Asymptotic shape for subadditve processes on groups of polynomial growth

Let $(\operatorname{\Upomega},\operatorname{\mathscr{F}},\operatorname{\mathbbm{P}})$ be a probability space and $(\operatorname{\Gamma},.)$ a finitely generated group with polynomial growth rate. Set $\operatorname{\upvartheta}:\operatorname{\Gamma}\curvearrowright(\operatorname{\Upomega},\operatorname{\mathscr{F}},\operatorname{\mathbbm{P}})$ to be a $\operatorname{\mathbbm{P}}$-preserving (p.m.p.) ergodic group action. Consider the family $\{c(x)\}_{x\in\operatorname{\Gamma}}$ of non-negative random variables such that, $\operatorname{\mathbbm{P}}$-a.s.,

Write $c(x,\omega)$ for $c(x)(\omega)$ and let $z\cdot\omega:=\operatorname{\upvartheta}_{z}(\omega)$. A function $c:\operatorname{\Gamma}\times\operatorname{\Upomega}\to\operatorname{\mathbb{R}}_{\geq 0}$ satisfying (1.1) is referred to as a subadditive cocycle. Once given a subadditive cocycle $c$, there is a correspondent random pseudo-quasi metric $d_{\omega}$ defined by

which is $\operatorname{\Gamma}$-right equivariant, i.e., for all $x,y,z\in\operatorname{\Gamma}$, and for every $\omega\in\operatorname{\Upomega}$,

The correspondence is one-to-one since given a $\operatorname{\Gamma}$-right equivariant random pseudoquasimetric $d_{\omega}$, one can easily verify that

is a subadditive cocycle.

To avoid dealing with unnecessary technicalities, we initially consider $\operatorname{\Gamma}$ as the group of polynomial growth, which is nilpotent and torsion-free. Later, we address the more general case where $\operatorname{\Gamma}$ is virtually nilpotent. The essential definitions and notation are introduced as we proceed with the text. The group will be associated with a finite symmetric generating set $S\subseteq\operatorname{\Gamma}$. We write $\|-\|_{S}$ and $d_{S}$ for a word length and a word metric, respectively. The following conditions will be needed throughout the paper. We assume the existence of $\upbeta>0$ and $\upkappa>1$ such that, for all $x\in\operatorname{\Gamma}$,

where $g(t)\in\mathcal{O}\left(1/t^{2D+\upkappa}\right)$ as $t\uparrow+\infty$.

Let $[\operatorname{\Gamma},\operatorname{\Gamma}]$ be the commutator subgroup of $\operatorname{\Gamma}$ and set $\|x\|_{S}^{\operatorname{\operatorname{ab}}}:=\inf_{y\in x[\operatorname{\Gamma},\operatorname{\Gamma}]}\|y\|_{S}$. Suppose that there exists $a>0$ such that, for all $x\in\operatorname{\Gamma}\setminus[\operatorname{\Gamma},\operatorname{\Gamma}]$ there is a sequence $\{n_{j}\}_{j\in\operatorname{\mathbb{N}}}$ of positive integers depending on $x[\operatorname{\Gamma},\operatorname{\Gamma}]$ with $\lim_{j\uparrow+\infty}n_{j}=+\infty$ and, for all $y\in x[\operatorname{\Gamma},\operatorname{\Gamma}]$ and every $j\in\operatorname{\mathbb{N}}$,

We say that the process grows at least linearly when condition (i) is satisfied. Condition (ii) provides a lower bound for the norm of the rescaled process $\phi$, which will be defined later.

To obtain the asymptotic result, we will introduce an innerness assumption. Specifically, for each $\upvarepsilon>0$, we require the existence of a finite generating set $F(\upvarepsilon)\subseteq\operatorname{\Gamma}\setminus[\operatorname{\Gamma},\operatorname{\Gamma}]$ such that, for $\operatorname{\mathbbm{P}}$-a.s. $\omega\in\operatorname{\Upomega}$ and for every $x\in\operatorname{\Gamma}$, we can write $x=z_{n}z_{n-1}\dots z_{1}$ with $z_{n},z_{n-1},\dots,z_{1}\in F(\upvarepsilon)$ satisfying

When considering First-Passage Percolation models where $S\subseteq\operatorname{\Gamma}\setminus[\operatorname{\Gamma},\operatorname{\Gamma}]$, condition (iii) is automatically fulfilled (see Section 2.5). Additionally, in the case where $\operatorname{\Gamma}$ is abelian, we can eliminate the need for hypothesis (iii) in the main theorem altogether.

The limit space $G_{\infty}$ is also known as a Carnot group and $d_{\infty}$ coincides with the Carnot-Carathéodory metric obtained by the asymptotic cone of $\operatorname{\Gamma}$ as the limit of $\frac{1}{n}d_{S}$. More details about its construction and properties will be given in Section 2 along with the definitions of $\delta_{t}$ and $d_{\phi}$. The usage of the pointed Gromov-Hausdorff convergence arises naturally from its correspondence with geometric group theory.

Let now $(\operatorname{\Gamma},.)$ be a finitely generated group with polynomial growth rate. Gromov’s Theorem [15] establishes the equivalence of polynomial growth and virtual nilpotency in finitely generated groups. Then there exists a normal nilpotent subgroup $N\unlhd\operatorname{\Gamma}$ with finite index $\kappa:=[\operatorname{\Gamma}:N]<+\infty$. Set $\operatorname{\operatorname{tor}}N$ to be the torsion subgroup of $N$ and define

Pansu [16] showed that $\operatorname{\Gamma}$ and $\operatorname{\Gamma}^{\prime}$ share the same asymptotic cone. Let us fix $z_{(j)}$ as a representative of the coset $N_{(j)}=z_{(j)}N$ such that $\operatorname{\Gamma}=\bigcup_{j=1}^{\kappa}N_{(j)}$. Consider $z_{(j)}=e$ when $N_{(j)}=N$. Set $\uppi_{N}:\operatorname{\Gamma}\to N$ to be given by $\uppi_{N}(x)=z_{(j)}^{-1}x$ for $x\in N_{(j)}$. Define now $\llbracket-\rrbracket:\operatorname{\Gamma}\to\operatorname{\Gamma}^{\prime}$ to be given by

To refine the first main theorem, let us introduce some new conditions. Suppose that there exists $a>0$ such that, for all $x\in\operatorname{\Gamma}$ there is a sequence $\{n_{j}\}_{j\in\operatorname{\mathbb{N}}}$ of positive integers depending on $\llbracket x\rrbracket.[\operatorname{\Gamma}^{\prime},\operatorname{\Gamma}^{\prime}]$ with $n_{j}\uparrow+\infty$ as $j\uparrow+\infty$,

Let $c^{\prime}:\operatorname{\Gamma}^{\prime}\times\operatorname{\Upomega}\to\operatorname{\mathbb{R}}_{\geq 0}$ by

Fix, for each $\llbracket x\rrbracket\in\operatorname{\Gamma}^{\prime}$, a $\upupsilon_{x}\in\llbracket x\rrbracket$ and consider $\theta:\operatorname{\Gamma}^{\prime}\curvearrowright(\operatorname{\Upomega},\operatorname{\mathscr{F}},\operatorname{\mathbbm{P}})$ given by $\theta_{\llbracket x\rrbracket}\equiv\operatorname{\upvartheta}_{\upupsilon_{x}}$ and $\theta_{z}(\omega)=z\ast\omega$ (see Sec. 3.3 and Remark 5 for a detailed discussion). We consider a similar innerness assumption to replace (iii). Suppose that, for each $\upvarepsilon>0$, there exists a finite $F(\upvarepsilon)\subseteq N\setminus[N,N]$ which is a generating set of $\operatorname{\Gamma}^{\prime}$ such that, $\operatorname{\mathbbm{P}}$-a.s., for every $x\in\operatorname{\Gamma}$, one can write $\llbracket x\rrbracket=z_{n}z_{n-1}\dots z_{1}$ with $z_{n},z_{n-1},\dots,z_{1}\in F(\upvarepsilon)$ satisfying

Similar to (iii), First-Passage Percolation models satisfy (iii^′) under specific conditions. In the case where $\operatorname{\Gamma}=N$ is nilpotent, it suffices to have $S\subseteq N\setminus\big{(}[N,N]\cup\operatorname{\operatorname{tor}}N\big{)}$ for an FPP model to satisfy (iii^′). The virtually nilpotent case is treated separately in Section 3.4 with additional conditions imposed on $\llbracket S\rrbracket$ and $\operatorname{\upvartheta}$. Moreover, when $\operatorname{\Gamma}^{\prime}$ is abelian, hypothesis (iii^′) is not required to verify the theorem below.

The primary technique employed in this work involves the approximation of admissible curves through the use of polygonal paths and ergodic theory. In Section 3, we introduce and delve into these tools, presenting their application in proving the theorems and a corollary for FPP models. Section 4 showcases examples dedicated to illustrating the applicability of the theorems.

The interplay among finitely generated groups, Cayley graphs, word metrics, and the convergence of metric spaces establishes a bridge between the algebraic properties of groups and geometric structures.

Let $(\operatorname{\Gamma},.)$ be a group generated by a finite set $S$. The associated Cayley graph $\mathcal{C}(\operatorname{\Gamma},S)$ represents elements of $G$ as vertices, with edges connecting $x$ and $y$ if and only if $y=sx$ for some $s\in S$. Formally, the right-invariant Cayley graph $\mathcal{C}(\operatorname{\Gamma},S)=(V,E)$ is defined by

Cayley graphs provide a visual representation of the group structure and are fundamental in the study of geometric group theory.

Denote by $u\sim v$ the relation $\{u,v\}\in E$. Let $\mathscr{P}(x,y)$ be the set of self-avoiding paths from $x$ to $y$, where each $\gamma\in\mathscr{P}(x,y)$ follows $\gamma=(x_{0},\dots,x_{m})$ with $m\in\operatorname{\mathbb{N}}$, $x_{i}\sim x_{i+1}$, $x_{0}=x$, $x_{m}=y$, and $x_{i}\neq x_{j}$ for all $i\neq j$. We write $\mathtt{e}\in\gamma$ for $\mathtt{e}=\{x_{i},x_{i+1}\}\in E$, and $|\gamma|=m$ represents the length of the path.

The word length on $\operatorname{\Gamma}$ with respect to $S$ is defined as follows: For any $x\in\operatorname{\Gamma}$, the length of the shortest word (or self-avoiding path) in $S$ that represents $x$ is its word length, denoted by $\|x\|_{S}=\inf_{\gamma\in\mathscr{P}(e,x)}|\gamma|$. The word metric $d_{S}$ on $\operatorname{\Gamma}$ is given by $d_{S}(x,y)=\|yx^{-1}\|_{S}$.

Throughout this article, various distinct metrics will be considered. Therefore, let us consider a (semi-pseudo-quasi) metric $d_{\diamondsuit}$ on a non-empty set $\mathbb{X}$, where the metric is indexed by $\diamondsuit$. We define $B_{\diamondsuit}(x,r):=\{y\in\mathbb{X}\colon d_{\diamondsuit}(x,y)<r\}$ as the open $d_{\diamondsuit}$-ball centered at $x\in\mathbb{X}$. To streamline notation, let $t\vee t^{\prime}:=\max\{t,t^{\prime}\}$ and $t\wedge t^{\prime}:=\min\{t,t^{\prime}\}$. Here, the set $\operatorname{\mathbb{N}}$ stands for $\{1,2,\dots\}$ and $\operatorname{\mathbb{N}}_{0}=\operatorname{\mathbb{N}}\cup\{0\}$.

A finitely generated group $\operatorname{\Gamma}$ has polynomial growth with respect to $S$ when $|B_{S}(e,r)|\in\mathcal{O}\big{(}n^{D^{\prime}}\big{)}$ for a $D^{\prime}\in\operatorname{\mathbb{N}}_{0}$ as $n\uparrow+\infty$. The growth is associated with the Cayley graph $\mathcal{C}(\operatorname{\Gamma},S)$. The polynomial growth rate of $\operatorname{\Gamma}$ is a constant $D\in\operatorname{\mathbb{N}}_{0}$ such that there exists $\mathtt{k}\in(1,+\infty)$ for all $r>1$ satisfying

Thus $D=\min\big{\{}D^{\prime}\in\operatorname{\mathbb{N}}_{0}\colon|B_{S}(e,r)|\in\mathcal{O}\big{(}n^{D^{\prime}}\big{)}\big{\}}$. Moreover, one can verify that the polynomial growth rate of $\mathcal{C}(\operatorname{\Gamma},S)$ does not depend on the choice of $S$.

Recall the definition of the commutator element $[x,y]=xyx^{-1}y^{-1}$ and the subgroup $[U,V]:=\big{\langle}[u,v]\colon u\in U,v\in V\big{\rangle}$ for any $U,V\subseteq\operatorname{\Gamma}$. Set $\operatorname{\Gamma}_{0}:=\operatorname{\Gamma}$ and let $\operatorname{\Gamma}_{n}:=[\operatorname{\Gamma},\operatorname{\Gamma}_{n-1}]$ for all $n\in\operatorname{\mathbb{N}}$. Thus $\{\operatorname{\Gamma}_{n}\}_{n\in\operatorname{\mathbb{N}}}$ forms a lower central series with $\operatorname{\Gamma}_{n}\unlhd\operatorname{\Gamma}_{n-1}$ for all $n\in\operatorname{\mathbb{N}}$. The group $\operatorname{\Gamma}$ is called nilpotent when there is an $n\in\operatorname{\mathbb{N}}$ such that $\operatorname{\Gamma}_{n}=\{e\}$, i.e., when the sequence stabilizes in the trivial group. More specifically, $\operatorname{\Gamma}$ is nilpotent of class $n$ when $n$ is the minimal value such that $\operatorname{\Gamma}_{n}$ is the trivial group. A group is abelian if and only if it is nilpotent of class $1$. The abelianization of a group $\operatorname{\Gamma}$ is given by $\operatorname{\Gamma}^{\operatorname{\operatorname{ab}}}\cong\operatorname{\Gamma}/[\operatorname{\Gamma},\operatorname{\Gamma}]$.

The group $\operatorname{\Gamma}$ is called virtually nilpotent when there exists a normal subgroup $N\unlhd\operatorname{\Gamma}$ that is nilpotent with finite index $\kappa=[\operatorname{\Gamma}\colon N]<+\infty$. A noteworthy result obtained by Gromov [15] is that a finitely generated group has polynomial growth exactly when it is virtually nilpotent. Therefore, the growth established by word metrics is strongly related to algebraic properties of the group.

The torsion subgroup of a group $H$ is denoted by $\operatorname{\operatorname{tor}}H$ and it is defined as the set of all elements with finite order. In other words, $\operatorname{\operatorname{tor}}H:=\big{\langle}h\in H\colon\exists n\in\operatorname{\mathbb{N}}(h^{n}=e)\big{\rangle}$. The group $H$ is called torsion-free when $\operatorname{\operatorname{tor}}H$ is the trivial group.

Let $[U]_{\varepsilon}$ to be the $\varepsilon$-neighborhood of $U\subseteq\mathbb{X}$ of in a metric space $(\mathbb{X},d_{\diamondsuit})$, i.e., the set $[U]_{\varepsilon}=\bigcup_{u\in U}B_{\diamondsuit}(e,\varepsilon)$. The Hausdorff distance $d_{H}$ detects the largest variations between sets with respect to the given metric

We define the convergence of metric spaces used in the main theorems employing the Hausdorff distance. Let $(\mathbb{X}_{n},d_{\diamondsuit_{n}},o_{n})_{n\in\operatorname{\mathbb{N}}}$ be a sequence of centered, locally compact metric spaces. Consider $\{\psi_{n}\}_{n\in\operatorname{\mathbb{N}}}$ as a family of isometric embeddings $\psi_{n}:(\mathbb{X}_{n},d_{\diamondsuit_{n}},o_{n})\to(\mathbb{X},d_{\diamondsuit},o)$.

The pointed Gromov-Hausdorff convergence of $(\mathbb{X}_{n},d_{\diamondsuit_{n}},o_{n})$ to $(\mathbb{X},d_{\diamondsuit},o)$ is denoted by

and it implies, for all $r>0$,

The definitions above are immediately extended to random semi-pseudo-quasi metrics, as employed in the main theorems. The assumption of almost sure local compactness is also maintained. We are now prepared to present Pansu’s theorem on the convergence of finitely generated virtually nilpotent groups.

Note that Theorems 1.1 and 1.2 are generalizations of the theorem above. Therefore, the shape theorems under investigation can be interpreted as the convergence of random metric spaces in large-scale geometry. The next subsection is dedicated to the construction of the asymptotic cone $G_{\infty}$ and related results.

Consider for now $\operatorname{\Gamma}$ as a nilpotent and torsion-free group, unless stated otherwise. We also assume that its abelianization is torsion-free. In this subsection, we use $\operatorname{\Gamma}$ instead of $\operatorname{\Gamma}^{\prime}$ to simplify notation, but we will subsequently extend the results to the more general case.

Let $G$ denote the real Mal’cev completion of $\operatorname{\Gamma}$. The group $G$ can be defined as the smallest simply connected real Lie group such that $\operatorname{\Gamma}\leq G$ and, for all $z\in\operatorname{\Gamma}$ and $n\in\operatorname{\mathbb{N}}$, there exists $\mathlcal{z}\in G$ with $\mathlcal{z}^{n}=z$. In this case, $G$ is nilpotent of the same order of $\operatorname{\Gamma}$ and it is uniquely defined. Furthermore, $G$ is simply connected it is associated with the Lie algebra $(\operatorname{\mathfrak{g}},[\cdot,\cdot]_{1})$ where $\operatorname{\Gamma}$ is cocompact in $G$. We write $\log:G\to\operatorname{\mathfrak{g}}$ for the Lie logarithm map.

Define $\operatorname{\mathfrak{g}}^{1}:=\operatorname{\mathfrak{g}}$ and $\operatorname{\mathfrak{g}}^{i+1}:=[\operatorname{\mathfrak{g}},\operatorname{\mathfrak{g}}^{i}]_{1}$. It follows from the nilpotency of $\operatorname{\Gamma}$ that threre exists $l\in\operatorname{\mathbb{N}}$ such that $\operatorname{\Gamma}_{l}=\{e\}$. Thus $\operatorname{\mathfrak{g}}^{l+1}=(0)$. Since $[\operatorname{\mathfrak{g}}^{i},\operatorname{\mathfrak{g}}^{j}]_{1}\subseteq\operatorname{\mathfrak{g}}^{i+j}$ and, in particular, $[\operatorname{\mathfrak{g}}^{i+1},\operatorname{\mathfrak{g}}^{j}]_{1},[\operatorname{\mathfrak{g}}^{i},\operatorname{\mathfrak{g}}^{j+1}]_{1}\subseteq\operatorname{\mathfrak{g}}^{i+j+1}$, the Lie bracket on $\operatorname{\mathfrak{g}}$ determines a bilinear map

which in turn defines a Lie bracket $[\cdot,\cdot]_{\infty}$ on

Consider the decomposition $\operatorname{\mathfrak{g}}=V_{1}\oplus\cdots\oplus V_{l}$ given by $\operatorname{\mathfrak{g}}^{i}:=V_{i}\oplus\cdots\oplus V_{l}$. Thus, $(\operatorname{\mathfrak{g}}_{\infty},[\cdot,\cdot]_{\infty})$ is a graded Lie algebra. Let us define a family of linear maps $\delta_{t}:\operatorname{\mathfrak{g}}_{\infty}\to\operatorname{\mathfrak{g}}_{\infty}$ given by

for each $t>0$ and $v_{i}\in\mathfrak{v}_{i}$ with $i\in\{1,\dots,l\}$. It follows from the definition of $\delta_{t}$ that $\delta_{t}([u,v]_{\infty})=[\delta_{t}(u),\delta_{t}(v)]_{\infty}$ and $\delta_{tt^{\prime}}=\delta_{t}\circ\delta_{t^{\prime}}$ for all $u,v\in\operatorname{\mathfrak{g}}_{\infty}$ and $t,t^{\prime}>0$. Hence, $\{\delta_{t}\}_{t>0}$ defines a family of automorphisms in the graded Lie group $G_{\infty}:=\exp_{\infty}\left[\operatorname{\mathfrak{g}}_{\infty}\right]$. Here we write $\exp_{\infty}:\operatorname{\mathfrak{g}}_{\infty}\to G_{\infty}$ and $\exp:\operatorname{\mathfrak{g}}\to G$ to differentiate the distinct exponential maps of $\operatorname{\mathfrak{g}}_{\infty}$ and $\operatorname{\mathfrak{g}}$. Similarly, $\log_{\infty}$ and $\log$ stand for their correspondent Lie logarithm maps.

Let $\operatorname{\mathfrak{g}}=V_{1}\oplus\cdots\oplus V_{l}$ be the decomposition given by $\operatorname{\mathfrak{g}}^{i}=V_{i}\oplus\cdots\oplus V_{l}$. Set $L:\operatorname{\mathfrak{g}}\to\operatorname{\mathfrak{g}}_{\infty}$ to be an linear map such that $L(V_{i})=\operatorname{\mathfrak{v}}_{i}$. Consider now $\sigma_{t}$ to be the linear automorphism on $\operatorname{\mathfrak{g}}$ so that $\sigma_{t}(v_{i})=t^{i}v_{i}$ for each $v_{i}\in V_{i}$ and $i\in\{1,\dots,l\}$. Define the Lie brackets $[\cdot,\cdot]_{t}$ on $\operatorname{\mathfrak{g}}$ by

thus $(\operatorname{\mathfrak{g}},[\cdot,\cdot]_{t})$ is isomorphic to $(\operatorname{\mathfrak{g}},[\cdot,\cdot]_{1})$ via $\sigma_{t}$. Furthermore,

since, given $v\in V_{i}$ and $w\in V_{j}$, one has that the main term belongs to $V_{i+j}$, the other terms of superior order belong to $V_{i+j+1}\oplus\cdots\oplus V_{l}$ and it makes them insignificant in the rescaled limit (see [7, 16] for a detailed discussion). Set

The convergence established by Theorem 2.1 determines the metric $d_{\infty}$ such that

Hence, $\lim_{n\uparrow+\infty}\frac{1}{t_{n}}\operatorname{{\scriptscriptstyle\bullet}}x_{n}=\mathlcal{g}$ exactly when $\frac{1}{t_{n}}\operatorname{{\scriptscriptstyle\bullet}}x_{n}$ converges to $\mathlcal{g}$ in $(G_{\infty},d_{\infty})$. The corresponding metric statement shows that, given sequences $\{x_{n}\}_{n\in\operatorname{\mathbb{N}}}$, $\{x_{n}^{\prime}\}_{n\in\operatorname{\mathbb{N}}}$ in $\Gamma$, and $t_{n}\uparrow+\infty$ as $n\uparrow+\infty$ with $\lim_{n\uparrow+\infty}\frac{1}{t_{n}}\operatorname{{\scriptscriptstyle\bullet}}x_{n}=\mathlcal{g}$ and $\lim_{n\uparrow+\infty}\frac{1}{t_{n}}\operatorname{{\scriptscriptstyle\bullet}}x_{n}^{\prime}=\mathlcal{g}^{\prime}$,

The abelianized Lie algebras are defined by $\operatorname{\mathfrak{g}}_{\infty}^{\operatorname{\operatorname{ab}}}:=\operatorname{\mathfrak{g}}_{\infty}/[\operatorname{\mathfrak{g}}_{\infty},\operatorname{\mathfrak{g}}_{\infty}]_{\infty}\cong\mathfrak{v}_{1}$ and $\operatorname{\mathfrak{g}}^{\operatorname{\operatorname{ab}}}:=\operatorname{\mathfrak{g}}/[\operatorname{\mathfrak{g}},\operatorname{\mathfrak{g}}]_{1}$. In particular, $\operatorname{\mathfrak{g}}^{\operatorname{\operatorname{ab}}}\cong\operatorname{\mathfrak{g}}_{\infty}^{\operatorname{\operatorname{ab}}}$. By the Frobenius integrability criterion, the integrable curves in $G_{\infty}$, the tangent vectors must belong to $\mathfrak{v}_{1}$ at each point of the curve. An admissible (or curve) in $G_{\infty}$ is a Lipschitz curve $\upgamma:[t_{0},t_{1}]\to G_{\infty}$ such that the tangent vector $\upgamma^{\prime}(t)\in\mathfrak{v}_{1}$ for all $t\in[t_{0},t_{1}]$. Let $\phi:\operatorname{\mathfrak{g}}_{\infty}^{\operatorname{\operatorname{ab}}}\to[0,+\infty)$ be a norm in the abelianized algebra. Then the $\ell_{\phi}$-length of the admissible $\upgamma$ is

Set $d_{\phi}$ to be the inner metric of the length space $(G,\ell_{\phi})$ given by

In fact, the construction of $d_{\phi}$ can be employed to define $d_{\infty}$. The bi-Lipschitz property is a consequence of the results in Section 2.4. One can also verify that the metric $d_{\infty}$ is right-invariant and homogeneous with respect to $\delta_{t}$. Let us define the projections

so that, if $v=\sum_{i=1}^{l}v_{i}\in\operatorname{\mathfrak{g}}_{\infty}$ with $v_{i}\in\mathfrak{v_{i}}$, then $\pi_{\infty}(v)=v_{1}$ and $\pi=L^{-1}\circ\pi_{\infty}\circ L$. The next lemma compiles several well-known results that will be employed throughout the text. We state the results and their proofs can be found in [8, 10, 16].

The item (iii) in Lemma 2.2 has direct implications for the application of subadditive ergodic theorems. To address this constraint, we overcome it by approximating the lengths using polygonal curves. We present, without proof, Lemma 3.7 from [8], which will be employed in the approximation.

One standout example that exemplifies several properties presented above is the discrete Heisenberg group. As a prime example of a nilpotent group, it offers valuable insights into the fusion of algebraic structures with geometric phenomena in both geometric group theory and metric geometry.

In this subsection, our focus shifts to examples of virtually nilpotent groups that can be constructed through direct and outer semidirect products. The discussion of the virtually nilpotent case will be explored more extensively later in the text.

Let $\mathrm{L}$ be a nilpotent group and consider $M$ a finite group. Then the direct product

is a group with the binary operation given by $(x,m).(y,m^{\prime})=(xy,mm^{\prime})$. Note that the commutator is $\big{[}(x,m),(y,m^{\prime})\big{]}=\big{(}[x,y],[m,m^{\prime}]\big{)}$. It follows that, for all $A,A^{\prime}\subseteq\mathrm{L}$ and $B,B^{\prime}\subseteq M$,

Hence, $K$ is a nilpotent group if, and only if, $M$ is nilpotent. On the other hand, for all finite group $M$, $K$ is virtually nilpotent.

Set $S_{\mathrm{L}}$ and $S_{M}$ to be finite symmetric generating sets of $\mathrm{L}$ and $M$, respectively.

is a finite generating set of $K$. We will consider another useful example of generating set of $K$. Let $S_{\square}^{e}$ stand for $S_{\square}\cup\{e\}$. Then

is also a symmetric generating set of $K$. Furthermore, if $\mathrm{L}$ is torsion-free, then $\llbracket S\rrbracket$ is analog to $S_{\mathrm{L}}$ while $\operatorname{\Gamma}^{\prime}\simeq\mathrm{L}$.