extension of $\mathbb{Z}^{5}$ over the trivial subgroup. Following [12], with
this relatively hyperbolic structure the Bowditch boundary consists of the set
of infinite words with normal form for the HNN-extension structure glued
together with one point at infinity for every left coset of $\mathbb{Z}^{5}$.
In particular, as $\mathbb{Z}^{5}$ is one-ended, we see that the Bowditch
boundary coincides with the set of ends. Thus, by [47] the Bowditch boundary
is homeomorphic to the Cantor set and is in particular totally disconnected.

Now, on the one hand, the measure $\mu$ is spectrally degenerate along
$\mathbb{Z}^{5}$, so by [33, Theorem 1.2] the $R$-Martin boundary is
homeomorphic to the Bowditch boundary, hence it is totally disconnected. On
the other hand, the measure $\mu^{\prime}$ is spectrally non-degenerate, so
[33, Theorem 1.2] shows that the $R$-Martin boundary contains embedded
4-spheres. Therefore, the $R$-Martin boundaries for $\mu$ and $\mu^{\prime}$
cannot be homeomorphic.

Next, we show that the ratio-limit boundaries generally depend on the random
walk.

######  Example 7.3.

Let $\mu$ and $\mu^{\prime}$ be probability measures on
$\Gamma=\mathbb{Z}^{5}*\mathbb{Z}$ as in Example 7.2 so that $\mu$ is
convergent and spectrally degenerate along $\mathbb{Z}^{5}$ and $\mu^{\prime}$
is spectrally non-degenerate. Then the ratio-limit boundaries
$\partial_{\rho}(\Gamma,\mu)$ and $\partial_{\rho}(\Gamma,\mu^{\prime})$
associated with $\mu$ and $\mu^{\prime}$ are not $\Gamma$-equivariantly
homeomorphic.

Indeed, by Proposition 6.1 and Proposition 7.1 for $\mu$ and $\mu^{\prime}$
respectively, we know that

$\frac{H(x,y)}{K_{R}(x,y)}\to 1$

as $y$ converges to a conical limit point. Now, as in the proof of Corollary
6.2, we deduce for both $\mu$ and $\mu^{\prime}$ that there is a bi-Lipschitz
$\Gamma$-equivariant map

$\iota:\partial_{M,R}\Gamma\rightarrow\partial_{\rho}\Gamma$

which is a homeomorphism onto its closed image, such that for every conical
limit point $\xi$ we have

$H(x,\iota(\xi))=K_{R}(x,\xi).$

The proof of Theorem 6.4 can then be applied to show that the $R$-Martin
boundary (which coincides with the closure of minimal points in the $R$-Martin
boundary) is the unique smallest closed $\Gamma$-invariant subspace of the
ratio limit boundary. Thus, if the ratio-limit boundaries were
$\Gamma$-equivariantly homeomorphic, it would follow that so are their unique
smallest closed $\Gamma$-invariant subspaces, in contradiction to Example 7.2.

##  8\. Co-universal quotients of Toeplitz C*-algebras.

Our goal in this final section is to identify a unique smallest equivariant
quotient of Toeplitz C*-algebra of a random walk, when $\Gamma$ is relatively
hyperbolic, and $\mu$ is a finitely supported, admissible, aperiodic and
symmetric probability measure on $\Gamma$ which is spectrally non-degenerate.

Throughout this section, we will denote Hilbert spaces by $\mathcal{H}$, and
by $\mathbb{B}(\mathcal{H})$ and $\mathbb{K}(\mathcal{H})$ the C*-algebras of
all bounded and compact operators on $\mathcal{H}$, respectively. When
referring to an ideal $\mathcal{I}\lhd\mathcal{A}$ in a C*-algebra
$\mathcal{A}$, we will always mean a two-sided norm-closed ideal.

The following notion of co-universality is the one that fits our context. It
is similar in spirit to other such notions that appear in the literature [16,
24, 62], particularly [25, Definition 4.7], but it does not require
injectivity on some specified subalgebra.

Recall that if $\mathcal{A}$ be a C*-algebra, a $G$-action
$\alpha:G\curvearrowright\mathcal{A}$ is a homomorphism
$G\rightarrow\operatorname{Aut}(\mathcal{A})$ such that $\alpha_{g}$ is a
$*$-automorphism, and for every element $a\in\mathcal{A}$ the function
$g\mapsto\alpha_{g}(a)$ is norm continuous. When $\mathcal{I}\lhd\mathcal{A}$
is a $G$-invariant ideal, we get an induced action
$\alpha^{\mathcal{I}}:G\curvearrowright\mathcal{A}/\mathcal{I}$ given by
$\alpha_{g}^{\mathcal{I}}(a+\mathcal{I})=\alpha_{g}(a)+\mathcal{I}$, which is
still point-norm continuous.

######  Definition 8.1.

Let $\mathcal{A}$ be a C*-algebra, and $G$ a locally compact Hausdorff group.
Suppose that $\alpha:G\curvearrowright\mathcal{A}$ is an action by $G$. Let
$\mathcal{I}$ be an $G$-invariant ideal in $\mathcal{A}$. We say that
$\mathcal{C}:=\mathcal{A}/\mathcal{I}$ is _$G$ co-universal_ if for every
$*$-representation $\pi:\mathcal{A}\rightarrow\mathbb{B}(\mathcal{H})$ such
that

  1. (1)

there is a group action $\beta:G\curvearrowright\mathcal{B}$, where
$\mathcal{B}:=\pi(\mathcal{A})$ and;

  2. (2)

$\pi:\mathcal{A}\rightarrow\mathcal{B}$ is $G$-equivariant,

there is a $G$-equivariant surjective $*$-homomorphism
$\tau_{\pi}:\mathcal{B}\rightarrow\mathcal{C}$.

The following shows that the above notion of co-universality coincides with
the existence of a unique smallest $G$-equivariant quotient of $\mathcal{A}$.

######  Proposition 8.2.

Let $\mathcal{A}$ be a C*-algebra, and $G$ a locally compact Hausdorff group,
with an action $\alpha:G\curvearrowright\mathcal{A}$. Let $\mathcal{I}$ be a
$G$-invariant ideal. Then the quotient $\mathcal{C}:=\mathcal{A}/\mathcal{I}$
is $G$ co-universal if and only if $\mathcal{I}$ is the unique largest
$G$-invariant ideal of $\mathcal{A}$.

###### Proof.

Suppose first that $\mathcal{C}$ is $G$ co-universal, and let
$\mathcal{J}\lhd\mathcal{A}$ be some $G$-invariant ideal. Then the surjective
$*$-homomorphism
$q_{\mathcal{J}}:\mathcal{A}\rightarrow\mathcal{A}/\mathcal{J}$ is
$G$-equivariant with the induced action
$\alpha^{\mathcal{J}}:G\curvearrowright\mathcal{A}/\mathcal{J}$ on the image,
and by Gelfand-Naimark-Segel theorem we may embed $\mathcal{A}/\mathcal{J}$ as
a norm-closed subalgebra of $\mathcal{B}(\mathcal{H})$ for some Hilbert space
$\mathcal{H}$, making $q_{\mathcal{J}}$ into a genuine $*$-representation. By
$G$ co-universality of $\mathcal{C}$ we have a $G$-equivariant surjective
$*$-homomorphism
$\tau_{\mathcal{J}}:\mathcal{A}/\mathcal{J}\rightarrow\mathcal{A}/\mathcal{I}$,
which implies that $\mathcal{J}\subseteq\mathcal{I}$. Hence, $\mathcal{I}$ is
the unique largest $G$-invariant ideal of $\mathcal{A}$.

Conversely, if $\mathcal{I}$ is the largest $G$-invariant ideal of
$\mathcal{A}$, and $\pi:\mathcal{A}\rightarrow\mathbb{B}(\mathcal{H})$ a
$G$-equivariant $*$-representation such that $\mathcal{B}:=\pi(\mathcal{A})$
admits a $G$-action $\beta:G\curvearrowright\mathcal{B}$, then the kernel
$\ker\pi$ is a $G$-invariant ideal, and hence $\ker\pi\lhd\mathcal{I}$. By
applying the first isomorphism theorem for C*-algebras to $\pi$, by
composition we obtain a $G$-equivariant map
$\tau_{\pi}:\mathcal{B}\cong\mathcal{A}/\ker\pi\rightarrow\mathcal{A}/\mathcal{I}$.
∎

Thus, in order to show that with respect to a group action
$G\curvearrowright\mathcal{A}$ a $G$ co-universal quotient exists, it will
suffice to show that there is a _unique_ smallest $G$-equivariant quotient
$\mathcal{C}$. The following C*-algebras associated with Markov chains were
studied by the first author and his collaborators in a sequence of papers [26,
27, 19, 23]. These works form part of a larger program to determine the
structure of natural operator algebras arising from subproduct systems,
pioneered by Shalit, Solel and Viselter [63, 66, 67].

Let $P$ be the transition kernel of a random walk on a discrete group $\Gamma$
induced by an admissible probability measure $\mu$ on $\Gamma$. Coarse
analogues of Toeplitz C*-algebras for random walks and their subaglebras were
first studied in [26, 27] (see also [19]), and the following standard version
of Toeplitz C*-algebras for random walks was defined and studied in [23]. For
$m\in\mathbb{N}$ we denote by $E(P^{m}):=\\{\ (x,y)\ |\ P^{m}(x,y)>0\ \\}$.
The Toeplitz C*-algebras that we consider are norm closed $*$-subalgebras of
bounded operators on the Hilbert space

$\mathcal{H}_{P}:=\oplus_{m=0}^{\infty}\ell^{2}(E(P^{m}))$

with the standard orthonormal basis $\\{e_{y,z}^{(m)}\\}_{(y,z)\in E(P^{m})}$.
Then, for any $n\in\mathbb{N}$ and $(x,y)\in E(P^{n})$ we define an operator
$S_{x,y}^{(n)}$ on $\mathcal{H}_{P}$ by specifying for $(y^{\prime},z)\in
E(P^{m})$

$S_{x,y}^{(n)}(e_{y^{\prime},z}^{(m)})=\delta_{y,y^{\prime}}\sqrt{\frac{P^{n}(x,y)P^{m}(y,z)}{P^{n+m}(x,z)}}e_{x,z}^{(n+m)}.$

For a fixed $z\in\Gamma$ we also denote
$\mathcal{H}_{P,z}=\overline{\operatorname{span}}\\{\ e_{y,z}^{(m)}\ |\
(y,z)\in E(P^{m})\\}$, which is a reducing subspace for all of the operators
$S_{x,y}^{(n)}$ defined above. Thus, we see that our operators are all in the
direct product $\prod_{z\in\Gamma}\mathbb{B}(\mathcal{H}_{P,z})$.

######  Definition 8.3.

Let $P$ be a random walk on a discrete group $\Gamma$ induced by a probability
measure $\mu$. The _Toeplitz C*-algebra_ of $(\Gamma,\mu)$ is given by

$\mathcal{T}(\Gamma,\mu):=C^{*}(\ S_{x,y}^{(n)}\ |\ (x,y)\in E(P^{n}),\
n\in\mathbb{N}\ ).$

The _Cuntz C*-algebra_ of $(\Gamma,\mu)$ is given by

$\mathcal{O}(\Gamma,\mu):=\mathcal{T}(\Gamma,\mu)/\mathcal{J}(\Gamma,\mu),$

where
$\mathcal{J}(\Gamma,\mu)=\mathcal{T}(\Gamma,\mu)\cap\prod_{z\in\Gamma}\mathbb{K}(H_{P,z})$.

These C*-algebras come equipped with natural $\Gamma\times\mathbb{T}$ actions.
Namely, we have a natural unit circle action
$\gamma:\mathbb{T}\curvearrowright\mathcal{T}(\Gamma,\mu)$ given by
$\gamma_{z}(S^{(n)}_{x,y})=z^{n}S^{(n)}_{x,y}$, as well as a $\Gamma$-action
$\delta:\Gamma\curvearrowright\mathcal{T}(\Gamma,\mu)$ given by
$\delta_{g}(S^{(n)}_{x,y})=S^{(n)}_{gx,gy}$. As these actions commute, we get
an action
$\lambda:\Gamma\times\mathbb{T}\curvearrowright\mathcal{T}(\Gamma,\mu)$, and
since $\prod_{z\in\Gamma}\mathbb{K}(H_{P,z})$ is
$\Gamma\times\mathbb{T}$-invariant, we get an induced action
$\overline{\lambda}:\Gamma\times\mathbb{T}\curvearrowright\mathcal{O}(\Gamma,\mu)$
which acts similarly on generators.

Recall that SRLP stands for the strong ratio limit property. One of the main
results of [23] is the computation of $\mathcal{O}(\Gamma,\mu)$ under the
assumption of SRLP. Namely, in [23, Theorem 4.10] it was shown that by
assuming SRLP, we get $\mathcal{O}(\Gamma,\mu)\cong
C(\Delta_{\rho}^{r}\Gamma\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))$.
Under this identification (See [23, Section 5]), the induced action
$\overline{\lambda}:\Gamma\times\mathbb{T}\curvearrowright\mathcal{O}(\Gamma,\mu)$
is given for $f\otimes K\in
C(\Delta_{\rho}^{r}\Gamma\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))$
by

$\overline{\lambda}_{z,g}(f\otimes K)=((g,z)\cdot f)\otimes S_{g}KS_{g}^{-1},$

where $S:\Gamma\rightarrow\mathbb{B}(\ell^{2}(\Gamma))$ is the left regular
representation of $\Gamma$, and for
$(\alpha,w)\in\Delta_{\rho}^{r}\Gamma\times\mathbb{T}$, $((g,z)\cdot
f)(\alpha,w)=f(g^{-1}\alpha,zw)$.

Now, suppose $\Gamma$ is non-elementary hyperoblic relative to a finite
collection of subgroups $\Omega$, and $\mu$ is a finitely supported,
admissible, aperiodic and symmetric probability measure on $\Gamma$. By
combining Corollary 6.2 and Corollary 6.3 we obtain a $\Gamma$-equivariant
injection
$\iota:\overline{\partial^{m}_{M,R}\Gamma}\rightarrow\partial_{\rho}^{r}\Gamma$.
Let $q_{P}:\mathcal{T}(\Gamma,\mu)\rightarrow\mathcal{O}(\Gamma,\mu)$ be the
canonical surjection, and define the $\lambda$ invariant ideal
$\mathcal{J}_{\lambda}$ of $\mathcal{T}(\Gamma,\mu)$ by setting

$\mathcal{J}_{\lambda}:=q_{P}^{-1}\Big{[}C([\Delta_{\rho}^{r}\Gamma\setminus\iota(\overline{\partial^{m}_{M,R}\Gamma})]\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))\Big{]}.$

The following result extends [23, Corollary 5.2] to relatively hyperbolic
groups with symmetric, aperiodic, spectrally non-degenerate random walks. This
partly answers the first part of [23, Question 5.6] for a large class of
random walks on relatively hyperbolic groups in the form of co-universality.

######  Theorem 8.4.

Suppose $\Gamma$ is non-elementary hyperoblic relative to a finite collection
of subgroups $\Omega$, and $\mu$ is aperiodic, symmetric, spectrally non-
degenerate probability measure on $\Gamma$ with finite support generating
$\Gamma$. Then $\mathcal{T}(\Gamma,\mu)$ admits a $\Gamma\times\mathbb{T}$ co-
universal quotient which coincides with

$\mathcal{T}(\Gamma,\mu)/\mathcal{J}_{\lambda}\cong
C(\overline{\partial^{m}_{M,R}\Gamma}\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma)).$

###### Proof.

By Proposition 8.2 it will suffice to show that $\mathcal{J}_{\lambda}$ is the
largest $\Gamma\times\mathbb{T}$-invariant ideal of $\mathcal{T}(\Gamma,\mu)$.
Let $\mathcal{J}$ be a proper $\Gamma\times\mathbb{T}$-invariant ideal of
$\mathcal{T}(\Gamma,\mu)$, and denote by
$\overline{\mathcal{J}}:=q_{P}(\mathcal{J})$. Then there are two
possibilities.

If $\mathcal{J}(\Gamma,\mu)\subseteq\mathcal{J}$, then
$\overline{\mathcal{J}}$ is a proper ideal of $\mathcal{O}(\Gamma,\mu)\cong
C(\Delta_{\rho}^{r}\Gamma\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))$.
Thus, by Gelfand-Naimark duality, there exists an open
$\Gamma\times\mathbb{T}$-invariant subset
$Y\subset\Delta_{\rho}^{r}\Gamma\times\mathbb{T}$ such that
$\overline{\mathcal{J}}=C(Y)\otimes\mathbb{K}(\ell^{2}(\Gamma))$. By Theorem
6.4 we know that
$\iota(\overline{\partial^{m}_{M,R}\Gamma})\cong\overline{\partial^{m}_{M,R}\Gamma}$
is the unique smallest closed $\Gamma$-invariant subspace of
$\partial_{\rho}\Gamma\cong\partial^{r}_{\rho}\Gamma$. Since the action of
$\Gamma$ on $\Gamma/R_{\mu}$ is transitive, we see that $\Gamma/R_{\mu}$ has
no closed $\Gamma$-invariant subsets, so that any proper closed
$\Gamma$-invariant subset of $\Delta_{\rho}^{r}\Gamma$ must be a subset of
$\partial^{r}_{\rho}\Gamma$. Thus,
$\iota(\overline{\partial^{m}_{M,R}\Gamma})$ is the unique smallest closed
$\Gamma$-invariant subspace of $\Delta_{\rho}^{r}\Gamma$.

Since $\mathbb{T}\curvearrowright\mathbb{T}$ acts minimally, we get that
$\iota(\overline{\partial^{m}_{M,R}\Gamma})\times\mathbb{T}$ is the unique
smallest closed $\Gamma\times\mathbb{T}$-invariant subspace of
$\Delta_{\rho}^{r}\Gamma\times\mathbb{T}$. Since $Y$ is proper, open and
$\Gamma\times\mathbb{T}$ invariant, its complement in
$\Delta_{\rho}^{r}\Gamma\times\mathbb{T}$ must contain
$\iota(\overline{\partial^{m}_{M,R}\Gamma})\times\mathbb{T}$. Thus, $Y$ is
contained in
$[\Delta_{\rho}^{r}\Gamma\setminus\iota(\overline{\partial^{m}_{M,R}\Gamma})]\times\mathbb{T}$,
and we get that $\mathcal{J}\subseteq\mathcal{J}_{\lambda}$.

Now suppose that $\mathcal{J}$ is a general $\Gamma\times\mathbb{T}$-invariant
ideal. Since $\overline{\mathcal{J}}$ is proper in $\mathcal{O}(\Gamma,\mu)$,
we get that $\mathcal{J}+\mathcal{J}(\Gamma,\mu)$ is also proper in
$\mathcal{T}(\Gamma,\mu)$. Hence, by the previous argument we get that
$\mathcal{J}\subseteq\mathcal{J}+\mathcal{J}(\Gamma,\mu)\subseteq\mathcal{J}_{\lambda}$.
∎

Although it is unknown if all relatively hyperbolic groups admit a spectrally
non-degenerate random walk, by Propositions 4.2 and 4.4, there are many
spectrally non-degenerate adapted random walks on arbitrary free products, and
any random walk on relatively hyperbolic groups with virtually nilpotent
subgroups of homogeneous dimension at most $4$ is spectrally non-degenerate.
Hence, beyond the class of hyperbolic groups, there is a wide class of
relatively hyperbolic groups for which there always exists a random walk such
that $\mathcal{T}(\Gamma,\mu)$ has a $\Gamma\times\mathbb{T}$ co-universal
quotient.

As a final application, we use of co-universality for two random walks on
$\Gamma=\mathbb{Z}^{5}*\mathbb{Z}$ as in Example 7.2 to come up with examples
of Toeplitz C*-algebras $\mathcal{T}(\Gamma,\mu)$ and
$\mathcal{T}(\Gamma,\mu^{\prime})$ for which no two non-trivial
$\Gamma\times\mathbb{T}$-equivariant quotients are isomorphic.

######  Example 8.5.

Let $\mu$ and $\mu^{\prime}$ be probability measures on
$\Gamma=\mathbb{Z}^{5}*\mathbb{Z}$ as in Example 7.2 so that $\mu$ is
convergent and spectrally degenerate along $\mathbb{Z}^{5}$ and $\mu^{\prime}$
is spectrally non-degenerate. Suppose that $\mathcal{C}_{\mu}$ and
$\mathcal{C}_{\mu^{\prime}}$ are non-trivial
$\Gamma\times\mathbb{T}$-equivariant quotients of $\mathcal{T}(\Gamma,\mu)$
and $\mathcal{T}(\Gamma,\mu^{\prime})$ respectively. Then $\mathcal{C}_{\mu}$
and $\mathcal{C}_{\mu^{\prime}}$ cannot be isomorphic. In particular, the
Toeplitz C*-algebras $\mathcal{T}(\Gamma,\mu)$ and
$\mathcal{T}(\Gamma,\mu^{\prime})$ as well as the Cuntz C*-algebras
$\mathcal{O}(\Gamma,\mu)$ and $\mathcal{O}(\Gamma,\mu)$ are not isomorphic.

Recall first that for both $\mu$ and $\mu^{\prime}$, the closures of minimal
points in their $R$-Martin boundaries coincide with their whole $R$-Martin
boundary. Suppose now that the quotients $C_{\mu}$ and $C_{\mu^{\prime}}$ are
$*$-isomorphic via some $*$-isomorphism $\varphi$. Then, by the same proof of
Theorem 8.4 for $\mu$ (using the same proof of Theorem 6.4 by appealing to
Proposition 7.1 instead of Proposition 6.1) as well as Theorem 8.4 for
$\mu^{\prime}$ there exist _unique_ smallest
$\Gamma\times\mathbb{T}$-equiavariant quotients
$q:\mathcal{C}_{\mu}\rightarrow
C(\partial_{M,R}(\Gamma,\mu)\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))$
and $q^{\prime}:\mathcal{C}_{\mu^{\prime}}\rightarrow
C(\partial_{M,R}(\Gamma,\mu^{\prime})\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))$.
Since these quotients are unique, the ideals $\ker q$ and $\ker q^{\prime}$
are the unique largest proper $\Gamma\times\mathbb{T}$ invariant ideals in
their repsective Toeplitz algebras. Therefore, as $\varphi$ is
$\Gamma\times\mathbb{T}$-equivariant, it must map $\ker q$ onto $\ker
q^{\prime}$. Thus, we obtain an induced $*$-isomorphism
$\widetilde{\varphi}:C(\partial_{M,R}(\Gamma,\mu)\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))\rightarrow
C(\partial_{M,R}(\Gamma,\mu^{\prime})\times\mathbb{T})\otimes\mathbb{K}(\ell^{2}(\Gamma))$.

By Gelfand-Naimark duality, we must then have that
$\partial_{M,R}(\Gamma,\mu)\times\mathbb{T}$ and
$\partial_{M,R}(\Gamma,\mu^{\prime})\times\mathbb{T}$ are homeomorphic.
However, in Example 7.2 we saw that $\partial_{M,R}(\Gamma,\mu)$ has
topological dimension $0$ while $\partial_{M,R}(\Gamma,\mu^{\prime})$ has
topological dimension at least $4$. Hence, we arrive at a contradiction
because $\partial_{M,R}(\Gamma,\mu)\times\mathbb{T}$ is at most $1$
dimensional while $\partial_{M,R}(\Gamma,\mu^{\prime})\times\mathbb{T}$ is at
least $4$ dimensional.

Let us conclude with a few words about possible ways to generalize our work.
According to Corollary 5.1 (2), whenever $\Gamma$ is hyperbolic relative to
virtually abelian subgroups, for any finitely supported, admissible,
aperiodic, symmetric and convergent random walk, the ratio limit kernel
$H(x,y)$ can be expressed as the limit of $I^{(s)}(x,y|r)/I^{(s)}(e,y|r)$ as
$r\to R$. In this situation, beyond the specific case of Example 8.5, the
proof of Proposition 6.1 might be adaptable to show that once again that for
every $\xi$ in the Martin boundary such that $\pi(\xi)$ is conical and for
every sequence $y_{n}$ converging to $\xi$ we still have that
$H(\cdot,y_{n})/K_{R}(\cdot,y_{n})$ converges to 1. Consequently, the
conclusions of Theorem 6.4 and Theorem 8.4 might still hold in this situation.
This raises the question how much the assumption of being spectrally non-
degenerate can be relaxed for our main results to hold.

However, as illustrated in Example 5.3, we know that there are limitations to
the overall strategy we used. Hence, new techniques are necessary to prove a
result analogous to Theorem 8.4 for any admissible, admissible, aperiodic and
symmetric random walk on arbitrary relatively hyperbolic group.

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