Cayley Graphs, Word Metrics and Quasi-Isometry — Examples

Given: The objects and hypotheses in the Example.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

The word length $\mathrm{\mid }g{\mathrm{\mid }}_S$ is the least $n$ such that $g$ is a product of $n$ elements of $S\cup S^{-1}$ (Word length of a group element with respect to a generating set).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph (The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph).

Define $d_{\mathbb{R}}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ by $d_{\mathbb{R}}(x,y):=\mathrm{\mid }x-y\mathrm{\mid }$. (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

A free abelian group on a set $X$ is an abelian group $A(X)$ together with a map $i:X\to A(X)$ such that, for every abelian group $B$ and every function $u:X\to B$, there is a unique group homomorphism $\widehat{u}:A(X)\to B$ satisfying (Free abelian group on a set).

Given: The objects and hypotheses in the Example.

The word length $\mathrm{\mid }g{\mathrm{\mid }}_S$ is the least $n$ such that $g$ is a product of $n$ elements of $S\cup S^{-1}$ (Word length of a group element with respect to a generating set).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

The identity map between the word metrics of two finite generating sets of a group is a bilipschitz equivalence (The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence).

A map is a bilipschitz embedding when $c^{-1}d(x,x^{\mathrm{\prime }})\le d(f(x),f(x^{\mathrm{\prime }}))\le cd(x,x^{\mathrm{\prime }})$ for some $c>0$, and a bilipschitz equivalence when it is a bijective such map with bilipschitz inverse (Bilipschitz embeddings and bilipschitz equivalences of metric spaces).

• $d$ and $d^{\mathrm{\prime }}$ are topologically equivalent if they have the same metric topology: ${\mathcal{T}}_d={\mathcal{T}}_{d^{\mathrm{\prime }}}\mathrm{.}$ - $d$ and $d^{\mathrm{\prime }}$ are uniformly equivalent if for every real $\epsilon >0$ there are reals $\delta >0$ and ${\delta }^{\mathrm{\prime }}>0$ such that, for all $x,y\in X$, $d(x,y)<\delta ⟹d^{\mathrm{\prime }}(x,y)<\epsilon \text{and}{d}^{\mathrm{\prime }}(x,y)<{\delta }^{\mathrm{\prime }}⟹d(x,y)<\epsilon \mathrm{.}$ - $d$ and $d^{\mathrm{\prime }}$ are Lipschitz equivalent if there are reals $\alpha ,\beta >0$ with $\alpha d(x,y)\le d^{\mathrm{\prime }}(x,y)\le \beta d(x,y)\text{for all }x,y\in X\mathrm{.}$ (Topologically, uniformly and Lipschitz equivalent metrics on a set).

Given: The objects and hypotheses in the Example.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

A free abelian group on a set $X$ is an abelian group $A(X)$ together with a map $i:X\to A(X)$ such that, for every abelian group $B$ and every function $u:X\to B$, there is a unique group homomorphism $\widehat{u}:A(X)\to B$ satisfying (Free abelian group on a set).

The word length $\mathrm{\mid }g{\mathrm{\mid }}_S$ is the least $n$ such that $g$ is a product of $n$ elements of $S\cup S^{-1}$ (Word length of a group element with respect to a generating set).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph (The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph).

$d_1(x,y):={\sum }_{k<n}\mathrm{\mid }{x}_k-y_k\mathrm{\mid },d_2(x,y):=\sqrt{{\sum }_{k<n}(x_k-y_k{)}^2},d_{\mathrm{\infty }}(x,y):=\max \{\mathrm{\mid }{x}_k-y_k\mathrm{\mid }:k<n\}\mathrm{.}$ (${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it).

Given: The objects and hypotheses in the Example.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

The Cayley graph of a free group with respect to a free basis is a tree (The Cayley graph of a free group with respect to a free basis is a tree).

Every vertex of a Cayley graph has the same degree, and the graph is locally finite exactly when the symmetrised generating set is finite (Cayley-graph neighbourhoods are equipotent, and local finiteness is equivalent to finiteness of the symmetrised subset).

A cycle is a closed walk of length at least three with distinct vertices apart from its endpoints; a forest is a simple graph with no cycle and a tree is a connected forest (Cycles, trees and forests in a simple graph on an arbitrary vertex set).

A free group on a set $X$ is a group $F(X)$ together with a map $i:X\to F(X)$ such that, for every group $G$ and every function $u:X\to G$, there is a unique group homomorphism $\widehat{u}:F(X)\to G$ satisfying (Free group on a set of generators).

The subset $B$ is a free basis of $F$ if $(F,i)$ is a free group on the set $B$ in the sense of. (A free basis of a group).

Given: The objects and hypotheses in the Example.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

Every vertex of a Cayley graph has the same degree, and the graph is locally finite exactly when the symmetrised generating set is finite (Cayley-graph neighbourhoods are equipotent, and local finiteness is equivalent to finiteness of the symmetrised subset).

For $D=\mathrm{Dih}(C_4)=⟨r,s⟩$, one has $r^4=s^2=1$, $sr{s}^{-1}=r^{-1}$, and every element is uniquely $r^i$ or $r^{i}s$ for $0\le i<4$ ( $\mathrm{Dih}(C_n)=C_n⋊C_2$ with inversion action has order $2n$ and the dihedral relations).

$\mathrm{Dih}(A)=A⋊C_2\mathrm{.}$ ( The generalized dihedral group $\mathrm{Dih}(A)=A⋊C_2$ for an abelian group $A$).

The degree of $v$ is ${\mathrm{deg}}_G(v):=\mathrm{\mid }{N}_G(v)\mathrm{\mid }$, equivalently the number of edges incident with $v$. A graph is $r$-regular when every vertex has degree $r$; it is cubic when it is $3$-regular. (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

Given: The objects and hypotheses in the Example.

A map is $(L,C)$-coarse Lipschitz when $d(f(x),f(x^{\mathrm{\prime }}))\le Ld(x,x^{\mathrm{\prime }})+C$, and an $(L,C)$-quasi-isometric embedding when in addition $L^{-1}d(x,x^{\mathrm{\prime }})-C\le d(f(x),f(x^{\mathrm{\prime }}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

A map is a bilipschitz embedding when $c^{-1}d(x,x^{\mathrm{\prime }})\le d(f(x),f(x^{\mathrm{\prime }}))\le cd(x,x^{\mathrm{\prime }})$ for some $c>0$, and a bilipschitz equivalence when it is a bijective such map with bilipschitz inverse (Bilipschitz embeddings and bilipschitz equivalences of metric spaces).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

It is written $\lfloor x\rfloor$ and called the integer part, or floor, of $x$. (Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$).

Define $d_{\mathbb{R}}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ by $d_{\mathbb{R}}(x,y):=\mathrm{\mid }x-y\mathrm{\mid }$. (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

Given: The objects and hypotheses in the Example.

A subgroup of finite index in a finitely generated group is finitely generated and its inclusion is a quasi-isometry (A subgroup of finite index in a finitely generated group is finitely generated, and its inclusion is a quasi-isometry).

$[G:H]:=\mathrm{\mid }G\mathrm{/}H\mathrm{\mid }$ (The coset set $G\mathrm{/}H$ and the index $[G:H]$ of a subgroup).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

A free abelian group on a set $X$ is an abelian group $A(X)$ together with a map $i:X\to A(X)$ such that, for every abelian group $B$ and every function $u:X\to B$, there is a unique group homomorphism $\widehat{u}:A(X)\to B$ satisfying (Free abelian group on a set).

Given: The objects and hypotheses in the Example.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

A map is a bilipschitz embedding when $c^{-1}d(x,x^{\mathrm{\prime }})\le d(f(x),f(x^{\mathrm{\prime }}))\le cd(x,x^{\mathrm{\prime }})$ for some $c>0$, and a bilipschitz equivalence when it is a bijective such map with bilipschitz inverse (Bilipschitz embeddings and bilipschitz equivalences of metric spaces).

The identity map between the word metrics of two finite generating sets of a group is a bilipschitz equivalence (The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence).

A finitely generated group is quasi-isometric to a metric space when its word metric for some, equivalently every, finite generating set is (The quasi-isometry type of a finitely generated group).

The group with presentation (Group presentation by generators and relations).

If the evaluation of every $r\in R$ under $u$ is $e_H$, then there is a unique homomorphism (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group).

Let $u,v\in F(X)$ and put $N=⟨\text{\hspace{0.17em}⁣}⟨R⟩\text{\hspace{0.17em}⁣}{⟩}_{F(X)}$. The words $u$ and $v$ represent the same element of $⟨X\mid R⟩$ if and only if (In $⟨X\mid R⟩$, the words $u$ and $v$ represent the same element if and only if $u^{-1}v\in ⟨\text{\hspace{0.17em}⁣}⟨R⟩\text{\hspace{0.17em}⁣}⟩$).

$\mathrm{Dih}(A)=A⋊C_2\mathrm{.}$ ( The generalized dihedral group $\mathrm{Dih}(A)=A⋊C_2$ for an abelian group $A$).

Given: The proposed claim together with the witness named in the Statement refuted.

The word length $\mathrm{\mid }g{\mathrm{\mid }}_S$ is the least $n$ such that $g$ is a product of $n$ elements of $S\cup S^{-1}$ (Word length of a group element with respect to a generating set).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

Balls of a word metric are finite if and only if the generating set is finite (Balls of a word metric are finite if and only if the generating set is finite).

The identity map between the word metrics of two finite generating sets of a group is a bilipschitz equivalence (The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence).

Bounded subset. $A$ is bounded if $A=\mathrm{\varnothing }$ or there are $x_0\in X$ and a real $r>0$ with $A\subseteq B(x_0,r)$. (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

Given: The proposed claim together with the witness named in the Statement refuted.

A map is $(L,C)$-coarse Lipschitz when $d(f(x),f(x^{\mathrm{\prime }}))\le Ld(x,x^{\mathrm{\prime }})+C$, and an $(L,C)$-quasi-isometric embedding when in addition $L^{-1}d(x,x^{\mathrm{\prime }})-C\le d(f(x),f(x^{\mathrm{\prime }}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

It is written $\lfloor x\rfloor$ and called the integer part, or floor, of $x$. (Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$).

Define $d_{\mathbb{R}}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ by $d_{\mathbb{R}}(x,y):=\mathrm{\mid }x-y\mathrm{\mid }$. (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

• (a) $f$ is continuous at every point of $X$ in the $\epsilon$-$\delta$ sense. (For a map of metric spaces the following agree: $\epsilon$-$\delta$ continuity everywhere, preimages of open sets are open, preimages of closed sets are closed, sequential continuity, and $f(\bar{A})\subseteq \overline{f(A)}$).

Given: The proposed claim together with the witness named in the Statement refuted.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

Every vertex of a Cayley graph has the same degree, and the graph is locally finite exactly when the symmetrised generating set is finite (Cayley-graph neighbourhoods are equipotent, and local finiteness is equivalent to finiteness of the symmetrised subset).

Given: The proposed claim together with the witness named in the Statement refuted.

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph (The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph).

Right translation by a fixed element displaces every point of a word metric space by exactly the word length of that element (Right translation by a fixed element displaces every point of a word metric space by exactly the word length of that element).

Given: The proposed claim together with the witness named in the Statement refuted.

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

A map is $(L,C)$-coarse Lipschitz when $d(f(x),f(x^{\mathrm{\prime }}))\le Ld(x,x^{\mathrm{\prime }})+C$, and an $(L,C)$-quasi-isometric embedding when in addition $L^{-1}d(x,x^{\mathrm{\prime }})-C\le d(f(x),f(x^{\mathrm{\prime }}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

It is written $\lfloor x\rfloor$ and called the integer part, or floor, of $x$. (Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$).

Define $d_{\mathbb{R}}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ by $d_{\mathbb{R}}(x,y):=\mathrm{\mid }x-y\mathrm{\mid }$. (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

Given: The proposed claim together with the witness named in the Statement refuted.

A finitely generated group is quasi-isometric to a metric space when its word metric for some, equivalently every, finite generating set is (The quasi-isometry type of a finitely generated group).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

Balls of a word metric are finite if and only if the generating set is finite (Balls of a word metric are finite if and only if the generating set is finite).

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

$B(x,r)$ is the open ball, $\bar{B}(x,r)$ the closed ball and $S(x,r)$ the sphere of centre $x$ and radius $r$. The radius is always a strictly positive real; a ball of radius $0$ or of negative radius is never written in this library. (Open ball, closed ball and sphere in a metric space).

A free group on a set $X$ is a group $F(X)$ together with a map $i:X\to F(X)$ such that, for every group $G$ and every function $u:X\to G$, there is a unique group homomorphism $\widehat{u}:F(X)\to G$ satisfying (Free group on a set of generators).

A set $A$ is finite when $A\approx n$ for some $n\in \mathbb{N}$. (The cardinality $\mid A\mid$ of a finite set).

A map is $(L,C)$-coarse Lipschitz when $d_Y(f(x),f(x^{\mathrm{\prime }}))\le L{d}_X(x,x^{\mathrm{\prime }})+C$, and an $(L,C)$-quasi-isometric embedding satisfies in addition $L^{-1}{d}_X(x,x^{\mathrm{\prime }})-C\le d_Y(f(x),f(x^{\mathrm{\prime }}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

In a free group with respect to a free basis, word length is reduced-word length (With respect to a free basis, the word length of an element is the length of its reduced word).

For every real number there is a larger natural number (Every complete ordered field is Archimedean).

Given: The proposed claim together with the witness named in the Statement refuted.

A geodesic of length $L$ in a metric space is an isometric embedding of the interval $[0,L]$, and the space is geodesic when every two points are the endpoints of one (Geodesics and geodesic metric spaces).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

A group with the word metric of any generating set is a $(1,1)$-quasi-geodesic space (A group with the word metric of any generating set is a $(1,1)$-quasi-geodesic space).

Given: The proposed claim together with the witness named in the Statement refuted.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

On a finite vertex set $V$, the empty graph has edge set $\mathrm{\varnothing }$ and the complete graph $K_V$ has edge set $[V{]}^2$. When $V$ is an $n$-element labelled set, these are also denoted ${\bar{K}}_n$ and $K_n$. (Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

If $G=⟨g⟩$ is cyclic, then exactly one of the following applies: (Every cyclic group is isomorphic to $(\mathbb{Z},+)$ or to $(\mathbb{Z}\mathrm{/}n,+)$ for its finite order $n\ge 1$).

A graph isomorphism is a bijection $\phi :V\to W$ such that, for all distinct $u,v\in V$, (Graph isomorphisms, automorphisms and graph complements).

Given: The proposed claim together with the witness named in the Statement refuted.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

A cycle is a closed walk of length at least three with distinct vertices apart from its endpoints; a forest is a simple graph with no cycle and a tree is a connected forest (Cycles, trees and forests in a simple graph on an arbitrary vertex set).

If no product of two members of a generating set is the identity and the Cayley graph is a tree, the set is a free basis (If no product of two members of a generating set is the identity and the Cayley graph is a tree, the set is a free basis).

The subset $B$ is a free basis of $F$ if $(F,i)$ is a free group on the set $B$ in the sense of. (A free basis of a group).

If $G=⟨g⟩$ is cyclic, then exactly one of the following applies: (Every cyclic group is isomorphic to $(\mathbb{Z},+)$ or to $(\mathbb{Z}\mathrm{/}n,+)$ for its finite order $n\ge 1$).

Given: The objects and hypotheses in the Example.

The quasi-isometry group of a metric space is the set of quasi-isometries of it modulo bounded distance (The quasi-isometry group of a metric space).

The quasi-isometry group is a group under composition, and a quasi-isometry induces an isomorphism between the quasi-isometry groups of its source and target (Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups).

Bounded subset. $A$ is bounded if $A=\mathrm{\varnothing }$ or there are $x_0\in X$ and a real $r>0$ with $A\subseteq B(x_0,r)$. (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

Two maps into a metric space are at bounded distance when the distance between their values is bounded uniformly (Bounded distance between two maps into a metric space).

Given: The objects and hypotheses in the Example.

The quasi-isometry group of a metric space is the set of quasi-isometries of it modulo bounded distance (The quasi-isometry group of a metric space).

The quasi-isometry group is a group under composition, and a quasi-isometry induces an isomorphism between the quasi-isometry groups of its source and target (Quasi-isometries modulo bounded distance form a group, and a quasi-isometry induces an isomorphism of these groups).

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

It is written $\lfloor x\rfloor$ and called the integer part, or floor, of $x$. (Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$).

Define $d_{\mathbb{R}}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ by $d_{\mathbb{R}}(x,y):=\mathrm{\mid }x-y\mathrm{\mid }$. (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$. (Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$).