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\hyphenation{
con-tinu-ity
hyper-bol-ic
par-ametris-ation
Loba-chev-sky
Min-kow-ski
}
\datereceived{2018-06-11}
\dateaccepted{2018-12-04}
\dateposted{2019-07-02}
\editor{S. Cantat}
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\title[Self-representations of the M\"obius group]{Self-representations of the M\"obius group}
\alttitle{Auto-repr\'esentations du groupe de M\"obius}
\author[\initial{N.} \lastname{Monod}]{\firstname{Nicolas} \lastname{Monod}}
\address{EPFL (Switzerland)}
\email{nicolas.monod@epfl.ch}
\author[\initial{P.} \lastname{Py}]{\firstname{Pierre} \lastname{Py}}
\address{Instituto de Matem\'aticas, Universidad Nacional Aut\'onoma de M\'exico (M\'exico)}
\curraddr{IRMA, Universit\'e de Strasbourg \& CNRS\\
67084 Strasbourg (France)}
\email{ppy@math.unistra.fr}
\begin{abstract}
Contrary to the finite-dimensional case, the M\"obius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations.
The proofs are obtained in the equivalent setting of isometries of Lobachevsky spaces and use kernels of hyperbolic type, in analogy with the classical concepts of kernels of positive and negative type.
\end{abstract}
\begin{altabstract}
Contrairement au cas usuel de dimension finie, le groupe de M\"obius admet des auto-repr\'esentations int\'eressantes lorsqu'il est de dimension infinie. Nous les construisons et classifions toutes.
Les d\'emonstrations sont conduites dans le cadre \'equivalent des groupes d'isom\'etries des espaces de Lobatchevski et reposent sur le concept de noyau de type hyperbolique, en analogie avec la notion classique de noyau de type positif ou n\'egatif.
\end{altabstract}
\subjclass{53A35, 57S25, 53C50}
\keywords{M\"obius group, Lobatchevsky space, hyperbolic space, infinite-dimensional space}
\thanks{PP was partially supported by the French project ANR AGIRA and by project PAPIIT IA100917 from DGAPA UNAM.}
\begin{document}
\maketitle
\section{Introduction}
\subsection{Context}
For an ordinary connected Lie group, the study of its continuous \emph{self-representations} is trivial in the following sense: every injective self-representation is onto, and hence an automorphism.
In the infinite-dimensional case, another type of ``tautological'' self-representations presents itself. Namely, the group will typically contain isomorphic copies of itself as natural proper subgroups. For instance, a Hilbert space will be isomorphic to most of its subspaces.
Remarkably, some infinite-dimensional groups also admit completely different self-representations which are not in any sense smaller tautological copies of themselves. This phenomenon has no analogue in finite dimensions and the simplest case is as follows.
Let $E$ be a Hilbert space and $\Isom(E)\cong E \rtimes \OO(E)$ its isometry group. To avoid the obvious constructions mentioned above, we only consider \emph{cyclic} self-representations (in the affine sense). It is well known that there is a whole wealth of such self-representations. They are described by \emph{functions of conditionally negative type}. More precisely, the question becomes equivalent to describing all \emph{radial} functions of conditionally negative type on $E$ because one can arrange, by conjugating, that $\OO(E)$ maps to itself. Thus, upon identifying a ray with $\RR_+$, the question completely reduces to the study of a fascinating space of functions $\Psi\colon \RR_+\to \RR_+$. Moreover, recall that new such functions can be obtained by composing a given $\Psi$ with any \emph{Bernstein function}. (Reference monographs for affine actions and Bernstein functions are~\cite{Bekka-Harpe-Valette} and~\cite{Schilling-Song-Vondracek_2}, respectively.)
We see that this first example, $\Isom(E)$, has many -- almost too many -- self-represen\-tations for a precise classification. How about other infinite-dimensional groups? Are they too rigid to admit any, or again so soft as to admit too many?
%\bigskip
Considering that $\Isom(E)$ sits in the much larger \emph{M\"obius group} $\Mob(E)$ of $E$, this article answers the following questions:
%\medskip
\begin{itemize}
\item Does any non-tautological $\Isom(E)$-representation extend to $\Mob(E)$?
\item Can the irreducible self-representations of $\Mob(E)$ be classified?
\item Among all Bernstein functions, which ones correspond to M\"obius representations?
\end{itemize}
%\medskip
In short, the answer is that the situation is much more rigid than for the isometries $\Isom(E)$, but still remains much richer than in the finite-dimensional case. Specifically, there is exactly a one-parameter family of self-representations. This appears as a continuous deformation of the tautological representation, given by the Bernstein functions $x\mapsto x^t$ where the parameter $t$ ranges in the interval $(0, 1]$.
\subsection{Formal statements}
Recall that the M\"obius group $\Mob(E)$ is a group of transformations of the conform\-al sphere $\widehat E = E\cup\{\infty\}$; it is generated by the isometries of $E$, which fix $\infty$, and by the inversions $v \mapsto (r/\|v\|)^2 v$, where $r>0$ is the inversion radius (see e.g.~\cite[I.3]{Reshetnyak}). In particular it contains all homotheties.
A first basic formalisation of the existence part of our results is as follows. Let $E$ be an infinite-dimensional separable real Hilbert space.
\begin{imain}\label{thm:Mobnew}
For every $0 0\big\},
\]
where $\sH$ is a Hilbert space of Hilbert dimension $\alpha$. The visual boundary $\partial\HH^\alpha$ can be identified with the space of $B$-isotropic lines in $\RR\oplus \sH$. We simply write $\HHI$ for our main case of interest, namely the separable infinite-dimensional Lobachevsky space $\HHI=\HH^{\aleph_0}$.
The distance function associated with the hyperbolic metric is characterized by
\[
\cosh d(x,y) = B(x,y)
\]
and is therefore compatible with the ambient topology and complete. The group $\OO(B)$ of invertible linear operators preserving $B$ acts projectively on $\HH^\alpha$, inducing an isomorphism $\PO(B)\cong\Isom(\HH^\alpha)$. Alternatively, $\Isom(\HH^\alpha)$ is isomorphic to the subgroup of index two $\OO_+(B)<\OO(B)$ which preserves the upper sheet $\HH^\alpha$. See Proposition~3.4 in~\cite{Burger-Iozzi-Monod}.
An isometric action on $\HH^\alpha$ is called \emph{elementary} if it fixes a point in $\HH^\alpha$ or in $\partial\HH^\alpha$, or if it preserves a line. Any non-elementary action preserves a \emph{unique} minimal hyperbolic subspace and $\HH^\alpha$ is itself this minimal subspace if and only if the associated linear representation is irreducible~\cite[\S4]{Burger-Iozzi-Monod}.
\subsection{Second model}\label{sec:model}
It is often convenient to use another model for the Minkowski space $\RR\oplus \sH$, as follows. Suppose given two points at infinity, represented by isotropic vectors $\xi_1, \xi_2$ such that $B(\xi_1, \xi_2)=1$. Define $E$ to be the $B$-orthogonal complement $\{\xi_1, \xi_2\}^\perp$. Then $-B$ induces a Hilbert space structure on $E$. We can now identify $(\RR\oplus \sH, B)$ with the space ${\RR^2\oplus E}$ endowed with the bilinear form $\bx$ defined by
\[
\bx\big((s_1,s_2)\oplus v, (s_1', s_2')\oplus v'\big)= s_1 s_2' + s_2 s_1' - \langle v,v'\rangle
\]
in such a way that the isomorphism takes $\xi_1, \xi_2$ to the canonical basis vectors of $\RR^2$ (still denoted $\xi_i$) and that $\HH^\alpha$ is now realised as
\[
\HH^\alpha=\big\{x=(s_1, s_2, v) : B'(x,x) = 1 \text{ and } s_1 > 0\big\},
\]
noting that $s_1>0$ is equivalent to $s_2>0$ given the condition ${B'(x,x) = 1}$. We can further identify $\partial \HH^\alpha$ with $\widehat E=E\cup\{\infty\}$ by means of the following parametrisation by $B'$-isotropic vectors:
\begin{equation}\label{eq:para:bord}
v \longmapsto \big(\tfrac12 \|v\|^2,1\big)\oplus v,\qquad \infty \longmapsto \xi_1.
\end{equation}
In particular, $0\in E$ corresponds to $\xi_2$. This parametrisation intertwines the action of $\Isom(\HH^\alpha)$ with the M\"obius group of $E$. For instance, the Minkowski operator exchanging the coordinates of the $\RR^2$ summand corresponds to the inversion in the sphere of radius~$\sqrt2$ around $0\in E$.
\subsection{Subspaces}\label{sec:subspace}
The hyperbolic subspaces of $\HH^\alpha$ are exactly all subsets of the form $\HH^\alpha \cap N$ where $N<\RR\oplus \sH$ is a closed linear subspace of $\RR\oplus\sH$. It is understood here that we accept the empty set, points and (bi-infinite) geodesic lines as hyperbolic subspaces.
\begin{defi}
The \emph{hyperbolic hull} of a subset of $\HH^\alpha$ is the intersection of all hyperbolic subspaces containing it. A subset is called \emph{hyperbolically total} if its hyperbolic hull is the whole ambient $\HH^\alpha$.
\end{defi}
Thus the hyperbolic hull of a subset $X\se \HH^\alpha$ coincides with $\HH^\alpha\cap\overline{\mathrm{span}}(X)$. It follows that $X$ is hyperbolically total if and only if it is total in the topological vector space $\RR\oplus \sH$.
%\medskip
There is a bijective correspondence, given by $\sH'\mapsto \HH^\alpha \cap (\RR\oplus \sH')$, between Hilbert subspaces $\sH'< \sH$ and hyperbolic subspaces that contain the point $1\oplus 0$.
In the second model, $E'\mapsto \HH^\alpha \cap (\RR^2\oplus E')$ is a bijective correspondence between Hilbert subspaces $E'0$ since the constant function~$1$ satisfies Definition~\ref{def:KHT} trivially. In view of Proposition~\ref{prop:KHT}, it suffices to prove that for any hyperbolic space $\HH^\alpha$ with distance $d$, where $\alpha$ is an arbitrary cardinal, the kernel
\[
(\cosh d)^t\colon \HH^\alpha\times \HH^\alpha \lra \RR
\]
is of hyperbolic type. Definition~\ref{def:KHT} considers finitely many points at a time, which are therefore contained in a finite-dimensional hyperbolic subspace of $\HH^\alpha$ (see e.g. Remark~3.1 in~\cite{Burger-Iozzi-Monod}). For this reason, it suffices to prove the above statement for $\HH^m$ with $m\in \NN$ arbitrarily large -- but fixed for the rest of this proof.
Given an integer $n\geq m$, we choose an isometric embedding $\HH^m\se \HH^n$ and consider the map
\[
f^n_t\colon \HH^n \lra \HHI
\]
that we provided in Theorem~C of~\cite{Monod-Py} (it was simply denoted by $f_t$ in that reference, but we will shortly let $n$ vary). Consider the kernel
\[
\beta_n\colon \HH^m \times \HH^m \lra \RR, \qquad \beta_n(x,y) = \cosh d\big(f^n_t(x), f^n_t(y)\big)
\]
obtained by restriction to $\HH^m\se \HH^n$; it is of hyperbolic type by Proposition~\ref{prop:KHT}. The proof will therefore be complete if we show that $\beta_n$ converges pointwise to $(\cosh d)^t$ on $\HH^m \times \HH^m$.
Choose thus $x,y\in \HH^m$. We computed an integral expression for the quantity $\beta_n(x,y)=\cosh d\big(f^n_t(x), f^n_t(y)\big)$ in \S3.B and \S3.C of~\cite{Monod-Py}. Namely, writing $u=d(x,y)$, we established
\[
\beta_n(x,y) =\int_{\bS^{n-1}}\big(\cosh(u)-b_{1}\sinh(u)\big)^{-(n-1+t)} \, db,
\]
where $db$ denotes the integral against the normalised volume on the sphere $\bS^{n-1}$, and $b_1$ is the first coordinate of $b$ when $b$ is viewed as a unit vector in $\RR^n$. We further recall (see~\cite[(3.vi)]{Monod-Py}) that
\[
\big(\cosh(u)-b_{1}\sinh(u) \big)^{-(n-1)}
\]
is the Jacobian of some hyperbolic transformation $g_u\inv$ of $\bS^{n-1}$. We can therefore apply the change of variable formula for $g_u$ and obtain
\begin{equation}\label{eq:I_u}
\begin{split}
\beta_n(x,y) &=\int_{\bS^{n-1}} \fhi(g_u b) \, \dd b, \quad\text{where} \\
\fhi(b) &=\big(\cosh(u)-b_{1}\sinh(u) \big)^{-t}.
\end{split}
\end{equation}
The transformation $g_u$ is given explicitly in~\cite{Monod-Py}, namely it is $g_{u}=g_{e^{u},0,\Id}$ as defined in \S2.A of~\cite{Monod-Py}. These formulas show that the first coordinate of $g_u b$ is
\[
(g_u b)_1 = \frac{\sinh(u)+ b_1 \cosh(u)}{\cosh(u) + b_1 \sinh(u) }.
\]
Entering this into~\eqref{eq:I_u}, we readily compute
\[
\beta_n(x,y) =\int_{\bS^{n-1}} \big(\cosh(u)+b_{1}\sinh(u) \big)^{t}\, \dd b.
\]
We are thus integrating on $\bS^{n-1}$ a continuous function depending only upon the first variable $b_1$, which is now independent of $n$. Therefore, when $n$ tends to infinity, the concentration of measure principle implies that this integral converges to the value of that function on the equator $\{b_1=0\}$. Since this equatorial value is $(\cosh(u))^t$, we have indeed proved that $\beta_n(x,y)$ converges to $(\cosh(d(x,y))^t$, as was to be shown.
\end{proof}
\section{On representations arising from kernels}
\subsection{General properties}
Let $G$ be a group and $F\colon G\to \RR$ a function of hyperbolic type. According to Corollary~\ref{cor:KHT-group}, this gives rise to an isometric $G$-action on a hyperbolic space $\HH^\alpha$ together with a point $p\in \HH^\alpha$ whose orbit is hyperbolically total in $\HH^\alpha$, and such that
\[
F(g) = \cosh d(gp, p) \qquad (\forall g\in G).
\]
We now investigate the relation between the geometric properties of this $G$-action and the properties of the function $F$.
The Cartan fixed-point theorem, in the generality presented, for example in~\cite[II.2.8]{Bridson-Haefliger}, implies the following.
\begin{lem}\label{lem:Cartan}
The function $F$ is bounded if and only if $G$ fixes a point in $\HH^\alpha$.
\end{lem}
Fixed points at infinity are a more subtle form of elementarity for the $G$-action; we begin with the following characterization for kernels.
\begin{prop}\label{prop:horo:neg}
Let $\beta$ be an unbounded kernel of hyperbolic type on a set~$X$ and consider the map $f\colon X\to\HH^\alpha$ granted by Proposition~\ref{prop:KHT}.
Then $f(X)$ is contained in a horosphere if and only if $\beta-1$ is of conditionally negative type.
\end{prop}
In particular we deduce the corresponding characterization for the $G$-actions.
\begin{coro}\label{cor:horo:neg}
Suppose $F$ unbounded. Then the orbit $G p$ is contained in a horosphere if and only if $F-1$ is of conditionally negative type.
\end{coro}
\begin{proof}[Proof of Proposition~\ref{prop:horo:neg}]
Suppose that $f(X)$ is contained in a horosphere. We can choose the model described in Section~\ref{sec:horo} in such a way that this horosphere is $\sigma_0(E)$ in the notations of that section. Therefore, equation~\eqref{eq:horo:CNT} implies for all $x,y\in X$ the relation
\[
\beta(x, y) = \cosh d(f(x), f(y)) = 1 + \tfrac 12 \, \big\| \sigma_0\inv (f(x)) - \sigma_0\inv (f(y))\big \|^2,
\]
where $\|\cdot\|$ is the norm of the Hilbert space $E$ parametrising the horosphere. This witnesses that $\beta-1$ is of conditionally negative type.
Conversely, if $\beta-1$ is of conditionally negative type, then the usual affine GNS construction (see e.g. \S C.2 in~\cite{Bekka-Harpe-Valette}) provides a Hilbert space $E'$ and a map $\eta\colon X\to E'$ such that $\eta(X)$ is total in $E'$, and such that
\[
\beta(x,y) -1 = \tfrac 12 \, \| \eta(x) - \eta(y) \|^2
\]
holds for all $x,y$. Now $\sigma_0\circ \eta$ is a map to a horosphere in the hyperbolic space $\HH'$ corresponding to $E'$ in the second model (Section~\ref{sec:model}), centred at $\xi_1\in\partial \HH'$. Let $\HH''\se\HH'$ be the hyperbolic hull of $\sigma_0\circ \eta(X)$ and observe that its boundary contains $\xi_1$ since $\beta$ is unbounded. Thus $\sigma_0\circ \eta(X)$ is contained in a horosphere of $\HH''$. By Theorem~\ref{thm:KHT-char}, $\sigma_0\circ \eta$ can be identified with $f$ and hence the conclusion follows.
\end{proof}
\subsection{Individual isometries}
The \emph{type} of an individual group element for the action defined by $F$ can be read from $F$. Recall first that the \emph{translation length} $\ell(g)$ associated to any isometry $g$ of any metric space $Y$ is defined by
\[
\ell(g) = \inf \big\{d(g y, y) : y\in Y\big\}.
\]
We now have the following trichotomy.
\begin{prop}\label{prop:tricho}
For any $g\in G$, the action defined by $F$ satisfies
\[
\ell(g) = \ln \Big(\lim_{n\to\infty} F(g^n)^{\frac1n} \Big).
\]
Moreover, exactly one of the following holds.
\begin{enumerate}
\item $F(g^n)$ is uniformly bounded over $n\in\NN$; then $g$ is \emph{elliptic}: it fixes a point in $\HH^\alpha$.\label{pt:elliptic}
\item $F(g^n)$ is unbounded and $\ell(g)=0$; then $g$ is \emph{neutral parabolic}: it fixes a unique point in $\partial \HH^\alpha$ and preserves all corresponding horospheres but has no fixed point in $\HH^\alpha$.\label{pt:parabolic}
\item $\ell(g)>0$; then $g$ is \emph{hyperbolic}: it preserves a unique geodesic line in $\partial \HH^\alpha$ and translates it by $\ell(g)$.\label{pt:hyperbolic}
\end{enumerate}
\end{prop}
\begin{rema}
Consider the Picard--Manin space associated to the Cremona group $\Bir(\mathbf{P}^{2})$ as mentioned in Example~\ref{ex:PM}. Recall that the limit $\lim_{n\to\infty} \deg(g^n)^{1/n}$ is the \emph{dynamical degree} of the birational transformation $g$. Thus we see that the translation length is the logarithm of the dynamical degree, which is a basic fact in the study of the Picard--Manin space.
\end{rema}
\begin{proof}[Proof of Proposition~\ref{prop:tricho}]
Since $\arcosh (F(g^n)) = d(g^n p, p)$, we see that
\[
\ln \Big(\lim_{n\to\infty} F(g^n)^{\frac1n} \Big) = \lim_{n\to\infty} \tfrac1n d(g^n p, p).
\]
Now the statements of the proposition hold much more generally. Recall that if $p$ is a point of an arbitrary \cat0 space $Y$ on which $G$ acts by isometries, then the translation length of $g\in G$ satisfies
\begin{equation}\label{eq:length}
\ell(g) = \lim_{n\to\infty} \tfrac1n d(g^n p, p),
\end{equation}
see e.g.~Lemma~6.6$\MK$(2) in~\cite{Ballmann-Gromov-Schroeder}. If in addition $X$ is complete and \cat{-1}, then the above trichotomy holds, see for instance \S4 in~\cite{Burger-Iozzi-Monod}.
\end{proof}
\section{Self-representations of \texorpdfstring{$\Isomi$}{Is(H)}}
\subsection{Definition of \texorpdfstring{$\ro^\infty_t$}{}}\label{defrot}
We choose a point $p_1\in\HHI$ and consider the corresponding function of hyperbolic type $F_1$ given by the tautological representation of $\Isom(\HHI)$ on $\HHI$. We denote by $\OO$ the stabiliser of $p_1$, which is isomorphic to the infinite-dimensional orthogonal group.
Fix $00$, independent of $v$. Conjugating once more by a homothety, we can assume that $c=1$.
It remains only to justify that the induced self-representation of $\Isom(E)$ is affinely irreducible: suppose that $F\se E$ is a closed affine subspace invariant under $\ro^\infty_t(\Isom(E))$. As in the proof of Theorem~\ref{thm:exunic}, we denote by $\HHI_n\se \HHI$ the unique minimal invariant hyperbolic subspace under $\Isom(\HH^n)$ seen as a subgroup of the source $\Mob(E)$. The previous discussion implies that $0\in F$, hence $F$ is linear. It further implies that $\HHI_n$ is of the form
\[
\HHI\cap \left(\RR^2\oplus E_n\right)
\]
for some increasing sequence of closed subspaces $E_n \se E$ with dense union. It thus suffices to show $E_n\se F$. This follows from Lemma~2.2 and Proposition~2.4 in~\cite{Monod-Py}.
\end{proof}
\section{Automatic continuity}\label{sec:auto}
Tsankov~\cite{Tsankov13} proved that every isometric action of $\OO$ on a Polish metric space is continuous. We do not know whether $\Isomi$ enjoys such a strong property. However, we shall be able to prove the automatic continuity of Theorem~\ref{thm:auto} by combining Tsankov's result for $\OO$ with the following fact about the local structure of $\Isomi$.
\begin{prop}\label{prop:auto}
Let $\OO$ be the stabiliser in $\Isomi$ of a point in $\HHI$ and let $g\notin\OO$. Let further $U\se \OO$ be a neighbourhood of the identity in $\OO$. Then $\OO g\inv U g \OO$ is a neighbourhood of $\OO$ in $\Isomi$.
\end{prop}
\begin{proof}
We claim that the set
\[
J=\big\{ d(g p, u g p) : u\in U \big\}
\]
contains the interval $[0, \epsilon)$ for some $\epsilon>0$. Indeed, consider a (totally geodesic) copy of $\HH^2$ in $\HHI$ containing both $p$ and $g p$ and denote by $\SO(2)< \OO$ a corresponding lift of the orientation-preserving stabiliser of $p$ in $\Isom(\HH^2)^\circ$. Note that $\SO(2)$ does not fix $gp$. Since $\SO(2)$ is locally connected, there is a connected neighbourhood $V$ of the identity in $\SO(2)$ with $V\se \SO(2)\cap U$. Moreover, since $\SO(2)$ is connected but does not fix $g p$, we deduce that $V$ cannot fix $g p$. Considering that $d(g p, v g p)$ is in $J$ when $v\in V$, and that $V$ is connected, the claim follows.
To establish the proposition, we shall prove that every $h\in\Isomi$ with $d(p, h p)<\epsilon$ lies in $\OO g\inv U g \OO$. By the claim, there is $u\in U$ with $d(g p, u g p)= d(p, h p)$. In other words, there is $q\in g\inv U g$ with $d(p, q p) = d(p, h p)$. By transitivity of $\OO$ on any sphere centred at $p$, there is $q'\in\OO$ with $q' q p = hp$. We conclude $h\in q' q\OO$, as claimed.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:auto}]
Let $\ro\colon\Isomi\to\Isomi$ be an irreducible self-rep\-re\-sen\-ta\-tion and let $\OO<\Isomi$ be the stabiliser of a point in $\HHI$ for its tautological representation. Then $\ro(\OO)$ has bounded orbits by~\cite[p.~190]{Ricard-Rosendal} and hence fixes a point $p\in\HHI$ by the Cartan fixed-point theorem~\cite[II.2.8]{Bridson-Haefliger}. We claim that the function
\[
D\colon \Isomi \longrightarrow \RR_+, \qquad D(g) = d\big(\ro(g) p, p\big)
\]
is continuous at $e\in\Isomi$. Thus let $(g_n)$ be a sequence converging to $e$ in $\Isomi$. We fix some $g\notin\OO$; then Proposition~\ref{prop:auto} implies that we can write $g_n = k_n g\inv u_n g k'_n$ for $k_n, u_n, k'_n\in\OO$ such that the sequence $(u_n)$ converges to~$e$. Since $\ro(\OO)$ fixes $p$, we have
\[
D(g_n) = d\big(\ro(u_n)\ro(g)p, \ro(g)p\big).
\]
Tsankov~\cite{Tsankov13} proved that every isometric action of $\OO$ on a Polish metric space is continuous; therefore the right-hand side above converges to zero as $n\to\infty$, proving the claim.
It now follows that $D$ is continuous on all of $\Isomi$, because if $g_n\to g_\infty$ in $\Isomi$, we can estimate
\[
\Big| d\big(\ro(g_n) p, p\big) - d\big(\ro(g_\infty) p, p\big) \Big| \leq d\big(\ro(g_n) p, \ro(g_\infty) p\big) = d\big(\ro(g_\infty\inv g_n) p, p\big).
\]
Equivalently, the function of hyperbolic type $\cosh D$ is continuous. Since $\ro$ is irreducible, the orbit of $p$ is hyperbolically total. It follows that $\ro$ is continuous, see Theorem~\ref{thm:KHT-char} and Corollary~\ref{cor:KHT-group}.
\end{proof}
\section{Finite-dimensional post-scripta}\label{sec:PS}
\begin{flushright}
\begin{minipage}[t]{\linewidth}\itshape\small
Et je vais te prouver par mes raisonnements\ldots\\
Mais malheur \`a l'auteur qui veut toujours instruire\,!\\
Le secret d'ennuyer est celui de tout dire.
\begin{flushright}
\upshape\small --- Voltaire, \itshape\ Sur la nature de l'homme, \upshape\ 1737\\ Vol.~I p.~953 of the 1827 edition by Jules Didot l'a\^{\i}n\'e (Paris).
\end{flushright}
\end{minipage}
\end{flushright}
\bigskip
Our previous work~\cite{Monod-Py} focused on representations $\ro^n_t$ into $\Isomi$ of the \emph{finite-dimensional} groups $\Isom(\HH^n) \cong \PO(1,n)$. A main application was the construction of exotic locally compact deformations of the classical Lobachevsky space $\HH^n$. Nonetheless, a significant part of that article was devoted to the ana\-lysis of an equivariant map
\begin{equation*}
f^n_t\colon \HH^n \lra \HHI
\end{equation*}
canonically associated to the representation with parameter $0