly annealed samples due to computational
constraints.

##  III Results

It has been long realized that the atomic structure and mechanical properties
of amorphous alloys strongly depend on the thermo-mechanical processing
history Greer16 . In particular, upon rapid cooling from the liquid state,
glasses freeze into highly unrelaxed configurations and, when externally
deformed, exhibit a gradual crossover from zero to a plateau level of stress
Wang2012 . On the other hand, more slowly cooled glasses settle at lower
energy states and become more brittle. Furthermore, under periodic strain with
an amplitude slightly above a critical value, slowly cooled glasses gradually
form a narrow shear band during a number of transient cycles and then undergo
a yielding transition, whereas rapidly quenched glasses first relax during
hundreds of cycles and then suddenly yield Priezjev17 ; Sastry17 ;
Sastry19band ; Priezjev18a ; Priez20ba ; NVP20altY . By contrast, repeated
cycling with an amplitude below the critical value typically leads to an
exploration of progressively lower energy states via structural relaxation
Sastry13 ; Sastry17 ; Priezjev18 ; NVP18strload ; PriezSHALT19 ; Jana20 .
These conclusions were obtained for binary glasses subjected to cyclic loading
at a fixed strain amplitude.

It can be hypothesized that periodic deformation with the strain amplitude
that is occasionally alternated between values slightly above and below the
critical amplitude, might lead to either enhanced rejuvenation or accelerated
relaxation, depending on the initial energy state. The rationale for this
hypothesis is as follows. In the case of well annealed glasses, one cycle with
an amplitude slightly larger than the critical value will increase the
potential energy via cooperative irreversible rearrangements of atoms.
Subsequently, one or several subyield cycles will relax the glass, thus
avoiding the formation of system-spanning shear bands and material failure.
Upon iteration, such deformation protocol might result in steady increase of
the potential energy, and, possibly, a well-defined energy level after many
cycles. On the contrary, poorly annealed glasses under periodic loading below
the yielding point tend to relax and ultimately get trapped in a local minimum
of the potential energy landscape. Thus, one cycle with a large strain
amplitude might relocate the system to an adjacent minimum with lower energy,
leading to accelerated relaxation upon iteration of the sequence of
alternating cycles.

In our simulations, these scenarios were tested for the well-studied KA binary
mixture model at the low temperature $T_{LJ}=0.01\,\varepsilon/k_{B}$ and
fixed density $\rho=1.2\,\sigma^{-3}$. Thus, it was recently found that the
critical value of the strain amplitude for rapidly cooled KA glasses at these
$T_{LJ}$ and $\rho$ is $\gamma_{0}\approx 0.067$ NVP20altY . Furthermore, well
annealed glasses under periodic shear with the strain amplitude
$\gamma_{0}=0.08$ were shown to undergo a yielding transition after about 20
transient cycles, while it might take many cycles to yield at
$\gamma_{0}=0.07$ Priezjev17 . Therefore, two values of the strain amplitude
slightly below and above the critical point were considered; namely,
$\gamma_{0}=0.06$ and $\gamma_{0}=0.08$. The deformation protocol consists of
one cycle with the strain amplitude $\gamma_{0}=0.08$, followed by $n-1$
consecutive cycles with $\gamma_{0}=0.06$. An example of the shear strain
variation for the sequence of alternating cycles, $n=10$, is displayed in Fig.
1. In this study, the simulations were carried out for the values of the
integer $n=2$, $5$, $10$, $20$, $50$, and $100$. For reference, the results
for periodic deformation with fixed strain amplitudes $\gamma_{0}=0.06$ and
$0.08$ are also reported.

The potential energy minima after each cycle at zero strain are plotted in
Fig. 2 for the well annealed glass subjected to periodic deformation with
periodicity $n=2$, $5$, $10$, $20$, $50$, and $100$. For comparison, the two
limiting cases with the fixed strain amplitudes $\gamma_{0}=0.06$ and
$\gamma_{0}=0.08$ are also included. Note that the potential energy at
$\gamma_{0}=0.06$ remains essentially constant during 3600 cycles, indicating
that mechanical annealing becomes inefficient for well annealed glasses even
in the presence of thermal fluctuations. It was previously found that glasses
under small-amplitude cyclic shear cannot access energy states below a certain
threshold KawBer20 ; Priez20del . Moreover, it can be clearly observed in Fig.
2 that upon increasing periodicity $n$, the yielding transition is delayed (up
to about 1600 cycles for $n=20$). After a sharp increase in potential energy
due to the formation of a shear band across the simulation domain, and two-
phase system continues steady state deformation.

By contrast, the energy curves for the well annealed glass appear to increase
and level out for cyclic loading with large periodicity, $n=50$ and $100$, as
shown in Fig. 2. In these cases, the energy series resemble a step-like
pattern, where a sudden increase after a cycle at $\gamma_{0}=0.08$ follows by
relaxation during $n-1$ cycles at the smaller amplitude $\gamma_{0}=0.06$,
thus avoiding the formation of a shear band. Hence, these results demonstrate
that periodic deformation with alternating amplitudes can lead to rejuvenation
in the absence of strain localization, provided that periodicity $n$ is
sufficiently large. It can be seen in Fig. 2 that the increase in potential
energy after 3600 cycles is about $0.007\,\varepsilon$, which is comparable to
the energy change reported for the KA binary glass subjected to prolonged
elastostatic loading PriezELAST19 ; PriezELAST20 .

In the case of the poorly annealed glass, the potential energy at the end of
each cycle is displayed in Fig. 3 for the modulated deformation $n=20$, $50$,
$100$ and the fixed strain amplitude $\gamma_{0}=0.06$. As is evident, all
deformation protocols initially result in a rapid decay of the potential
energy since many unstable clusters of atoms can be mechanically driven to
lower energy configurations. When the strain amplitude $\gamma_{0}=0.08$ is
applied every 10-th cycle, or more frequently (not shown), the binary glass
undergoes a yielding transition via the formation of a shear band across the
system. As shown in Fig. 3, the yielding transition for $n=10$ occurs after
about 800 shear cycles when the glass is mechanically annealed to
$U\approx-8.28\,\varepsilon$. These results are consistent with the critical
behavior reported in the previous MD studies Priezjev18a ; Sastry19band ;
Priez20ba ; NVP20altY . Furthermore, the potential energy continue to
gradually decay for $n=20$, 50, 100, and when the strain amplitude is always
fixed at $\gamma_{0}=0.06$ (shown by the black curve in Fig. 3). It can be
seen that upon including a cycle with the strain amplitude $\gamma_{0}=0.08$
($n=20$, 50, and 100), the energy levels are reduced on average by roughly
$0.002\,\varepsilon$ with respect to the black curve. This trend can be
explained by noticing relatively large spikes along the energy series due to
large-scale plastic deformation at $\gamma_{0}=0.08$, which increase the
probability of relocating the system between local minima of the potential
energy landscape.

The changes in mechanical properties for the well annealed glass subjected to
variable-amplitude deformation with $n=100$ (the violet curve in Fig. 2) were
evaluated by carrying out mechanical tests at selected time intervals. Thus,
following a certain number of loading cycles, the binary glass, initially at
zero strain, was continuously strained at a fixed rate
$\dot{\gamma}=10^{-5}\,\tau^{-1}$ along the $xy$, $xz$, and $yz$ planes. In
each case, the shear modulus $G$ and the peak value of the stress overshoot
$\sigma_{Y}$ were computed from the linear slope and the maximum of the
stress-strain curve. The results for $G$ and $\sigma_{Y}$ are shown in Fig. 4
during 3600 loading cycles. The data are somewhat noisy, since it were
collected for only one realization of disorder, but, nevertheless, one can
clearly see that the shear modulus reduces and acquires directional
anisotropy, and the yielding peak tends to decrease with the cycle number,
which is consistent with gradual rejuvenation reported in Fig. 2. We also
comment that it was previously shown for cyclically annealed binary glasses
that anisotropy in mechanical properties is reduced when the loading direction
is alternated in two or three spatial dimensions PriezSHALT19 .

We next perform a microscopic analysis of plastic deformation quantified via
the so-called nonaffine displacements of atoms. In disordered solids, the
decomposition of the total displacement of an atom into affine and nonaffine
components can be used to estimate its relative displacement with respect to
neighboring atoms. More specifically, the nonaffine measure for an atom $i$
can be computed using the matrix $\mathbf{J}_{i}$ that linearly transforms a
group of atoms and minimizes the following expression:

$D^{2}(t,\Delta
t)=\frac{1}{N_{i}}\sum_{j=1}^{N_{i}}\Big{\\{}\mathbf{r}_{j}(t+\Delta
t)-\mathbf{r}_{i}(t+\Delta
t)-\mathbf{J}_{i}\big{[}\mathbf{r}_{j}(t)-\mathbf{r}_{i}(t)\big{]}\Big{\\}}^{2},$
(3)

where $\Delta t$ is the time between two configurations, and the summation is
carried over neighbors located within $1.5\,\sigma$ from the position of the
$i$-th atom at $\mathbf{r}_{i}(t)$. This definition was first used by Falk and
Langer in the analysis of localized shear transformations in sheared
disordered solids Falk98 . Typically, if the nonaffine displacement of an atom
becomes greater than the cage size, then the local rearrangement is
irreversible, and it is often involves a group of atoms. In recent years, the
spatial and temporal correlations of nonaffine displacements were extensively
studied in amorphous materials under startup continuous HorbachJR16 ; Schall07
; Pastewka19 ; Priez20tfic ; Priez19star ; Ozawa20 ; ShiBai20  and time
periodic Priezjev16 ; Priezjev18 ; Priezjev18a ; PriezSHALT19 ; Priez20ba ;
Jana20 ; NVP20altY ; Priez20del  deformation. In particular, it was
demonstrated that the yielding transition is accompanied by the formation of a
system-spanning shear band that can be clearly identified by plotting atoms
with large nonaffine displacements.

The atomic configurations of the well annealed glass subjected to variable-
amplitude oscillatory deformation are shown in Fig. 5 for periodicity $n=20$
and in Fig. 6 for $n=100$. The system snapshots are taken at selected cycle
numbers, and, for clarity, only atoms with relatively large nonaffine
displacements during one full cycle are displayed. First, it can be observed
in Fig. 5 (a, b) that, upon continued loading, the typical size of clusters of
atoms with large nonaffine displacements increases but remains smaller than
the system size. The appearance of compact clusters during $500$ and $1500$-th
cycles, when the strain amplitude is $\gamma_{0}=0.08$, correlates well with
the gradual increase in potential energy shown by the brown curve in Fig. 2.
By contrast, the sharp increase in potential energy (after about 1600 cycles)
signals a yielding transition and formation of a shear band along the $xy$
plane, which can be clearly seen in Fig. 5 (c, d). Second, as illustrated in
Fig. 6, the situation is qualitatively different when the strain amplitude is
changed to $\gamma_{0}=0.08$ less frequently (i.e., $n=100$; see also the
violet curve in Fig. 2). In this case, the clusters remain finite but they
become larger during cycles with $\gamma_{0}=0.08$, shown in Fig. 6 (a, c),
while the deformation is nearly reversible during subsequent relaxation at
$\gamma_{0}=0.06$, see Fig. 6 (b, d). Hence, we conclude that the oscillatory
deformation with alternating excitation and relaxation periods leads to a
controllable rejuvenation without shear banding.

Lastly, the sequences of stroboscopic snapshots of the poorly annealed glass
under periodic deformation are presented in Figures 7 and 8 for $n=10$ and
$100$, respectively. The first two snapshots in Fig. 7 (a, b) show
interconnected networks during 100 and 800-th cycles when the strain amplitude
is $\gamma_{0}=0.08$. These large-scale irreversible rearrangements correspond
to structural relaxation after rapid cooling, as indicated by the brown curve
in Fig. 3. In this loading protocol, the strain amplitude is changed to
$\gamma_{0}=0.08$ frequently enough ($n=10$) to induce a yielding transition
after about 900 alternating cycles. Thus, one can clearly observe a single
shear band (across periodic boundaries) in Fig. 7 (c, d) that is formed after
yielding and remains stable upon continued loading. In the case $n=100$, the
poorly annealed glass continues relaxation during every 99 cycles with
$\gamma_{0}=0.06$ [ e.g., see Fig. 8 (b, d) ], which is interrupted by one
cycle with $\gamma_{0}=0.08$ that induces large-scale plastic events, shown
for example, in Fig. 8 (a, c). The occasional perturbation with the strain
amplitude $\gamma_{0}=0.08$ facilitates exploration of the potential energy
landscape and relocates the system to even lower energy states than cyclic
loading with the fixed amplitude $\gamma_{0}=0.06$ (see the red and black
curves in Fig. 3). Altogether, the results for well and poorly annealed
glasses under cyclic deformation with periodicity $n=100$, shown in Figs. 6
and 8, demonstrate that glasses become either highly rejuvenated or better
annealed, thus extending the range of accessible energy states.

##  IV Conclusions

In summary, molecular dynamics simulations were carried out to study the
effect of variable-amplitude periodic deformation on relaxation and
rejuvenation of disordered solids. The model glass former in three dimensions
was represented via the binary mixture, which was cooled from the liquid state
deep into the glass phase with relatively slow and fast rates, producing well
and poorly annealed samples. The binary glass was subjected to cyclic loading
where the strain amplitude is fixed below the critical value but occasionally
changed slightly above the critical strain. During such deformation protocol,
one shear cycle with the large strain amplitude typically induces collective
plastic rearrangements but not shear bands if the frequency of large-amplitude
cycles is sufficiently low.

It was found that well annealed samples can be rejuvenated during a sequence
of shear cycles provided that the strain amplitude is rarely changed above the
critical value. The increase in potential energy is reflected in mechanical
properties and leads to enhanced ductility. Interestingly, the same
deformation protocol drives poorly annealed glasses to progressively lower
energy states, since large-amplitude cycles occasionally perturb the system
and prevent trapping in local minima of the potential energy landscape. On the
other hand, both well and poorly annealed glasses undergo a yielding
transition after a number of transient cycles when the strain amplitude is
frequently changed above and below the critical value. These conclusions were
confirmed by visualizing spatial configurations of atoms with large nonaffine
displacements that are organized either in compact clusters or planar shear
bands.

## Acknowledgments

Financial support from the National Science Foundation (CNS-1531923) is
gratefully acknowledged. The article was prepared within the framework of the
HSE University Basic Research Program and funded in part by the Russian
Academic Excellence Project ‘5-100’. The simulations were performed at Wright
State University’s Computing Facility and the Ohio Supercomputer Center using
the LAMMPS code developed at Sandia National Laboratories Lammps .

Figure 1: An example of the imposed shear strain along the $xz$ plane during
20 cycles with the oscillation period $T=5000\,\tau$. The strain amplitude is
$\gamma_{0}=0.06$, except that during every $n$-th cycle the amplitude is
changed to $\gamma_{0}=0.08$. In this study, the following values were
considered $n=2$, $5$, $10$, $20$, $50$, and $100$.  Figure 2: (Color online)
The potential energy minima at zero strain as a function of the number of
shear cycles with the strain amplitude $\gamma_{0}=0.06$ during $n-1$ periods.
The strain amplitude is changed to $\gamma_{0}=0.08$ during every $n$-th
cycle. The values of periodicity $n$ are listed in the legend. The oscillation
period is $T=5000\,\tau$. The black and orange curves indicate cyclic loading
with the strain amplitudes $\gamma_{0}=0.06$ and $0.08$, respectively. The
well annealed glass was initially prepared via cooling from the liquid state
to $T_{LJ}=0.01\,\varepsilon/k_{B}$ with the rate
$10^{-5}\varepsilon/k_{B}\tau$.  Figure 3: (Color online) The time dependence
of the potential energy (at zero strain) for the indicated values of
periodicity $n$. The black curve denotes cyclic loading with the strain
amplitude $\gamma_{0}=0.06$. The time is expressed in oscillation periods
$T=5000\,\tau$. The poorly annealed sample was initially prepared by cooling
with the rate $10^{-2}\varepsilon/k_{B}\tau$ to
$T_{LJ}=0.01\,\varepsilon/k_{B}$ at $\rho=1.2\,\sigma^{-3}$.  Figure 4: (Color
online) The shear modulus $G$ (in units of $\varepsilon\sigma^{-3}$) and
yielding peak $\sigma_{Y}$ (in units of $\varepsilon\sigma^{-3}$) versus cycle
number for periodic deformation with $n=100$. The startup continuous shear
deformation with the strain rate $\dot{\gamma}=10^{-5}\,\tau^{-1}$ was imposed
along the $xy$ plane (blue circles), $xz$ plane (red squares), and $yz$ plane
(green diamonds). The variation of the potential energy for this loading
protocol is indicated by the violet curve in Fig. 2. The glass was initially
cooled with the rate $10^{-5}\varepsilon/k_{B}\tau$ to
$T_{LJ}=0.01\,\varepsilon/k_{B}$.  Figure 5: (Color online) Snapshots of the
well annealed glass subjected to oscillatory shear deformation with
periodicity $n=20$. The potential energy for the same protocol is denoted by
the brown curve in Fig. 2. The nonaffine measure, as defined by Eq. (3), is
(a) $D^{2}(500\,T,T)>0.04\,\sigma^{2}$, (b)
$D^{2}(1500\,T,T)>0.04\,\sigma^{2}$, (c) $D^{2}(1600\,T,T)>0.04\,\sigma^{2}$,
and (d) $D^{2}(2500\,T,T)>0.04\,\sigma^{2}$. The oscillation period is
$T=5000\,\tau$. The color in the legend indicates the magnitude of $D^{2}$.
Figure 6: (Color online) Plastic rearrangements in the well annealed glass
cyclically driven with periodicity $n=100$. The corresponding energy series
are indicated by the violet curve in Fig. 2. The nonaffine displacements are
(a) $D^{2}(1000\,T,T)>0.04\,\sigma^{2}$, (b)
$D^{2}(1050\,T,T)>0.04\,\sigma^{2}$, (c) $D^{2}(3000\,T,T)>0.04\,\sigma^{2}$,
and (d) $D^{2}(3050\,T,T)>0.04\,\sigma^{2}$, where $T=5000\,\tau$. The color
shows the magnitude of $D^{2}$.  Figure 7: (Color online) The sequence of
snapshots of poorly annealed glass subjected to variable-amplitude periodic
deformation with periodicity $n=10$. The same loading protocol as in Fig. 3
(the brown curve). The nonaffine measure is (a)
$D^{2}(100\,T,T)>0.04\,\sigma^{2}$, (b) $D^{2}(800\,T,T)>0.04\,\sigma^{2}$,
(c) $D^{2}(900\,T,T)>0.04\,\sigma^{2}$, and (d)
$D^{2}(1400\,T,T)>0.04\,\sigma^{2}$. The colorcode in the legend defines the
magnitude of $D^{2}$.  Figure 8: (Color online) Spatial configurations of
atoms with large nonaffine displacements after one cycle for the poorly
annealed glass under periodic deformation with $n=100$. See the red curve in
Fig. 3. The nonaffine quantity in Eq. (3) is (a)
$D^{2}(100\,T,T)>0.04\,\sigma^{2}$, (b) $D^{2}(150\,T,T)>0.04\,\sigma^{2}$,
(c) $D^{2}(1900\,T,T)>0.04\,\sigma^{2}$, and (d)
$D^{2}(1950\,T,T)>0.04\,\sigma^{2}$. The magnitude of $D^{2}$ is defined in
the legend.

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