$\eta_{\bf y}(h_{1}th_{2}y_{0})$, which from the previous case (but swapping
$\bf l$ and $\bf y$) is close to $\eta_{\bf y}(h_{1}y_{0})+\eta_{\bf
l}(h_{2}x_{0})$ and so to $-B(h_{1})+B(h_{2})$. But $\eta_{\bf
l}(th_{2}x_{0})+\eta_{\bf l}(h_{1}x_{0})$ is close to $\eta_{\bf
y}(h_{2}y_{0})+B(h_{1})$ and so to $-B(h_{2})+B(h_{1})$, thus we have covered
all cases.
$\Box$

###  9.4 Distinguishing isometric actions

Returning now to quasi-actions, suppose that we have a quasi-action $\alpha$
of a group $G$ on a metric space where quasi-orbits are quasi-isometric to
$\mathbb{R}$, so that we can reduce it to a cobounded isometric action on some
quasi-line. We now consider when $\alpha$ can be reduced to a cobounded
isometric action on $\mathbb{R}$. Note that if this action is of type (3)+
(respectively (3)-) then any reduction of $\alpha$ to an action on some
hyperbolic space will also be of type (3)+ (respectively (3)-). Theorem 9.3
and Corollary 9.5 tell us it is sufficient that the corresponding Busemann
function of this reduced isometric action is a genuine homomorphism to
$\mathbb{R}$. This is because if we have a reduction of $\alpha$ to an action
$\gamma$ on a quasi-line where the Busemann quasi-morphism $B$ is genuinely a
homomorphism $\theta$ then these results applied to $\gamma$ say that the
corresponding translation/dihedral quasi-action on $\mathbb{R}$ given by the
Busemann quasi-morphism $B$ will be an isometric action that is a reduction of
$\gamma$, therefore also a reduction of $\alpha$.

Conversely if a reduction of $\alpha$ to an isometric action on $\mathbb{R}$
exists, this reduced action will be by translations or will be dihedral with
$g\mapsto g(0)$ the Busemann quasi-morphism. So for this reduction of
$\alpha$, the Busemann quasi-morphism will be a homomorphism of $G$ to
$\mathbb{R}$. However this answer misses two important points: first, do we
obtain the same or a similar Busemann quasi-morphism whenever we reduce
$\alpha$ to an isometric action on some hyperbolic space? Second, even if we
have a positive answer to the above question, how do we characterise which
quasi-actions can be made into isometric actions on $\mathbb{R}$ without first
having to turn them into an isometric action on a particular hyperbolic space
(which might not be an easy process)? We deal with the second point first,
where our answer depends directly on our given quasi-action.

###### Theorem 9.6

Let $\alpha$ be a quasi-action of a group $G$ on a metric space which can be
reduced to an isometric action of type (3)+ on some hyperbolic space.
$\bullet$ If the set of elliptic elements of $\alpha$ is equal to the kernel
of a homomorphism from $G$ to $\mathbb{R}$ then there is a reduction of
$\alpha$ to an isometric translation action of $G$ on $\mathbb{R}$.
$\bullet$ If the set of elliptic elements of $\alpha$ is not equal to the
kernel of any homomorphism from $G$ to $\mathbb{R}$ then $\alpha$ cannot be
reduced to any isometric action on a hyperbolic space that is proper or is
CAT(0).
Let $\alpha$ be a quasi-action of a group $G$ on a metric space which can be
reduced to an isometric action of type (3)- on some hyperbolic space.
$\bullet$ If there is an index 2 subgroup $H$ of $G$ and a homomorphism from
$H$ to $\mathbb{R}$ such that the set of elliptic elements of the quasi-action
$\alpha$ consists of the union of this kernel and $G\setminus H$ then there is
a reduction of $\alpha$ to a isometric dihedral action of $G$ on $\mathbb{R}$.
$\bullet$ If there is no index 2 subgroup $H$ of $G$ having a homomorphism to
$\mathbb{R}$ where the set of elliptic elements of the quasi-action $\alpha$
consists of the union of this kernel and $G\setminus H$ then $\alpha$ cannot
be reduced to any isometric action on a hyperbolic space that is proper or is
CAT(0).

Proof. First suppose that we are in the type (3)+ case. Say there is no
homomorphism $\theta:G\rightarrow\mathbb{R}$ with $ker(\theta)$ equal to the
set of elliptic elements of $\alpha$. Elements of $G$ have the same type under
any reduction, with type (3) actions having no parabolic elements. Thus if
there were a reduction to an isometric translation action on $\mathbb{R}$ then
this Busemann quasi-morphism would just be the homomorphism obtained by this
translation action, with the kernel equal to the set of elliptic elements of
this action and so of $\alpha$ too. Furthermore in [6] Corollary 3.9 it is
shown that for any action on a proper hyperbolic metric space with a fixed
point on the boundary, the Busemann quasi-morphism so obtained is a
homomorphism. It is also mentioned there that this is true for any hyperbolic
space (not necessarily proper) that is CAT(0).

Now suppose the set of elliptic elements of $\alpha$ agrees exactly with the
kernel of a homomorphism $\theta:G\rightarrow\mathbb{R}$. We reduce $\alpha$
to any isometric action on a hyperbolic space, which will be of type (3)+. We
take the Busemann quasi-morphism $B$ of this action, which will be homogeneous
and whose zero set is exactly $ker(\theta)$, because the elliptic and non
elliptic elements are unchanged under reduction. In particular we have
$B(ghg^{-1}h^{-1})=0$ for every commutator of $G$ because these elements all
lie in the kernel. But by Barvard ([4] Lemma 3.6) we have for $B$ a
homogeneous quasi-morphism that the supremum of $|B(ghg^{-1}h^{-1})|$ over
$g,h\in G$ is the defect of $B$, so here it is zero implying that $B$ is a
homomorphism. So on now applying Theorem 9.3, the resulting quasi-action on
$\mathbb{R}$ is an isometric action.

Finally say that the quasi-action can be reduced to some type (3)- action on a
hyperbolic space. The first part of the proof above goes through as well for
this case by setting $H$ in the statement of our theorem to be the index two
orientation preserving subgroup $G^{+}$ of the action. This also works for the
second part by applying Corollary 9.5 instead of Theorem 9.3, except that we
need to show that $G^{+}$ is equal to the given index 2 subgroup $H$. If not
then we can take some element $h$ in $H\setminus G^{+}$. As all elements in
$G\setminus G^{+}$ are elliptic in a type (3)- action, we must have
$\theta(h)=0$ for $\theta$ our given homomorphism from $H$ to $\mathbb{R}$.
But on taking some $h_{1}\in H$ with $\theta(h_{1})=\theta(hh_{1})\neq 0$
(such an $h_{1}$ exists because there are loxodromic elements in $G$ and their
square will lie in $H$), we have that $h_{1}$ and $hh_{1}$ are not elliptic so
they must lie in $G^{+}$. Thus $hh_{1}h_{1}^{-1}$ is in $G^{+}$ which is a
contradiction.
$\Box$

###### Example 9.7

On taking the free group $F_{2}$ and a quasi-morphism $q$ which is not within
bounded distance of any homomorphism, say a Brooks quasi-morphism, we have
that the seemingly innocuous translation quasi-action $\alpha$ of $F_{2}$ on
$\mathbb{R}$ given by $\alpha(g,x)=x+q(g)$ cannot be reduced to any isometric
action on a CAT(0) space or on a proper hyperbolic space. (Indeed it cannot be
reduced to an isometric action on any proper metric space, say by [19]
Proposition 4.5 and Theorem 4.38.)

We would now like to deal with our first point: if we turn a quasi-action of
some group $G$ into an isometric action of type (3) then do we always get the
same Busemann quasi-morphism $B$ (once we have homogenised)? Of course we can
always rescale the metric, so that $B$ and $\lambda B$ can occur for any
$\lambda\neq 0$. However [1] Corollary 4.16 gives plenty of examples of
homogeneous quasi-morphisms of a group where the zero set is just the
identity. Indeed this can happen with homomorphisms, for instance take
$G=\mathbb{Z}^{2}=\langle x,y\rangle$ and homomorphisms
$\theta_{\pm}:G\rightarrow\mathbb{R}$ given by $\theta_{\pm}(x)=1$,
$\theta_{\pm}(y)=\pm\sqrt{2}$. Then $\theta_{+}$ and $\theta_{-}$ do not look
equivalent but both have zero kernel.

It turns out that scaling is the only ambiguity.

###### Proposition 9.8

Suppose that $\alpha$ is a quasi-action of a group $G$ on a metric space which
can be reduced to an isometric action of type (3) on a hyperbolic space. If
$\beta$ and $\gamma$ are two such reductions (not necessarily to the same
hyperbolic space) then the homogeneous Busemann quasi-morphisms
$B_{\beta},B_{\gamma}$ are related by scaling.

Proof. First assume that $\beta$ is of type (3)+, in which case so is
$\gamma$. Applying Theorem 9.3 to both $\beta$ and $\gamma$, each can be
reduced to the translation quasi-actions on $\mathbb{R}$ by the Busemann
quasi-morphisms $B_{\beta}$ and $B_{\gamma}$ respectively, which we will call
$\beta^{\prime}$ and $\gamma^{\prime}$. As these quasi-actions on $\mathbb{R}$
are both cobounded (unless one quasi-morphism is zero, in which case $\alpha$
is bounded and so both are), they are equivalent by Proposition 3.16 (ii).
Thus we have a $(K,\epsilon,C)$ quasi-isometry
$F:\mathbb{R}\rightarrow\mathbb{R}$ which is $G$-coarse equivariant from
$\beta^{\prime}$ to $\gamma^{\prime}$. The latter point means we have $M\geq
0$ such that for all $g\in G$ and $x\in\mathbb{R}$,

$d_{\mathbb{R}}(F\beta^{\prime}(g,x),\gamma^{\prime}(g,Fx))\leq M\mbox{ so
}|F(x)+B_{\gamma}(g)-F(x+B_{\beta}(g))|\leq M.$

Putting $x=0$ in this equation allows us to establish our key property, which
is that if we have a subset $S$ of $G$ such that the values of
$|B_{\beta}(g)|$ over $g\in S$ form a bounded subset of $\mathbb{R}$ then
$|F(0)+B_{\gamma}(g)-F(B_{\beta}(g))|\leq M$ implies

$|B_{\gamma}(g)|\leq M+K|B_{\beta}(g)|+\epsilon$

and so the values of $|B_{\gamma}(g)|$ over the same subset $S$ are bounded
too.

Now take any $g\in G$ with $B_{\beta}(g)\neq 0$ (so that $B_{\gamma}(g)\neq 0$
either as equivalent quasi-actions have the same loxodromic elements). We
rescale $B_{\beta},B_{\gamma}$ so that both have value 1 on $g$. For any $h\in
G$ and $n\in N$, we choose $m_{n}\in\mathbb{Z}$ to be the integer below
$B_{\beta}(h^{n})$, so that we have

$m_{n}\leq B_{\beta}(h^{n})<m_{n}+1.$

As $B_{\beta}$ and $B_{\gamma}$ are both homogeneous quasi-morphisms, we
obtain

$|B_{\beta}(g^{-m_{n}}h^{n})+m_{n}-nB_{\beta}(h)|\leq D_{\beta}$

and the equivalent equation for $\gamma$. As $|m_{n}-nB_{\beta}(h)|\leq 1$ by
construction, we have that $|B_{\beta}(g^{-m_{n}}h^{n})|$ is bounded over
$n\in\mathbb{N}$. Therefore so is $|B_{\gamma}(g^{-m_{n}}h^{n})|$ by our point
above, which implies that $|m_{n}-nB_{\gamma}(h)|$ is bounded too. Combining
this with $|m_{n}-nB_{\beta}(h)|\leq 1$ tells us that
$n|B_{\beta}(h)-B_{\gamma}(h)|$ is also bounded and hence
$B_{\beta}(h)=B_{\gamma}(h)$, implying that $B_{\beta}=B_{\gamma}$ after
rescaling.

As for the case where $\beta$ is an action of type of (3)-, the Busemann
quasi-morphism of this action is by definition the Busemann quasi-morphism of
the index 2 orientation preserving subgroup $G^{+}$, so we restrict $\beta$ to
$G^{+}$ whereupon it becomes a type (3)+ action. We do the same for $\gamma$
and its orientation preserving subgroup $G^{*}$, but the argument at the end
of Theorem 9.6 works to show that $G^{+}$ and $G^{*}$ are the same index 2
subgroup even if $\theta$ is a homogeneous quasi-morphism, provided our
loxodromic element $h_{1}$ has $\theta(h_{1})$ big enough, so we can take
$h_{1}$ to be a large power of itself if necessary. We now restrict $\beta$
and $\gamma$ to $G^{+}$, whereupon they are still both reductions of the same
quasi-action, namely $\alpha$ restricted to $G^{+}$. Now we can apply the
first part of the proof to these restricted actions.
$\Box$

Note that for two translation (respectively dihedral) quasi-actions on
$\mathbb{R}$, if their Busemann quasi-morphisms are related by scaling then
they are equivalent quasi-actions. This then gives us a complete and concrete
description of all type (3)+ and type (3)- cobounded actions, up to
equivalence, of a given group $G$ over all hyperbolic metric spaces: for type
(3)+ actions, it is the projective vector space of homogeneous quasi-morphisms
of $G$. For type (3)- actions it is the disjoint union over each index 2
subgroup $H$ of the projective vector space of homogeneous quasi-morphisms of
$H$ which are antisymmetric in $G$.

###  9.5 Simplicial actions

In this paper we have been interested in when quasi-actions of a group $G$ on
a hyperbolic space $X$ can be reduced to genuine actions on a simplicial tree.
These actions have always been isometric actions but we might want to ask
whether they can be taken to be simplicial actions. This is only relevant to
type (3) actions because for all trees other than the simplicial line $L$
(namely $\mathbb{R}$ with the simplicial structure where the vertices are
$\mathbb{Z}$), any isometry is a simplicial automorphism. However for $L$ the
simplicial automorphisms form a proper subgroup of of the isometry group and
it could be for some applications we only want to allow the former. Therefore
we give here the equivalent version of Theorem 9.6, where the obvious
condition of taking homomorphisms to $\mathbb{Z}$ rather than to $\mathbb{R}$
does indeed hold.

###### Corollary 9.9

Given the same hypotheses as Theorem 9.6, we have that a reduction of $\alpha$
to a simplicial translation/dihedral action of $G$ on the simplicial line $L$
exists or does not exist exactly under the same conditions as Theorem 9.6,
other than that all homomorphisms from $G$ or an index 2 subgroup are to
$\mathbb{Z}$ rather than to $\mathbb{R}$.

Proof. Here we take both type (3)+ and type (3)- actions together. The proof
of the existence of such a homomorphism being a necessary condition is exactly
as in the proof of Theorem 9.6, other than on changing $\mathbb{R}$ to
$\mathbb{Z}$.

As for sufficiency, the proof of Theorem 9.6 again gives us that our Busemann
quasi-morphism $B$ is a genuine homomorphism from $G$ to $\mathbb{R}$, so that
Theorem 9.3 or Corollary 9.5 gives us an isometric translation/dihedral action
of $G$ on $\mathbb{R}$ by $B$. But moreover the kernel of $B$ agrees with the
kernel of some homomorphism $\theta$ from $G$ (or from $G^{+}$) to
$\mathbb{Z}$. This means that $B$ is just a rescaling of $\theta$ and so we
can rescale this isometric action of $G$ on $\mathbb{R}$ to make the Busemann
quasi-morphism equal to $\theta$, whereupon the action on $L$ is now
simplicial.
$\Box$

##  10 Applications

###  10.1 Splittings of groups from quasi-actions

We finish by applying our results to some general situations, with an emphasis
on finitely generated groups. Indeed by results in [21] we know that a
finitely generated group has an unbounded action on a tree (equivalently has
an action without a global fixed point) if and only if it splits as an
amalgamated free product or HNN extension. In this statement the action is
assumed to be simplicial (we can subdivide all edges if any are inverted).
Thus Theorem 8.4 now becomes:

###### Theorem 10.1

Let $G$ be any finitely generated group having an unbounded quasi-action on
any metric space where the quasi-orbits both quasi-isometrically embed in some
tree and also into some proper metric space. Either the quasi-orbits are
quasi-isometric to $\mathbb{R}$ or $G$ splits non trivially as a finite graph
of groups, where the image of any edge group under inclusion into a vertex
group has finite index.

Proof. By Proposition 3.19 (i) the quasi-orbits are coarse connected, so that
Theorem 8.4 applies. If the quasi-orbits are not quasi-isometric to
$\mathbb{R}$ then we must be in the case where the quasi-action reduces to a
cobounded isometric action on a bounded valence bushy tree. As the original
quasi-action is unbounded, so is this action. Moreover a bushy tree is not the
simplicial line, so this action is also by simplicial automorphisms and thus
Serre’s result gives us a splitting. As the tree is of bounded valence and the
action is cobounded, the quotient graph is finite. The statement about edge
groups is because the tree is of bounded valence.
$\Box$

###### Corollary 10.2

Suppose that a finitely generated group $G$ has the property that every quasi-
morphism of $G$ and of its index 2 subgroups is within bounded distance of a
homomorphism (for instance amenable groups). If $G$ has an unbounded quasi-
action on any metric space where the quasi-orbits both quasi-isometrically
embed in some tree and also into some proper metric space then $G$ splits non
trivially as a finite graph of groups, where the image of any edge group under
inclusion into a vertex group has finite index.

Proof. This follows by Theorem 10.1 unless the quasi-orbits are quasi-
isometric to $\mathbb{R}$. If so then by Theorem 9.3 or Corollary 9.5 we have
that the quasi-action reduces to a translation/dihedral quasi-action with
respect to the relevant Busemann quasi-morphism $B$ of $G$ or an index 2
subgroup. But Busemann functions are homogeneous, thus $B$ must be a non
trivial homomorphism to $\mathbb{R}$ by the hypothesis on $G$. Now a finitely
generated group with such a homomorphism to $\mathbb{R}$ will also have a
surjective homomorphism $\theta$ say to $\mathbb{Z}$. Thus in the orientation
preserving case $G$ has a non trivial simplicial action on the line. In the
orientation reversing case with $G^{+}$ the index 2 orientation preserving
subgroup of $G$ and $t$ any fixed element of $G\setminus G^{+}$ as before, we
have our non trivial homomorphism $B$ from $G^{+}$ to $\mathbb{R}$ which we
know satisfies $B(tgt^{-1})=-B(g)$ for all $g\in G^{+}$. As the image
$B(G^{+})$ is a copy of $\mathbb{Z}^{n}$ in $\mathbb{R}$ for some $n\geq 1$,
we can compose with a surjective homomorphism $\chi$ from $B(G^{+})$ to
$\mathbb{Z}$, whereupon $\chi B$ is a homomorphism from $G^{+}$ onto
$\mathbb{Z}$ which is also antisymmetric in $G$. Then we have seen that
sending $g$ in $G^{+}$ to $x\mapsto x+\chi B(g)$ and $tg$ to $-x-\chi B(g)$ is
a homomorphism of $G$ to $Isom(\mathbb{R})$. But as the image of $\chi B$ is
$\mathbb{Z}$, this is also a homomorphism to the dihedral group
$Isom(\mathbb{Z})$ which is onto. Thus $G$ has a non trivial simplicial action
on the line in this case too.
$\Box$

We now convert this into a statement about actions of groups on proper quasi-
trees. This differs vastly from the case without the finiteness condition of
being proper, which was considered in [2].

###### Corollary 10.3

Suppose that $G$ is any finitely generated group with an unbounded isometric
action on any proper quasi-tree $X$. Then $G$ acts by automorphisms and
without a global fixed point on some simplicial tree with bounded valence.
Thus $G$ has the structure of a fundamental group of a (non trivial) finite
graph of groups where all edge groups have finite index in all of the
respective vertex groups.

Proof. This is again Theorem 10.1 unless quasi-orbits are quasi-isometric to
$\mathbb{R}$. If so then we can also apply Theorem 9.3 or Corollary 9.5 to
obtain a reduction of this action to a translation/dihedral quasi-action of
$G$ on $\mathbb{R}$ with respect to the relevant Busemann quasi-morphism $B$
of $G$ or an index 2 subgroup. But $B$ is obtained as the Busemann quasi-
morphism of the action of $G$ on $X$ at either of the two limit points in
$\partial X$ of an orbit. This is a homomorphism if $X$ is a proper hyperbolic
space by [6] Corollary 3.9. We can now follow the proof of Corollary 10.2.
$\Box$
Example 1: We cannot replace $X$ is proper with $X$ is quasi-isometric to some
proper space. To see this, take a hyperbolic group $H$ with property (T) and a
homogeneous quasi-morphism $q$ of $H$ which is not a homomorphism (ie not the
trivial homomorphism). By [1] Lemma 4.15, we can obtain from $q$ an infinite
generating set $S$ for $H$ such that the Cayley graph $Cay(H,S)$ is quasi-
isometric to $\mathbb{R}$ (and $q$ is indeed the Busemann quasi-morphism of
the isometric action of $H$ on $Cay(H,S)$ by Proposition 9.8).

Example 2: Just because our finitely generated group $G$ splits with respect
to an action on a bounded valance tree, this does not imply that any edge or
vertex group will itself be finitely generated. Note that these subgroups are
commensurable in $G$, so if one is finitely generated then they all are. But
let $G$ be a closed orientable surface group of genus at least 2. Then $G$
surjects to $F_{2}$, so certainly has a splitting where each edge group embeds
with finite index in its respective vertex groups. Note that any vertex group
is commensurable to any of its conjugates. But $G$ is a torsion free
hyperbolic group where any finitely generated subgroup is quasi-convex. In [7]
Theorem 3.15 it is shown that an infinite quasi-convex subgroup of a
hyperbolic group has finite index in its commensurator. Hence if such an
action on a bounded valence tree existed with finitely generated vertex/edge
stabilisers then these would either have finite index in $G$, so that actually
the splitting is trivial, or the stabilisers are themselves trivial in which
case $G$ would be a free group.

Example 3: To give an example of a finitely presented group that has unbounded
actions on trees but no unbounded actions on locally finite trees, take a
finitely presented group $G$ which has property (FA) so it has no unbounded
actions on any tree. Suppose that $G$ has a finitely generated subgroup $H$ of
infinite index in $G$ with the following property: if $H$ is contained in a
finite index subgroup $L$ of $G$ then $L=G$. (This is sometimes known in the
literature as $H$ is not engulfed in $G$ or $H$ is dense in the profinite
topology of $G$.) For instance if $G$ is an infinite finitely presented group
having property (T) but with no proper finite index subgroups then we can take
$H$ to be trivial. Then form the amalgamation $A=G_{1}*_{H_{1}=H_{2}}G_{2}$
where $G_{1}$ and $G_{2}$ are isomorphic copies of $G$ with $H_{1},H_{2}$ the
corresponding copies of the subgroup $H$. Now $A$ is finitely presented and
has an unbounded action on the tree given by this splitting. But suppose that
$A$ acts on a locally finite tree. Then $G_{1}$ fixes a vertex $v_{1}$ of this
tree (subdividing if necessary) and $G_{2}$ fixes some vertex $v_{2}$. On
taking a path from $v_{1}$ to $v_{2}$, the finite valence means that the
subgroup $L_{1}$ of $G_{1}$ which fixes the path between $v_{1}$ and $v_{2}$
has finite index in $G_{1}$. Now $H=H_{1}=H_{2}$ is a subgroup of both $G_{1}$
and $G_{2}$ and so fixes both $v_{1}$ and $v_{2}$, thus $H$ is contained in
$L_{1}$. But our chosen property of $H_{1}$ means that $L_{1}$ is equal to
$G_{1}$. Thus $G_{1}$ fixes the path between $v_{1}$ and $v_{2}$, so certainly
fixes $v_{2}$. But so does $G_{2}$, meaning that $A=\langle
G_{1},G_{2}\rangle$ does as well.

We now have a brief consideration of groups that are not finitely generated.
In our main theorem 8.4, if the initial metric space $X$ is itself a
simplicial tree $T$ and the quasi-action is an action by automorphisms on $T$
then it might seem that the conclusion would give nothing new. Recall that by
standard facts, if there exists a loxodromic element in this action then there
is a unique minimal $G$-invariant subtree $T_{G}$ of $T$. (If there is no
loxodromic element then either we have a bounded action, or a type (2) action
whereupon orbits cannot be coarse connected.) Now if $G$ is finitely generated
then (say by [9] Proposition I.4.13) the quotient graph $G\backslash T_{G}$ is
finite and so certainly this is a cobounded action. In this case we know the
orbits are coarse connected and this generalises to groups that need not be
finitely generated.

###### Proposition 10.4

Let $G$ be an arbitrary group acting by isometries on a simplicial tree $T$
and with a loxodromic element. Then the action of $G$ on the minimal invariant
subtree $T_{G}$ is cobounded if and only if the orbits are coarse connected.

Proof. If the orbits are not coarse connected then the action on $T$, on
$T_{G}$, or indeed on any $G$-invariant geodesic subspace of $T$ cannot be
cobounded by Lemma 3.15 (ii). Now take any vertex $v$ in $T_{G}$. The convex
closure of $Orb(v)$ must be equal to $T_{G}$ by minimality and Theorem 6.1
tells us that if $Orb(v)$ is coarse connected then it is coarse dense in its
convex closure $T_{G}$.
$\Box$
Note that for $G$ acting by automorphisms on a tree $T$, saying the action is
cobounded is the same as saying that the quotient graph $G\backslash T$ is a
bounded graph, even if this graph is not locally finite.

###### Corollary 10.5

Suppose that the group $G$ acts unboundedly on the locally finite simplicial
tree $T$ by isometries with coarse connected orbits. Then the minimal
invariant subtree has bounded valence.

Proof. We can assume that $G$ acts by automorphisms as otherwise
$T=\mathbb{R}$. There will be a loxodromic element so the action of $G$ on
$T_{G}$ is cobounded by Proposition 10.4. But then there are only finitely
many orbits of vertices, each of finite valence, so $T$ has bounded valence.
$\Box$

We finish this subsection by giving a counterexample to this if the group
(necessarily infinitely generated) does not have coarse connected orbits. We
will find a group $G$ with a type (2) action on a locally finite tree, but
where every action of $G$ on any bounded valence tree has a global fixed
point. This action will even be metrically proper and it is a variation on
Section 5 Example 2.

Example 4: Consider the restricted direct product $G=C_{2}\times C_{3}\times
C_{5}\times C_{7}\times\ldots$. By regarding $G$ as a direct union of the
increasing sequence of subgroups $C_{2},C_{2},\times C_{3},C_{2}\times
C_{3}\times C_{5},\ldots$ we can use the coset construction as before to
obtain a type (2) action of $G$ on a locally finite tree but with unbounded
valence. As all stabilisers are finite, this is a proper action.

Now suppose $T$ is any tree with valence bounded by $N$ and consider an
element $g$ acting on $T$ with $g$ having order coprime to the numbers
$2,3,4,\ldots,N$. Then $g$ must fix a vertex $v_{0}$ as $T$ is a tree but it
must then also fix the vertices adjacent to $v_{0}$ by Orbit - Stabiliser and
so on, thus $g$ acts as the identity. This means that if $p^{+}$ is the
smallest prime which is greater than $N$ and $p^{-}$ the largest prime less
than $N$ then any element in the subgroup $C_{p^{+}}\times\ldots$ acts
trivially on $T$. Thus the quotient of $G$ by this subgroup which is in the
kernel of the action leaves us only the finite group $H=C_{2}\times
C_{3}\times\ldots\times C_{p^{-}}$ and so any action of $G$ on such a $T$ will
have bounded orbits and hence a global fixed point.

###  10.2 Metrically proper actions

Throughout this paper we have been considering quasi-actions of arbitrary
groups with very weak conditions on the quasi-action, so this level of
generality does not allow us to identify the group from the quasi-action. For
instance in our earlier results, whenever a group $Q$ has a quasi-action
$\alpha$ satisfying our conditions and $Q=G/N$ is a quotient of some other
group $G$ then setting $\alpha^{\prime}(g,x)=\alpha(gN,x)$ results in the
quasi-action $\alpha^{\prime}$ of $G$ also satisfying these conditions.

However we will end by showing that if in addition the quasi-action is
metrically proper then this is enough. Moreover no finiteness condition is
needed here on the space.

###### Theorem 10.6

Suppose that an arbitrary group $G$ has a metrically proper quasi-action on
some metric space where a quasi-orbit quasi-isometrically embeds in some
simplicial tree.
$\bullet$ If the quasi-orbits are not coarse connected then $G$ is not
finitely generated.
$\bullet$ If the quasi-orbits are coarse connected then $G$ is a finitely
generated, virtually free group.

Proof. The division into the finitely and infinitely generated cases is
Proposition 3.19 (ii). Assuming therefore that quasi-orbits are coarse
connected, the quasi-orbits quasi-isometrically embed into trees by the
hypothesis and they are also proper metric spaces, because the metrically
proper condition implies that closed balls in any such quasi-orbit are finite
sets. Therefore the conditions for Theorem 8.4 apply. Consequently we obtain a
reduction of $\alpha$ to a (quasi-)action $\beta$ say, whereupon $\beta$ is
still metrically proper.

Running through the conclusions in turn, if $\beta$ is the trivial action on a
point but also metrically proper then $G$ must be finite.

If $\beta$ is a cobounded quasi-action on $\mathbb{R}$, so that quasi-orbits
are quasi-isometric to $\mathbb{R}$ then by Theorem 3.13 (ii) we can change
this into an equivalent isometric action on a geodesic metric space $Z$ that
is quasi-isometric to $\mathbb{R}$. This action of $G$ will be both cobounded
and metrically proper, so by Svarc - Milnor we have that the group $G$ is
quasi-isometric to $Z$ and hence to $\mathbb{R}$. Thus $G$ has two ends which
means that it is virtually cyclic (this result goes back to Hopf in [13] from
1944).

Otherwise $G$ acts coboundedly by isometries on a bounded valence bushy tree
and hence also by automorphisms. This action is equivalent to $\alpha$ and so
is metrically proper. Thus all stabilisers are finite. Coboundedness and
bounded valence mean that the quotient must be a finite graph, whereupon [21]
II.2.6 Proposition 11 states that the fundamental group of a finite graph of
groups with all vertex groups finite is virtually a free group.
$\Box$
Note that the above has the following consequence.

###### Corollary 10.7

If $G$ is a finitely generated group which quasi-isometrically embeds in some
simplicial tree $T$ then $G$ is virtually free.

Proof. First assume that $G$ is quasi-isometric to $T$. As $G$ acts on itself
by left multiplication, it quasi-acts on $T$ by Example 3.2. This quasi-action
is also metrically proper and cobounded, thus quasi-orbits are quasi-isometric
to $T$ (and certainly coarse connected) so Theorem 10.6 applies.

If $G$ quasi-isometrically embeds in $T$ then $G$ is certainly coarse
connected (with the word metric/Cayley graph from a finite generating set),
thus its image under this embedding is coarse connected and hence is also
quasi-isometric to some tree by Theorem 6.1, so $G$ is too.
$\Box$
This result is well known but would normally be proven using the Stallings
Theorem and Dunwoody’s Accessibility Theorem (see [10] Chapter 20 for a
thorough treatment). Our argument is not completely independent of these
approaches because the proof of Mosher, Sageev and Whyte’s Theorem 1 in 8.1
utilises Dunwoody tracks. Nevertheless we have not relied on either of these
big results to conclude that our group is virtually free. We have relied on
Hopf’s result in [13] to cover the two ended case, but this seems perfectly
reasonable given its vintage.

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Selwyn College, University of Cambridge, Cambridge CB3 9DQ, UK

E-mail address<EMAIL_ADDRESS>


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# Regimes of electronic transport in doped InAs nanowire.

A A Zhukov1 and I E Batov1,2 1 Institute of Solid State Physics, Russian
Academy of Science, Chernogolovka, 142432 Russia  2 Faculty of Physics,
National Research University Higher School of Economics, Moscow 101000, Russia
<EMAIL_ADDRESS>
###### Abstract

We report on the low temperature measurements of the magnetotransport in Si-
doped InAs quantum wire in the presence of a charged tip of an atomic force
microscope serving as a mobile gate, i.e. scanning gate microscopy (SGM). By
altering the carrier concentration with back gate voltage, we transfer the
wire through several transport regimes: from residual Coulomb blockade to
nonlinear resonance regime, followed by linear resonance regime and, finally,
to almost homogeneous diffusion regime. We demonstrate direct relations
between patterns measured with scanning gate microscopy and spectra of
universal conductance fluctuations. A clear sign of fractal behavior of
magnetoconductance dependence is observed for non-linear and linear resonance
transport regimes.

###### pacs:

73.23.Hk, 73.40.Gk, 73.63.Nm

##  1 Introduction

One of the peculiarities of one-dimensional and quasi-one-dimensional
diffusion electronic transport is the presence in these type of systems an
electron scattering from so-called resonance scatters. These scatters, such as
weak links or any other potential barriers, influence on electrons in all
channels. If these scatters are strong enough and temperature $T$ is low, the
transport can demonstrate Coulomb blockade [1] ($h/\tau,k_{B}T<e^{2}/c$) or
zero-bias anomaly [2], here $h$ is Planck’s constant, $k_{B}$ is Boltzmann’s
constant, $\tau$ is the tunneling time through barriers, $e$ is elementary
charge and $c$ is capacitance of the system section in between two nearest
weak links. In the case that potential barriers are weak, the transport is
linear but well-defined resonances in magnetoresistance can be observed due to
over-barrier reflections [3]. Finally, if the role of such barriers/scatters
is negligible, the homogeneous diffusive transport regime is realized [4].

In samples with the phase coherence length ($l_{\phi}$) comparable to the
sample size ($l_{wire}$ the universal conductance fluctuations (UCF) are
observed [5]. Using the correlator $F(\Delta B)=\left\langle\delta
R^{-1}(B)\delta R^{-1}(B+\Delta B)\right\rangle$, where $\delta
R^{-1}(B)=\left\langle R^{-1}\right\rangle-R^{-1}(B)$, it is possible to
extract the value of $l_{\phi}=\Phi_{0}/dB_{c}$ from the correlation field
$B_{c}$ defined as $F(B_{c})=0.5F(0)$ [6, 7], $\Phi_{0}=e/h$ is the quantum
flux and $d$ is a constant of the order of unity [7].

According to the work [8], the deviation of correlation function $\Delta
F(\Delta B)=F(0)-F(\Delta B)\propto\Delta B^{\gamma}$, and the value of
exponent $\gamma$ defines the dimension of the fractional Brownian motion
$D=2-\gamma/2$ [9, 10, 11] of magnetoconductance dependence. Fractional
Brownian motion has been found in quasiballistic Au-nanowires [12], different
types of semiconductor nanowires [13], and graphene stripes [14].

Here, we present a detailed investigation of magnetotransport in Si-doped InAs
nanowire using SGM mapping and standard magnetotransport measurements in a
wide range of carrier concentrations. The system passed through four transport
regimes from residual Coulomb blockade to non-linear resonance scattering,
then to linear resonance scattering and finally to quasi-homogenious diffusion
regime. The role of resonant scatters in formation of fractional Brownian
motion of magnetoconductance dependence is discussed.

##  2 Experimental

The nanowires were grown on GaAs (111)B substrates by low-pressure metal
organic vapor phase epitaxy (MOVPE). Nitrogen gas (N2) was used as the carrier
gas to transport trimethylindium (TMIn) and arsine (AsH3) in the reactor at a
working pressure of 20 mbar and a total gas flow rate of 3100 ml/min. The
growth temperature was $650^{\circ}$C. For silicon doping during growth, the
disilane (Si2H6) flux was adjusted to achieve various n-type doping levels. To
quantify the supply of doping species more easily, doping factor, consisting
of the partial pressure ratio of dopant versus group III precursor, is defined
as a ratio of p(Si2H6)/p(TMIn). The ratio of $7.5\times 10^{-4}$ was used for
growth the batch of the nanowires investigated in the current experiment.
Additional details of the Si-doped nanowires growth procedure can be found
elsewhere [15, 16].

The diameter of the wire was $d_{wire}=100$ nm. The wire was placed on an
$n$-type doped Si (100) substrate covered by a thermal grown 100 nm thick SiO2
insulating layer. The Si substrate served as the back-gate electrode. The
Ti/Au contacts to the wire as well as the markers of the search pattern were
defined by electron-beam lithography. The distance between the contacts was
$l_{wire}=2.9\mu$m. A scanning electron beam micrograph of the sample is shown
in Fig. 1a). The source and drain metallic electrodes connected to the wire
are marked as S and D.

All measurements were performed at $T=\,4.2$ K. The charged tungsten tip of a
home-built scanning probe microscope [17] was used as a mobile gate during
scanning gate imaging measurements, see Fig. 1b). All scanning gate
measurements were performed by keeping the potential of the scanning probe
microscope tip ($V_{t}$=0 V) as well as the back-gate voltage ($V_{BG}$)
constant. The conductance of the wire during the scan was measured in a two-
terminal circuit by using a standard lock-in technique. The tip to SiO2
surface distance chosen for the scanning process was $h_{tip}=250$ nm. During
SGM scans and linear magnetotransport measurements a driving AC current with
an amplitude of $I_{AC}=0.8$ nA at a frequency of 231 Hz was applied while the
voltage was measured by a differential amplifier. External magnetic field was
directed perpendicular to the wire axis and Si substrate surface.

##  3 Experimental results

In Fig. 2 the dependence of conductance ($R^{-1}$) of the nanowire as a
function of back gate voltage is presented. The overall linear dependence of
conductance on back gate voltage (carrier density) is typical for Si-doped
InAs nanowires [15]. Non-regular fluctuations are universal conductance
fluctuations (UCF) which arise because the phase coherence length in InAs is
comparable to the length of the wire. The magnified low back gate section of
$R^{-1}(V_{BG})$ curve is shown in the top-left inset of Fig. 2. Additionally
to the UCF, there are oscillations with period of $\Delta V_{BG}=36$ mV,
marked with blue oval. These oscillations come from the residual Coulomb
blockade of a quantum dot positioned in the mid of the nanowire [18].

According to finite element calculations of the capacitance of the nanowire
[19], the specific capacitance of our sample is $c_{s}\simeq 80$pF/m. Thus, it
is possible to calculate the carrier density of the nanowire as a function of
the back gate voltage $n_{e}=c_{s}(V_{BG}-V_{Th})/\pi e(d_{wire}/2)^{2}$, here
$V_{Th}=-5.2$ V is the threshold voltage, and the elastic mean free path
$l_{e}=(hl_{wire}/2\pi^{2}e^{2}R(d_{wire}/2)^{2})(2\pi^{2}/n_{e}^{2})^{1/3}$.
The low-right inset in Fig. 2 shows the dependencies of $l_{e}$ (black curve,
left scale) and $l_{e}k_{F}$ (blue curve, right scale) on back-gate voltage,
here $k_{F}=(3\pi^{2}n_{e})^{1/3}$ is the Fermi vector. Two green arrows point
to the $l_{e}$ curve at $V_{BG}=-1$ V and 1 V. It is worth noting that the
variations of $l_{e}$ and $l_{e}k_{F}$ as $V_{BG}$ changes from -1 V to 1 V is
just 25-30%.

The results of SGM mapping of the nanowire are presented in Figs. 3 – 8. Each
figure demonstrates the evolution of the scanning results due to weak
variation of the back gate voltage with a step of 10 mV (Figures a) to f)),
and applied external magnetic field ($B=0.1$ T, 0.2 T, 0.3 T, 0.5 T, 1 T, and
2 T, Figures g) to l)) for six back gate voltages, i.e. $V_{BG}=-3$ V, $-2.1$
V, $-1$ V, 1 V, 4 V, and 10 V, respectively. The step of $V_{BG}$ of 10 mV was
chosen to exceed correlation back gate voltage $V_{cBG}=hD/el_{\phi}^{2}<5$ mV
for all applied $V_{GB}$, here $D$ is the diffusion coefficient.

Mapping results of the SGM scanning in the vicinity of $V_{BG}=-3$ V presented
in Fig. 3 demonstrate quite complex structure typical for nanowires or
nanotubes with a number of the blocking barriers of different opacity [20].
The key feature of scans in Fig. 3 is well defined concentric ovals, see Figs.
3f) and 3l) for example. The presence and the position of such ovals means the
formation of the quantum dot in the mid-section of the nanowire and ovals are
the result of the Coulomb blockade realized in this dot [20]. The period of
Coulomb blockade oscillations of $\Delta V_{BG}=36$ mV obtained from
$R^{-1}(V_{BG})$ curve allows to estimate the length of electronic system of
this quantum dot $l_{dot}\simeq e/c_{s}\Delta V_{BG}\simeq 60$ nm. Taking into
account the lengths of depletion zones of around 100 nm, the distance between
two strong blocking barriers forming this dot can be estimated as
$l_{BB}\simeq 250$ nm. Thus, this quantum dot is positioned approximately in
the center of the wire with barrier-to-barrier distance $l_{BB}\ll l_{wire}$.
It means that the InAs nanowire is divided into three sections. The Section I
extends from the source contact to the first blocking barrier, the Section II
is the quantum dot itself and the Section III lays between the second blocking
barrier and the drain contact.

The scanning gate microscopy mapping performed in the vicinity of
$V_{BG}=-2.1$ V is shown in Fig. 4. At this back gate voltage the SGM mapping
results look still quite complex, see Fig. 4a), but they demonstrate a
response from all three sections of the nanowire. Here equicapacitance ovals
and circles are not resulted from the Coulomb blockade, because the shape of
them changes dramatically with application of external magnetic field, see
Figs. 4g) – 4l). The equicapacitance ovals and circles in SGM data come from
the alteration of the opacity of the blocking barriers or, probably, due to
the variation of the density of states [2].

Fig. 5 shows the scanning gate microscopy mapping carried out in the vicinity
of $V_{BG}=-1$ V. No well-defined response from each of the Sections can be
resolved, but the mapping results are extremely stable against the back gate
voltage variation and application of magnetic field up to $B=0.3$ T, see Figs.
5a) – 5h). Higher magnetic fields from $B=0.5$ T to 2 T deform the SGM mapping
results, similar to Fig. 4, see Figs. 5j) – 5l).

The scanning gate microscopy mapping done in the vicinity of $V_{BG}=1$ V is
presented in Fig. 6. Both the variation of back gate voltage and the magnetic
field as small as 0.1 T alter the SGM mapping pictures. The smallest scale
features of SGM scans have the size of 250 – 300 nm. This size is comparable
to the tip to SiO2 surface distance and it is the spatial resolution of the
current experimental setup [21]. Three well-defined resistance minima are
visible in scans and marked with arrows in Fig. 6c). As it has been estimated
previously, the distance between two barriers forming the central quantum dot
is 250 nm and each of the barrier cannot be resolved separately, so they are
visualized as a single one. Thus, these three minima are related to the source
contact barrier, quantum dot double barrier, and the drain contact barrier.
The positions of them are stable against variation of $V_{BG}$ (Figs. 6a) –
6f)). The double barrier sign (the red spot marked with arrow) is visible even
at SGM scan done at $B=2$ T, see Fig. 6l). The scans shown in Fig. 6
additionally confirm the position of the quantum dot allocated from SGM
mapping, see Fig. 4.

The SGM mapping done in the vicinity of $V_{BG}=4$ V and 10 V is shown in Fig.
7 and 8, respectively. These scans demonstrate the interplay of the features
come from UCF which vary their positions and the residual impact of the
blocking barriers on the wire conductivity. It is worth noting that some minor
influence of these barriers in SGM scans remains both at $V_{BG}=4$ V (Fig.
7i)) and at $V_{BG}=10$ V (Fig. 8a)). This behavior of blocking barrier
influence on SGM mapping is the same as observed in InN nanowires previously
[3].

Figure 9 shows the dependence of the wire conductance ($R^{-1}$) on magnetic
field $B$ ($-1$ T$\leq B\leq 7$ T) measured at $V_{BG}=-3$ V, $-2$ V, $-1$ V,
1 V, 4 V, and 10 V, i.e almost at the same values of the back gate voltages as
for SGM scans. Curves, except one measured at $V_{BG}=$10 V, are shifted for
clarity. Non-regular reproducible oscillations are UCF. The peak in
conductivity around $B=0$ T for $V_{BG}=$10 V comes from weak antilocalization
quantum correction due to spin-orbit interaction in InAs and it transforms to
the localization dip for $V_{BG}\leq-2$ V. This transition from weak
anilocalization to localization with decreasing the carrier concentration is
typical for InAs nanowires [22, 23, 24, 25, 26, 27].

The top panel of Fig. 10 shows universal conductance fluctuations spectra
calculated from $R^{-1}(B)$ data measured at $V_{BG}=-3$ V, $-2$ V, and $-1$
V. The bottom panel of Fig. 10 presents the spectra measured at $V_{BG}=$1 V,
4 V, and 10 V. The essential suppression of the fluctuation spectra for
$1/B>1$ (T-1) for top panel data in comparison to bottom panel data is clearly
visible.

The normalized correlator dependencies $F(\Delta B)/F(0)$ calculated from
magnetoconductance data obtained at $V_{BG}=-3$ V, $-2$ V, $-1$ V, 1 V, 4 V,
and 10 V are integrated in Fig. 11. The smallest value of the correlation
field $B_{c}=0.11$ T extracted from data measured at $V_{BG}=$10 V gives
$l_{\phi}=200$ nm. This value of the phase correlation length is within the
range from 200 nm to 500 nm of previously obtained values [22, 23, 24, 25, 26,
27]. The significant increase of $l_{\phi}$ value at higher carrier density
has been obtained previously as well [22, 23, 24]. The correlator $F(\Delta
B)/F(0)$ is calculated from $R^{-1}(V_{BG})$ data measured in magnetic fields
from $B=0.65$ T to 7 T to eliminate any influence of low-field quantum
corrections.

The dependencies of deviation of the correlator $\Delta F(\Delta B)=F(\Delta
B)-F(0)$ in double logarithmic scale for $V_{BG}=-3$ V, $-2$ V, $-1$ V, 1 V, 4
V, and 10 V are shown in Fig. 12. Two straight lines show the slopes of
$\Delta B^{\gamma}$ dependence for two exponent values $\gamma=1.5$ (red) and
$\gamma=2.0$ (black).

The normalized differential resistance $(dV/dI)/(dV/dI(0))$ as a function of
driving current measured at $V_{BG}=-3$ V, $-2$ V, $-1$ V, and 1 V are shown
in Fig. 13. The dots on the curves mark values of current where
$eV_{SD}=k_{B}T$. The well-defined non-linear behavior is seen for $V_{BG}=-3$
V and $-2$ V only. This experiment is done at $B=0.65$ T to eliminate the
quantum corrections influence.

##  4 Discussion

Transitions from Coulomb blockade to Fabri-Pérot interference in ballistic
one-dimensional and quasi-one-dimensional systems were recently observed and
discussed in detail in [30, 31]. In inhomogeneous diffusive quasi-one-
dimensional systems, the transport phenomena are more complex. Presented data
help to illustrate all relevant transport regimes focusing on their
peculiarities in magnetotransport data and the SGM mapping results obtained in
the same run.

The doped InAs nanowire is chosen because of its homogeneous radial carrier
density [15]. Thus, any influence on the transport resulted from cylindrical
shape of the electronic system [32] is minor.

As it has been mentioned previously, the dependence $R^{-1}(V_{BG})$ (Fig. 2)
and SGM data (Fig. 3) obtained at $V_{BG}=-3$ V allow to estimate the position
and the size of the quantum dot formed in the mid-section of the wire. The
spectrum of the UCF (see Fig. 10, top panel) confirms the statement that the
dominating role belongs to the small area loops ($90\times 50$ nm2). The
transport at this value of the back gate voltage is non-linear, see Fig. 13.

By increasing the carrier density concentration ($V_{BG}=-2$ V), we transfer
the system to non-linear resonant regime (see Fig. 13). In this regime, the
strongest resonant scatters (blocking barriers) define three quasi-localized
states formed in corresponding Sections of wire. Maps obtained with the SGM
technique depend weakly on variation of the back gate voltage and application
of the small magnetic field, see Fig. 3. New energy scale of the order of 100
meV is the characteristic energy of variation of opacity of the blocking
barriers and fluctuation of the density of states. The contribution from small
area loops is dominant in the UCF spectra (Fig. 10, top panel).

Further increase of the carrier concentration ($V_{BG}=-1$ V) gives rise to
the reduced influence of the blocking barriers. Transport becomes linear, see
Fig. 13. No well-defined patterns allocating three Sections of the nanowire
can be observed in SGM scans, but the resonant reflections of electrons still
play an important role altering the local carrier density of states and ruling
the weak dependence of the SGM mapping results on the back gate voltage and
the magnetic field (Fig. 4), similarly to previously described non-linear
resonant regime. Contribution from small area loops is still dominant in the
UCF spectrum (Fig. 10, top panel). The size of the area of these loops allows
to estimate the maximum characteristic length of segments into which the
nanowire is divided $l_{seg}\sim 50-100$ nm .

At $V_{BG}=1$ V we observe important changes in the SGM scans. The SGM mapping
results (Fig. 5) become sensitive to the small (10 mV) increase of the back
gate voltage as well as to application of the small ($B=0.1$ T) external
magnetic field. This situation is similar to the one obtained previously in
undoped InAs nanowires [33] and can be considered as diffusive regime with
diminished influence of resonant scatters. Additionally, the spectrum of the
UCF extends to the higher frequencies, thus the larger area loops start to
come into play (Fig. 10, bottom panel). Transport is linear as indiated in
Fig. 13.

Further increasing the back gate voltage results in even more homogeneous
transport according to SGM data (Figs. 6 and 7) with wide spectra of UCF (Fig.
10, bottom panel). At $V_{BG}=10$ V correlation field $B_{c}$ has reached the
value related to the phase coherence length of 200 nm, finally.

The transiton from linear resonant regime to more homogenous diffusive one
occurs at back gate voltages between $V_{BG}=-1$ V and 1 V (see two arrows in
the bottom-right inset of Fig. 2). As it has been mentioned previously, no
essential variations of $l_{e}$ or $k_{F}l_{e}$ happen in this range of back
gate voltages. Therefore, it is not possible to find any sign of this
transition just from $R^{-1}(V_{BG})$ dependence. But the correlated switch of
the behavior of SGM mapping results, as well as sudden increase of the high
frequency spectrum of UCF can be considered as an undoubtful evidence of this
transition.

The most exciting feature of the obtained data is non-trivial behavior of the
exponent $\gamma$ in $\Delta F(\Delta B)\propto\Delta B^{\gamma}$ dependencies
obtained for different back gate voltages, see Fig. 13. This exponent is
$\gamma=2.1\pm 0.1$ for Coulomb blockade regime when the transport is defined
mostly by quantum dot. Then, the value of exponent drops to values close to
$\gamma=1.4$ in non-linear and linear resonant scattering regimes, see Table
1. The transport depends strongly on resonant scatters in the wire forming a
set of quasilocalized states resulting in nonhomogeneous fractional Bownian
motion of magnetoconductance dependence with dimensionality greater than one
($Dim=2-\gamma/2=1.3$) [8, 12] for both non-linear and linear resonant
regimes. The reason of weak variation of $\gamma$ in this regime is presently
not clear and might be the subject for further investigations. At $V_{BG}\geq
1$ V when the role of the resonance scatters diminishes, the value of $\gamma$
starts to rise toward to the standard value of 2, see Table. 1.

Table 1: The dependence of the correlation field $B_{c}$ and $\gamma$ on back gate voltage. $V_{BG}(V)$ |  $B_{c}$(T) | $\gamma$
---|---|---
-3 | 0.48 | 2.1$\pm 0.1$
-2 | 0.25 | 1.38$\pm 0.1$
-1 | 0.27 | 1.42$\pm 0.1$
1 | 0.21 | 1.78$\pm 0.1$
4 | 0.21 | 1.71$\pm 0.1$
10 | 0.11 | 1.90$\pm 0.1$

According to the underlying paper by Altshuler, Gefen, Kamenev and Levitov
(AGKL), there are three regimes of many-body localization [34, 35]. The first
one is realized when the energy $k_{B}T$ is less than the characteristic
energy of interacting quantum dot $E_{QD}$ ($E_{QD}(V_{BG}=-3$ V$)=e\Delta
V_{BG}\sqrt{g/ln(g)}\sim 10$ meV, and it is the Coulomb blockade regime in
this paper). In this regime, one-particle states are very similar to the exact
many-body states. Here $g=h/e^{2}R_{QD}\sim 1$ is the normalized conductance
of the central quantum dot. In the intermediate regime when
$k_{B}T<E_{Th,seg}\sim 1.7$ meV at $V_{BG}=-1$ V, the quasiparticle states are
delocalized, but they are fractal and non-ergodic [34, 35]. Here
$E_{Th,seg}=hD/l_{seg}^{2}$ is the Thouless energy of the typical largest
segment of the wire formed by resonant scatters.

Fractal structure of wave function of the electronic system and overall non-
ergodic behavior of this regime correlates well with in the current experiment
at non-linear and linear resonant regimes demonstrating fractional Bownian
motion of magnetoconductance behavior. It is worth noting that the value of
Fermi length is $\lambda_{F}\simeq 37$ nm at $V_{BG}=-1$ V, this means that
the number of channels in the nanowire is $N_{channel}\simeq\pi
d_{wire}^{2}/\lambda_{F}^{2}\simeq 24$, and the localization length is
$l_{loc}\simeq N_{channel}l_{e}\simeq 240$ nm. This value is comparable to
$l_{seg}$ and to the actual phase-coherence length for $V_{BG}=-1$ V, i.e.
$l_{\phi}(V_{BG}=-1$V$)=[(l_{e}(V_{BG}=-1$V$)/l_{e}(V_{BG}=10$V$)]^{0.5}\cdot
200$ nm$=160$ nm. Thus, the many body localization is possible in each segment
($l_{\phi}(V_{BG}=-1$ V$)\simeq l_{loc}$). Finally, when $k_{B}T$ is the
largest energy scale ($k_{B}T>E_{Th,wire}\sim 3\mu$eV), a simple exponential
decay of the quasiparticle life time is set again, here $E_{Th,wire}$ is the
Thouless energy of the whole wire. This regime corresponds to the homogenous
diffusive transport regime in the current work.

Thus, as in AGKL theoretical picture so in the current experiment there is a
special regime with non-trivial behavior of the electronic system wave
function or fractal Bownian motion of magnetoconductance curve lying in
between Coulomb blockade and homogenous diffusion transport regimes.

There are quite a number of theoretical papers in which quasi-localized states
formed by resonant scatters are considered as a possible origin of formation
of the observed fractional Bownian motion of magnetoconductance dependence in
the intermediate regime [36, 37, 38]. However, there have been no clear
experimental evidences or visualizations of such kind of states. Here we show
as the SGM mapping scans visualize them and hence the fractional Bownian
motion they initiate.

##  5 Conclusion

We performed the low temperature measurements of the magnetotransport in doped
InAs nanowire in the presence of a charged tip of an atomic force microscope
serving as a mobile gate. By varying the carrier concentration, the system
under investigation passes through four transport regimes from Coulomb
blockade to homogeneous diffusive transport. Fractional Bownian motion of
magnetoconductance dependence is observed in non-linear and linear resonant
transport regimes. Two key experiments for demonstration of transition from
the linear resonant transport regime to the diffusive one are presented. The
experimental results are in general agreement with the theoretical model
proposed by AGKL.

##  6 Acknowledgements

The authors would like to thank Chrisian Blömers for samples preparation and
Raffaella Calarco for nanowire growth. This work is supported by the RSF grant
23-22-00141, https://rscf.ru/project/23-22-00141/.

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Figure 1: (color online) a) False color scanning electron microscope image of
the InAs nanowire. The scale bar corresponds to 1 $\mu$m. b) Schematic diagram
of the experimental setup. In the Figures, the nanowire has a blue color and
two metallic contact pads are yellow, source and drain contacts are labelled S
and D, respectively. Figure 2: (color online) Conductance $R^{-1}$ of the InAs
nanowire versus back gate voltage ($V_{BG}$). Non-regular oscillations come
from universal conductance fluctuations. The upper-left insert shows the
dependence of the conductance on the back gate voltage in the range indicated
by the dashed frame in the Figure. Small period oscillations within the range
marked with blue circle come from residual Coulomb blockade of the quantum dot
positioned in the mid part of the nanowire (see main text for details). The
lower-right insert presents the elastic scattering length as a function of the
back gate voltage (black curve, left scale). Blue curve in the insert shows
the dependence of $k_{F}l_{e}$ on the back gate voltage (right scale). Two
green arrows point to $V_{BG}=-1$ V and $V_{BG}=1$ V. Figure 3: (color online)
a)-f) Scanning gate microscopy maps of InAs nanowire carried out at zero
magnetic field and in range of back gate voltages varied from $V_{BG}=-3.00$ V
to $-2.95$ V with step of 10 mV, respectively. g)-l) Scanning gate microscopy
maps gained at $V_{BG}=-3.0$ V and $B=0.1$ T, 0.2 T, 0.3 T, 0.5 T, 1.0 T, and
2.0 T, respectively. Concentric circles are the sign of the Coulomb blockade
in quantum dot positioned at the mid part of the nanowire. The scale bar in
Figure a) corresponds to 1 $\mu$m. The scale and the area of scanning gate
microscopy maps are the same for all scans. The black solid lines mark the
borders of contact pads and the dotted line marks the axis of nanowire.
Figure 4: (color online) a)-f) Scanning gate microscopy maps of InAs nanowire
obtained at zero magnetic field and in range of back gate voltages varied from
$V_{BG}=-2.10$ V to $-2.15$ V with step of 10 mV, respectively. g)-l) Scanning
gate microscopy maps gained at $V_{BG}=-2.10$ V and $B=0.1$ T, 0.2 T, 0.3 T,
0.5 T, 1.0 T, and 2.0 T, respectively. The scale bar in Figure 4a) corresponds
to 1 $\mu$m. The scale and the area of scanning gate microscopy maps are the
same for all scans. The black solid lines mark the borders of contact pads and
the dotted line marks the axis of nanowire. Ovals and circles of
equicapacitance for all sections of the nanowire are shown in the Figure a).
Two yellow ovals relate to the Sections I and III, two green circles mark
equicapacitance curve for central quantum dot (Section II). Figure 5: (color
online) a)-f) Scanning gate microscopy maps of InAs nanowire made at zero
magnetic field and in range of back gate voltages varied from $V_{BG}=-1.00$ V
to $-1.05$ V with step of 10 mV, respectively. g)-l) Scanning gate microscopy
maps carried out at $V_{BG}=-1.00$ V and $B=0.1$ T, 0.2 T, 0.3 T, 0.5 T, 1.0
T, and 2.0 T, respectively. The scale bar in Figure a) corresponds to 1
$\mu$m. The scale and the area of scanning gate microscopy maps are the same
for all scans. The black solid lines mark the borders of contact pads and the
dotted line marks the axis of nanowire. Figure 6: (color online) a)-f)
Scanning gate microscopy maps of InAs nanowire obtained at zero magnetic field
and in range of back gate voltages varied from $V_{BG}=1.00$ V to $1.05$ V
with step of 10 mV, respectively. g)-l) Scanning gate microscopy maps gained
at $V_{BG}=1.00$ V and $B=0.1$ T, 0.2 T, 0.3 T, 0.5 T, 1.0 T, and 2.0 T,
respectively. The scale bar in Fig. 6a) corresponds to 1 $\mu$m. The scale and
the area of scanning gate microscopy maps are the same for all scans. The
black solid lines mark the borders of contact pads and the dotted line marks
the axis of nanowire. Arrows in Figure c) point to the positions of all
potential barriers. The arrow in Figure l) points to the position of the
central double barrier.  Figure 7: (color online) a)-f) Scanning gate
microscopy maps of InAs nanowire carried out at zero magnetic field and in
range of back gate voltages varied from $V_{BG}=4.00$ V to $4.05$ V with step
of 10 mV, respectively g)-l) Scanning gate microscopy maps performed at
$V_{BG}=4.00$ V and $B=0.1$ T, 0.2 T, 0.3 T, 0.5 T, 1.0 T, and 2.0 T,
respectively. The scale bar in Figure a) corresponds to 1 $\mu$m. The scale
and the area of scanning gate microscopy maps are the same for all scans. Tbe
black solid lines mark the borders of contact pads and the dotted line marks
the axis of nanowire. The arrow in Fig. i) points to the position of the
central double barrier. Figure 8: (color online) a)-f) Scanning gate
microscopy maps of InAs nanowire obtained at zero magnetic field and in range
of back gate voltages varied from $V_{BG}=10.00$ V to $9.95$ V with step of 10
mV, respectively. g)-l) Scanning gate microscopy maps carried out at
$V_{BG}=10.00$ V and $B=0.1$ T, 0.2 T, 0.3 T, 0.5 T, 1.0 T, and 2.0 T,
respectively. The scale bar in Figure a) corresponds to 1 $\mu$m. The scale
and the area of scanning gate microscopy maps are the same for all scans. The
black solid lines mark the borders of contact pads and the dotted line marks
the axis of nanowire. The arrow in Figure a) marks the position of the double
barrier.  Figure 9: (color online) Magnetic field dependencies of the
conductance of the nanowire ($R^{-1}$) measured in the range of magnetic
fields from -1 T to 7 T and at different back gate voltages as indicated in
Figure.  Figure 10: (color online) Top panel is power spectra of
magnetoconductance for $V_{BG}=-3$ V, -2 V, and -1 V, see Fig. 9. Bottom panel
is power spectra of magnetoconductance for $V_{BG}=1$ V, 4 V, and 10 V, see
Fig. 9. The significant suppression of all spectra for $1/B>1$ T-1 presented
in the top panel is clearly visible.  Figure 11: (color online) Normalized
correlation functions $F(\Delta B)/F(0)$ calculated from magnetoconductance
dependencies presented in Fig. 9 for six different back gate voltages.  Figure
12: (color online) Deviation of the correlation function for six different
back gate voltages. The black and the red straight lines show slopes related
to $\gamma=2.0$ and 1.5, respectively. The arrow points at $B_{c}$ for
$V_{BG}=10$ V .  Figure 13: (color online) a) Normalized differential
resistance ((dV/dI)/(dV/dI(0))) as a function of the driving current
($I_{SD}$) measured at $V_{BG}=-3$ V, -2 V, -1 V, and 1 V. The dot in each
curve points to the value of the bias current where $eU_{SD}=k_{B}T$.
Measurements were done at $B=0.65$ T.