Proof. We consider in this proof $X={{{\mathbb H}}^2_{{\mathbb R}}}$ , where ${{{\mathbb H}}^n_{{\mathbb R}}}$ is the upper half-space model of the real hyperbolic space of dimension n (with constant sectional curvature $-1$ ). We again denote by $\iota :T^1{{{\mathbb H}}^n_{{\mathbb R}}}{\rightarrow } T^1{{{\mathbb H}}^n_{{\mathbb R}}}$ the antipodal map $v\mapsto -v$ . We normalize, as we may, the Patterson density $(\mu _x)_{x\in X}$ of the (non-uniform arithmetic) lattice $\Gamma $ of the orientation-preserving isometry group G of X to consist of probability measures. The critical exponent of $\Gamma $ is

(18) $$ \begin{align} \delta_\Gamma=1. \end{align} $$

We start the proof by recalling precisely a bijection between G and the unit tangent bundle of ${{{\mathbb H}}^2_{{\mathbb R}}}$ . We denote by $\cdot $ the action of G by homographies on ${{{\mathbb H}}^2_{{\mathbb R}}}\cup \partial _\infty {{{\mathbb H}}^2_{{\mathbb R}}}$ , as well as its derived action on $T^1{{{\mathbb H}}^2_{{\mathbb R}}}$ . We fix $v^\bullet =(i,-i)\in T^1{{{\mathbb H}}^2_{{\mathbb R}}}$ , which is the unit tangent vector at the base point i of ${{{\mathbb H}}^2_{{\mathbb R}}}$ pointing vertically down (its length is not adequate in the picture below, but this makes the picture easier to understand).

We denote by ${\widetilde {\varphi }}:G{\rightarrow } T^1{{{\mathbb H}}^2_{{\mathbb R}}}$ the orbital map at $v^\bullet $ , defined by $g\mapsto g\cdot v^\bullet $ , which is a G-equivariant (for the left actions) homeomorphism, and by $\varphi : \Gamma {\backslash } G{\rightarrow } \Gamma {\backslash } T^1{{{\mathbb H}}^2_{{\mathbb R}}}$ its quotient homeomorphism. We define $S=\big[\!\begin {smallmatrix} 0 & -1\\1&0\end {smallmatrix}\!\big]$ , which is an order-two element of $\Gamma $ . The involution S satisfies the following remarkable properties, in the connected center-free semisimple real Lie group G, that it anticommutes with the standard Cartan subgroup $\Phi ^{\mathbb R}= \{\Phi ^t:t\in {\mathbb R}\}$ of G and that the conjugation by S is the standard Cartan involution $g\mapsto \;^tg^{-1}$ of G:

(19) $$ \begin{align} \text{for all }g\in G, \text{we have}\quad ^t{\kern-1.5pt}g^{-1}=SgS^{-1}\quad \mathrm{and}\quad \text{for all }s\in {\mathbb R}, \text{we have}\quad S\Phi^sS^{-1}=\Phi^{-s}. \end{align} $$

Hence, with $\Theta $ defined just before the statement of Corollary 4.1, for all $x\in \Gamma {\backslash } G$ and $s\in {\mathbb R}$ , we have

$$ \begin{align*} \Theta(x\Phi^s)=\Theta(x)\Phi^{-s}. \end{align*} $$

The element S represents a generator of the standard Weyl group $N_G(\Phi ^{\mathbb R})/Z_G(\Phi ^{\mathbb R})$ (whose order is $2$ ). The following properties say that the action of the geodesic flow ${\texttt{g}^{t}}$ on $T^1{{{\mathbb H}}^2_{{\mathbb R}}}$ corresponds to multiplication on the right by $\Phi ^t$ in G, and that the antipodal map on $T^1{{{\mathbb H}}^2_{{\mathbb R}}}$ corresponds to multiplication on the right by S in G:

(20) $$ \begin{align} \text{for all }t\in {\mathbb R}\text{ and } g\in G,\text{we have}\quad {\texttt{g}^{t}}{\widetilde{\varphi}}(g)={\widetilde{\varphi}}(g\Phi^t) \quad\mathrm{and}\quad \iota\;{\widetilde{\varphi}}(g)={\widetilde{\varphi}}(gS). \end{align} $$

By the above two centred formulas and since $S\in \Gamma $ , the homeomorphism $\varphi $ relates the antipodal map $\iota $ on $\Gamma {\backslash } T^1{{{\mathbb H}}^2_{{\mathbb R}}}$ to the Cartan involution $\Theta $ on $\Gamma {\backslash } G$ by

$$ \begin{align*} \iota\circ\varphi =\varphi\circ\Theta. \end{align*} $$

Let ${\cal H}_\infty =\{z\in {{{\mathbb H}}^2_{{\mathbb R}}}:{\operatorname {Im}} \;z\geq 1\}$ , which is a (closed) horoball centred at $\infty $ in ${{{\mathbb H}}^2_{{\mathbb R}}}$ . The subgroup $\Gamma _H$ is equal to the stabilizer $\Gamma \!_{{\cal H}_\infty }$ of ${\cal H}_\infty $ in $\Gamma $ . We define

(21) $$ \begin{align} {\cal D}^-={\cal D}^+=(\gamma\cdot{\cal H}_\infty)_{\gamma\in\Gamma}, \end{align} $$

which are locally finite $\Gamma $ -equivariant families of horoballs. The map from $\Gamma -\Gamma \!_{{\cal H}_\infty }$ to ${\mathbb R}$ defined by $\gamma = \big[\!\begin {smallmatrix} p & r\\q&s\end {smallmatrix}\!\big]\mapsto \gamma \cdot \infty ={p}/{q}$ (where we assume, as we may, that $q>0$ ) induces a bijection from $\Gamma \!_{{\cal H}_\infty }{\backslash }(\Gamma -\Gamma \!_{{\cal H}_\infty })/ \Gamma \!_{{\cal H}_\infty }$ to the additive group ${\mathbb Q}/{\mathbb Z}$ such that we have $d({\cal H}_\infty ,\gamma \cdot {\cal H}_\infty ) =2\ln q$ (see the above picture). In particular, for all $t,t_0\in {\mathbb R}$ , we have

(22) $$ \begin{align} d({\cal H}_\infty,\gamma\cdot{\cal H}_\infty)\leq t-t_0\quad\mathrm{if~and~only~if}\quad q\leq e^{({t-t_0})/{2}}. \end{align} $$

Identifying geodesic rays in ${\cal G}_{+,0}X$ and geodesic lines in ${\cal G} X$ with their unit tangent vector at time $0$ , we have

$$ \begin{align*} {\partial^1_{+}} {\cal H}_\infty=W^-(v^\bullet)={\widetilde{\varphi}}(H), \end{align*} $$

so that, by the left equivariance of ${\widetilde {\varphi }}$ , the orbits of the right action of H on G correspond to the strong unstable leaves for the geodesic flow on $T^1{{{\mathbb H}}^2_{{\mathbb R}}}$ . Similarly, using equation (20), we have

$$ \begin{align*} {\partial^1_{-}}{\cal H}_\infty=W^+(-v^\bullet)=\iota W^-(v^\bullet)= {\widetilde{\varphi}}(H\,S)\quad\mathrm{and}\quad W^{0+}(-v^\bullet)={\widetilde{\varphi}}(H\Phi^{\mathbb R}\,S). \end{align*} $$

More precisely, using the right-hand parts of equations (19) and (20),

(23) $$ \begin{align} \text{for all }s,r\in{\mathbb R},\text{we have}\quad {\widetilde{\varphi}}({\mathfrak n}_-(r)\,\Phi^{-s}\,S)= {\widetilde{\varphi}}({\mathfrak n}_-(r)\,S\,\Phi^{s})= {\texttt{g}^{s}} \;\iota\,{\widetilde{\varphi}}({\mathfrak n}_-(r)). \end{align} $$

The endpoint map ${\widetilde {\psi }}:{\partial ^1_{+}}{\cal H}_\infty {\rightarrow }{\mathbb R}$ defined by $\rho \mapsto \rho _+$ is a $\Gamma _H$ -equivariant homeomorphism, such that we have ${\widetilde {\varphi }}^{-1}({\widetilde {\psi }}^{-1}(r))={\mathfrak n}_-(r)$ for all $r\in {\mathbb R}$ . We denote by $\psi :\Gamma _H{\backslash }{\partial ^1_{+}}{\cal H}_\infty {\rightarrow }{\mathbb R}/{\mathbb Z}$ the quotient homeomorphism, and we identify $\Gamma _H{\backslash }{\partial ^1_{+}} {\cal H}_\infty $ with its image in $\Gamma {\backslash } T^1{{{\mathbb H}}^2_{{\mathbb R}}}$ . For every $\gamma =\big[\!\begin {smallmatrix} p & r\\q&s \end {smallmatrix}\!\big] \in \Gamma -\Gamma _H$ , with $\rho _\gamma \in {\partial ^1_{+}}{\cal H}_\infty $ the geodesic ray entering perpendicularly in $\gamma \cdot {\cal H}_\infty $ , we have

(24) $$ \begin{align} {\widetilde{\varphi}}^{-1}(\rho_\gamma)={\mathfrak n}_-(\gamma\cdot\infty) \quad\mathrm{and}\quad \psi_*(\Delta_{\Gamma\rho_\gamma})=\Delta_{\gamma\cdot\infty\, \mod\! 1}= \Delta_{{p}/{q}\, \mod\! 1}. \end{align} $$

Furthermore, by [Reference Parkkonen, Paulin, Aravinda, Farrell and LafontPaP4, Theorem 9.11] or [Reference Parkkonen and PaulinPaP5, Proposition 20(2)] with $n=2$ , the skinning measure ${\widetilde {\sigma }}^\pm _{{\cal H}_\infty }=\iota_*{\widetilde {\sigma }}^\mp_{{\cal H}_\infty}$ is equal to twice the Riemannian volume of ${\partial ^1_{\mp }} {\cal H}_\infty $ , so that

(25) $$ \begin{align} \psi_*(\sigma^+_{{\cal D}^-})=2\,dx\quad\mathrm{and}\quad (\varphi^{-1})_*(\sigma^+_{{\cal D}^-})=2\,\mu_{\Gamma_H{\backslash} H}. \end{align} $$

By for instance [Reference Parkkonen, Paulin, Aravinda, Farrell and LafontPaP4, Theorem 9.10] or [Reference Parkkonen and PaulinPaP5, Proposition 20(1)] with $n=2$ , we have

$$ \begin{align*} \|m_{\mathrm{BM}}\|=4\pi\operatorname{vol}(\Gamma{\backslash}{{{\mathbb H}}^2_{{\mathbb R}}})=\frac{4\pi^2}3. \end{align*} $$

Mertens’s formula [Reference Hardy and WrightHaW, Theorem 330] (see also [Reference Parkkonen, Paulin, Dal’Bo and LecuirePaP3, §3] for a geometric proof) implies that, as $t{\rightarrow }+\infty $ ,

$$ \begin{align*} {\operatorname{Card}}\,{\mathcal{F}}_{t-t_0}\sim\frac{3}{\pi^2}e^{2({t-t_0})/{2}}= \frac{3}{\pi^2}e^{t-t_0}. \end{align*} $$

Since no element of $\Gamma $ pointwise fixes a non-trivial geodesic segment of ${{{\mathbb H}}^2_{{\mathbb R}}}$ , for every $\gamma \in \Gamma $ such that $d({\cal H}_\infty ,\gamma \cdot {\cal H}_\infty )>0$ , we have

$$ \begin{align*} m_\gamma=1. \end{align*} $$

For every $t_0\in {\mathbb R}$ , let us consider the truncation $\Phi ^{\geq t_0} =\{\Phi ^t:t\geq t_0\}$ of the Cartan subgroup $\Phi ^{\mathbb R}$ . For all $t\in {\mathbb R}$ and $\gamma \in \Gamma -\Gamma _H$ , by the two left-hand parts of equations (20) and (24), we have

(26) $$ \begin{align} (\varphi^{-1})_*(\Delta_{\Gamma{\texttt{g}^{t}}\rho_\gamma})= \Delta_{\Gamma n_{-}(\gamma\cdot\infty)\Phi^{t}}. \end{align} $$

By equation (23), the homeomorphism $\varphi ^{-1}$ maps the truncated stable leaf

$$ \begin{align*} \Gamma W^{0+}_{t_0}(-v^\bullet)= \bigcup_{s\geq t_0} \Gamma{\texttt{g}^{s}}\partial^1_-{\cal H}_\infty = \bigcup_{s\geq t_0} \Gamma{\texttt{g}^{s}}W^{+}(-v^\bullet) =\bigcup_{s\geq t_0} \Gamma {\texttt{g}^{s}} \iota W^{-}(v^\bullet) \end{align*} $$

to the truncated orbit $\Gamma H(\Phi ^{\geq t_0})^{-1}S$ in $\Gamma {\backslash } G$ of the lower triangular subgroup of G. Furthermore, by the left-hand part of equation (19) and since $S\in \Gamma $ for the first equality, by equation (23) for the third equality, by equations (7) and (18) for the fourth equality, and since $\iota _*\sigma ^-_{{\cal D}^+}=\sigma ^+_{{\cal D}^-}$ and by the right-hand part of equation (25) for the last equality, for all $s,r\in {\mathbb R}$ with $s\geq t_0$ , we have

$$ \begin{align*} &d((\varphi^{-1})_*(\mu^{0+}_{{\cal D}^+,t_0})) (\Theta(\Gamma n_{-}(r)\Phi^{-s})) = d((\varphi^{-1})_*(\mu^{0+}_{{\cal D}^+,t_0})) (\Gamma n_{-}(r)\Phi^{-s}S)\\&\quad= d\mu^{0+}_{{\cal D}^+,t_0}(\Gamma{\widetilde{\varphi}}(n_{-}(r)\Phi^{-s}S)) = d\mu^{0+}_{{\cal D}^+}(\Gamma{\texttt{g}^{s}}\iota\, {\widetilde{\varphi}}(n_{-}(r))) = e^{-s}d\sigma^-_{{\cal D}^+}(\Gamma\iota\, {\widetilde{\varphi}}(n_{-}(r)))\,ds \\ &\quad= e^{-s}(\varphi^{-1})_*\,\iota_*\,d\sigma^-_{{\cal D}^+}(\Gamma n_{-}(r))\,ds =2\;d\mu_{\Gamma_H{\backslash} H}(\Gamma n_{-}(r))\,e^{-s}\,ds. \end{align*} $$

Therefore, by the left-hand part of equation (25), for all $x\in {\mathbb R}/{\mathbb Z}$ , $y\in \Gamma _H{\backslash } H$ and $s\geq t_0$ , we have

(27) $$ \begin{align} d( (\psi\times \varphi^{-1})_*(\sigma^+_{{\cal D}^-}\otimes \mu^{0+}_{{\cal D}^+,t_0}))(x,\Theta(y\,\Phi^{-s}))= 4\;dx\;d\mu_{\Gamma_H{\backslash} H}(y)\,e^{-s}\,ds. \end{align} $$

By the linearity of the pushforward of measures and by equation (22), the left-hand part of equation (24), and equation (26), as $t{\rightarrow }+\infty $ , we have

(28) $$ \begin{align} &(\psi\times \varphi^{-1})_*\bigg(\|m_{\mathrm{BM}}\|\;e^{-\delta_\Gamma\, t} \sum_{\substack{\gamma\,\in\;\Gamma\!_{{\cal H}_\infty}\!\backslash \Gamma/\Gamma\!_{{\cal H}_\infty})\\ 0<d({\cal H}_\infty,\gamma \cdot {\cal H}_\infty)\leq\, t-t_0}} \; m_\gamma \;\Delta_{\Gamma\rho_\gamma}\otimes\Delta_{{\texttt{g}^{t}}\Gamma\rho_\gamma}\bigg) \nonumber\\&\quad=\frac{4\pi^2}3\;e^{-t}\sum_{r\in{\mathcal{F}}_{t-t_0}} \Delta_{r}\otimes\Delta_{\Gamma {\mathfrak n}_-(r)\Phi^t}\nonumber\\ &\quad\sim 4\,e^{-t_0}\;\frac 1{{\operatorname{Card}}\,{\mathcal{F}}_{t-t_0}}\sum_{r\in{\mathcal{F}}_{t-t_0}} \Delta_{r}\otimes\Delta_{\Gamma {\mathfrak n}_-(r)\Phi^t}. \end{align} $$

Since the product map $\psi \times \varphi ^{-1}$ is a homeomorphism from $(\Gamma W^-(v^\bullet )) \times (\Gamma W^{0+}_{t_0}(-v^\bullet ))$ to $({\mathbb R}/{\mathbb Z}) \times (\Gamma H(\Phi ^{\geq t_0})^{-1})$ , its pushforward map on measures is continuous for the weak-star convergence. Hence, Corollary 4.1 follows from equations (27) and (28) by Theorem 3.3(1) applied to the families ${\cal D}^\pm $ defined in equation (21).

Proof. We mostly indicate the differences with the proof of Corollary 4.1. We now consider $X={{{\mathbb H}}^3_{{\mathbb R}}}$ with coordinates $(z,u)\in {\mathbb C}\times \; ]0,+\infty [$ . The critical exponent of the (non-uniform arithmetic) lattice $\Gamma $ of the orientation-preserving isometry group G of X is now

$$ \begin{align*} \delta_\Gamma=2. \end{align*} $$

We denote by $\cdot $ the action of G by homographies on $ \partial _\infty {{{\mathbb H}}^3_{{\mathbb R}}}={\mathbb C}\cup \{\infty \}$ , by isometries on ${{{\mathbb H}}^3_{{\mathbb R}}}$ through the Poincaré extension, and by the derived action on $T^1{{{\mathbb H}}^3_{{\mathbb R}}}$ . We now fix the unit tangent vector $v^\bullet =((0,1), (0,-1)) \in T^1{{{\mathbb H}}^3_{{\mathbb R}}}$ . The stabilizer of $v^\bullet $ in G is equal to M and is hence centralized by $\Phi ^{\mathbb R}$ . The orbital map ${\widetilde {\varphi }}:g\mapsto g\cdot v^\bullet $ now defines a homeomorphism $\varphi : \Gamma {\backslash } G/M{\rightarrow } \Gamma {\backslash } T^1{{{\mathbb H}}^3_{{\mathbb R}}}$ . The order-two element $S=\big[\!\begin {smallmatrix} 0 & -1\\1&0\end {smallmatrix}\!\big]$ still belongs to $\Gamma $ . It normalizes M and $\Phi ^{\mathbb R}$ , and formulae (19) and (20) are still satisfied.

Now let ${\cal H}_\infty =\{(z,u)\in {{{\mathbb H}}^3_{{\mathbb R}}}:u\geq 1\}$ . With $\Phi ^{\geq t_0} =\{\Phi ^t:t\geq t_0\}$ , we again have

(30) $$ \begin{align} {\partial^1_{+}}{\cal H}_\infty=W^-(v^\bullet)={\widetilde{\varphi}}(H)\quad\mathrm{and}\quad W^{0+}_{t_0}(-v^\bullet) = \bigcup_{s\geq t_0} {\texttt{g}^{s}}{\partial^1_{-}}{\cal H}_\infty={\widetilde{\varphi}}(H(\Phi^{\geq t_0})^{-1}S). \end{align} $$

The subgroup $\Gamma _H$ is again equal to the stabilizer $\Gamma \!_{{\cal H}_\infty }$ of the horoball ${\cal H}_\infty $ in $\Gamma $ . We again consider the locally finite $\Gamma $ -equivariant families of horoballs

$$ \begin{align*} {\cal D}^+={\cal D}^-=(\gamma\cdot{\cal H}_\infty)_{\gamma\in\Gamma}. \end{align*} $$

The map $\gamma = \big[\!\begin {smallmatrix} p & r\\q&s\end {smallmatrix}\!\big]\mapsto \gamma \cdot \infty = {p}/{q}$ now induces, for every $t\in {\mathbb R}$ , a bijection from the set $\{[\gamma ]\in \Gamma \!_{{\cal H}_\infty }{\backslash } (\Gamma -\Gamma \!_{{\cal H}_\infty }) /\Gamma \!_{{\cal H}_\infty }: d({\cal H}_\infty , \gamma \cdot {\cal H}_\infty )\leq t\}$ to ${\mathcal {F}}_t$ . With $\rho _\gamma $ the element of ${\partial ^1_{+}}{\cal H}_\infty $ whose point at infinity is $\gamma \cdot \infty $ , the endpoint map ${\widetilde {\psi }}: {\partial ^1_{+}} {\cal H}_\infty {\rightarrow }{\mathbb C}$ now induces a homeomorphism $\psi : \Gamma _H{\backslash } {\partial ^1_{+}}{\cal H}_\infty {\rightarrow }{\mathcal {O}}^{\prime }_K{\backslash } {\mathbb C}$ , such that

$$ \begin{align*} \psi_*(\Delta_{\Gamma\rho_\gamma})=\Delta_{{\mathcal{O}}^{\prime}_K\gamma\cdot\infty}. \end{align*} $$

Let us compute the total mass of the induced Lebesgue measure $d\operatorname {Leb}_{{\mathcal {O}}^{\prime }_K {\backslash } {\mathbb C}}$ on ${\mathcal {O}}^{\prime }_K{\backslash } {\mathbb C}$ , yielding $dx$ after renormalization to a probability measure. Since the branched cover ${\mathcal {O}}_K{\backslash }{\mathbb C}{\rightarrow }{\mathcal {O}}^{\prime }_K{\backslash }{\mathbb C}$ is $({|{\mathcal {O}}_K^\times |}/{2})$ -sheeted outside the singular part and since ${\mathcal {O}}_K$ is generated as a ${\mathbb Z}$ -lattice of ${\mathbb C}$ by $1$ and $(D_K+i\sqrt {|D_K|})/2$ , we have

$$ \begin{align*} \|d\operatorname{Leb}_{{\mathcal{O}}^{\prime}_K{\backslash} {\mathbb C}}\!\|=\frac{2}{|{\mathcal{O}}_K^\times|}\; \|d\operatorname{Leb}_{{\mathcal{O}}_K{\backslash} {\mathbb C}}\!\|= \frac{\sqrt{|D_K|}}{|{\mathcal{O}}_K^\times|}. \end{align*} $$

Again by [Reference Parkkonen, Paulin, Aravinda, Farrell and LafontPaP4, Theorem 9.11] or [Reference Parkkonen and PaulinPaP5, Proposition 20(2)], now with $n=3$ , we have

$$ \begin{align*} \psi_*(\sigma^+_{{\cal D}^-})=4\,d\operatorname{Leb}_{{\mathcal{O}}^{\prime}_K{\backslash} {\mathbb C}}= \frac{4\,\sqrt{|D_K|}}{|{\mathcal{O}}_K^\times|}\,dx \quad \mathrm{and}\quad (\varphi^{-1})_*(\sigma^+_{{\cal D}^-}) =\frac{4\,\sqrt{|D_K|}}{|{\mathcal{O}}_K^\times|}\,d\mu_{\Gamma_H{\backslash} H}. \end{align*} $$

Again by [Reference Parkkonen, Paulin, Aravinda, Farrell and LafontPaP4, Theorem 9.10] or [Reference Parkkonen and PaulinPaP5, Proposition 20(1)], now with $n=3$ and with Humbert’s volume formula (see, for instance, [Reference Elstrodt, Grunewald and MennickeEGM, §§8.8 and 9.6]), we have

$$ \begin{align*} \|m_{\mathrm{BM}}\|=4\operatorname{Vol}({\mathbb S}^2)\operatorname{Vol}(\Gamma{\backslash}{{{\mathbb H}}^3_{{\mathbb R}}}) =\frac{4}{\pi}\;|D_K|^{3/2}\;\zeta_K(2). \end{align*} $$

Mertens’s formula for the quadratic imaginary number fields (see also [Reference Parkkonen, Paulin, Dal’Bo and LecuirePaP3, Theorem 3.1]) gives, using the action of $k\in {\mathcal {O}}_K$ on $(p,q)\in {\mathcal {O}}_K\times {\mathcal {O}}_K$ by horizontal shears $k\cdot (p,q)= (p+kq)$ , as $t{\rightarrow }+\infty $ ,

$$ \begin{align*} &{\operatorname{Card}}\,{\mathcal{F}}_{t-t_0}\\&\quad\sim\frac{2}{|{\mathcal{O}}_K^\times|} {\operatorname{Card}}\bigg({\mathcal{O}}_K{\backslash}\bigg\{\frac pq:\;p,q\in{\mathcal{O}}_K, p{\mathcal{O}}_K+q{\mathcal{O}}_K={\mathcal{O}}_K, 0<|q|\leq e^{(t-t_0)/2}\bigg\}\bigg) \\&\quad=\frac{2}{|{\mathcal{O}}_K^\times|^2} {\operatorname{Card}}({\mathcal{O}}_K{\backslash}\{(p,q)\in {\mathcal{O}}_K\times{\mathcal{O}}_K:p{\mathcal{O}}_K+q{\mathcal{O}}_K={\mathcal{O}}_K, 0<|q|^2\leq e^{t-t_0}\}) \\&\quad\sim\frac{2\,\pi}{|{\mathcal{O}}_K^\times|^2\;\zeta_K(2)\sqrt{|D_K|}}\; e^{2t-2t_0}. \end{align*} $$

Since ${\mathcal {O}}_K$ has finite index in ${\mathcal {O}}^{\prime }_K$ , there are only finitely many elliptic elements in $\Gamma $ up to conjugation by $\Gamma \cap H$ whose fixed point set contains $\infty $ as a point at infinity. There are only finitely many $\Gamma \!_{{\cal H}_\infty }$ -orbits of images of ${\cal H}_\infty $ by $\Gamma $ meeting ${\cal H}_\infty $ . Hence, there exists a finite subset F of the set of double cosets $\Gamma \!_{{\cal H}_\infty }{\backslash }\Gamma / \Gamma \!_{{\cal H}_\infty }$ such that for every element $[\gamma ]\in \Gamma \!_{{\cal H}_\infty }{\backslash }\Gamma / \Gamma \!_{{\cal H}_\infty }-F$ , we have

$$ \begin{align*} d({\cal H}_\infty,\gamma\cdot{\cal H}_\infty)>0 \quad \mathrm{and}\quad m_\gamma=1. \end{align*} $$

We have, similarly to equation (26), for all $\gamma \in \Gamma -\Gamma \!_{{\cal H}_\infty }$ and $t\in {\mathbb R}$ ,

$$ \begin{align*} (\varphi^{-1})_*(\Delta_{\Gamma{\texttt{g}^{t}}\rho_\gamma})= \Delta_{\Gamma n_{-}(\gamma\cdot\infty)\Phi^{t}M} \end{align*} $$

and, for all $y\in \Gamma _H{\backslash } H$ and $s\in {\mathbb R}$ with $s\geq t_0$ ,

$$ \begin{align*} d((\varphi^{-1})_*(\mu^{0+}_{{\cal D}^+,t_0})) (\Theta(y\,\Phi^{-s})) &=\|\sigma^-_{{\cal D}^+}\|\; d\mu_{\Gamma_H{\backslash} H}(y)\,e^{-2s}\,ds\\ & =\frac{4\;\sqrt{|D_K|}}{|{\mathcal{O}}_K^\times|}\; d\mu_{\Gamma_H{\backslash} H}(y)\,e^{-2s}\,ds. \end{align*} $$

The end of the proof of Corollary 4.2 now proceeds like that of Corollary 4.1.

Proof. We mostly indicate the differences with the proof of Corollary 4.1. We refer to [Reference GoldmanGol] as well as [Reference Parkkonen and PaulinPaP1, §6.1], [Reference Parkkonen and PaulinPaP6, §3] for background on complex hyperbolic geometry. We follow the conventions of the latter reference concerning the normalization of the sectional curvature and the choice of the Hermitian form with signature $(1,2)$ .

We now consider $X={{{\mathbb H}}^2_{{\mathbb C}}}$ the Siegel domain model of the complex hyperbolic plane, that is, the complex manifold

$$ \begin{align*} \{(w_0,w)\in{\mathbb C}^2: 2\,{\operatorname{Re}}\; w_0 -|w|^2>0\}, \end{align*} $$

endowed with the Riemannian metric

(32) $$ \begin{align} ds^2_{\,{{{\mathbb H}}^2_{{\mathbb C}}}}=\frac{1}{(2\,{\operatorname{Re}}\; w_0 -|w|^2)^2} ((dw_0-dw\, \overline{w})(\overline{dw_0}-w\,\overline{dw})+ (2\,{\operatorname{Re}}\; w_0 \kern1.6pt{-}\kern1.6pt|w|^2)\;dw\, \overline{dw}\kern1.5pt). \end{align} $$

This metric is normalized so that its sectional curvatures are in $[-4,-1]$ . The boundary at infinity of ${{{\mathbb H}}^2_{{\mathbb C}}}$ is

$$ \begin{align*} \partial_\infty{{{\mathbb H}}^2_{{\mathbb C}}}=\{(w_0,w)\in {\mathbb C}^2 : 2\,{\operatorname{Re}}\; w_0 -|w|^2=0\}\cup\{\infty\}. \end{align*} $$

Using homogeneous coordinates, we identify ${{{\mathbb H}}^2_{{\mathbb C}}}\cup \partial _\infty {{{\mathbb H}}^2_{{\mathbb C}}}$ with its image in ${\mathbb P}^2({\mathbb C})$ by the map $(w_0,w)\mapsto [w_0:w:1]$ and $\infty \mapsto [1:0:0]$ . We denote by $\cdot $ the projective action of G on ${{{\mathbb H}}^2_{{\mathbb C}}}\cup \partial _\infty {{{\mathbb H}}^2_{{\mathbb C}}}$ , as well as its derived action on $T^1{{{\mathbb H}}^2_{{\mathbb C}}}$ . The holomorphic isometry group of ${{{\mathbb H}}^2_{{\mathbb C}}}$ is G (acting projectively on ${\mathbb P}^2({\mathbb C})$ ).

The critical exponent of the (non-uniform arithmetic) lattice $\Gamma $ of G is now (see, for instance, [Reference Corlette and IozziCI, §6])

$$ \begin{align*} \delta_\Gamma=4. \end{align*} $$

We now fix $v^\bullet =((1,0),(-2,0))\in T^1{{{\mathbb H}}^2_{{\mathbb C}}}$ , which is indeed a unit tangent vector with footpoint $x^\bullet =(1,0)$ by equation (32). The stabilizer of $v^\bullet $ in G is equal to M and hence is centralized by $\Phi ^{\mathbb R}$ . The orbital map ${\widetilde {\varphi }}: g\mapsto g\cdot v^\bullet $ now defines a homeomorphism $\varphi : \Gamma {\backslash } G/M {\rightarrow } \Gamma {\backslash } T^1{{{\mathbb H}}^2_{{\mathbb C}}}$ .

For every $t\in {\mathbb R}$ , the element $\Phi ^t$ acts on ${{{\mathbb H}}^2_{{\mathbb C}}}$ by the map $(w_0,w)\mapsto (e^{-2t}w_0,e^{-t}w)$ . The geodesic line $\ell $ in ${{{\mathbb H}}^2_{{\mathbb C}}}$ such that $\ell (0)=x^\bullet $ and $\ell '(0)= v^\bullet $ is $t\mapsto (e^{-2\,t},0)$ . Hence, ${\texttt{g}^{t}} v^\bullet =\ell '(t)=(-2\,e^{-2\,t},0)= d_{x^\bullet }\Phi ^t(v^\bullet ) =\Phi ^t \cdot v^\bullet $ . Therefore, by equivariance,

$$ \begin{align*} \text{for all }t\in{\mathbb R} \text{ and }g\in G,\text{we have}\quad {\texttt{g}^{t}}{\widetilde{\varphi}}(g)= {\widetilde{\varphi}}(g\Phi^t). \end{align*} $$

The order-two element $S=\bigg [\!\begin {smallmatrix} \ \ 0 & 0&-1\\ \ \ 0&1&\ \ 0\\-1 & 0 & \ \ 0 \end {smallmatrix}\!\bigg ]\in \Gamma $ acts by the map $(w_0,w)\mapsto ({1}/{w_0}, -{w}/{w_0})$ on ${{{\mathbb H}}^2_{{\mathbb C}}}$ . It thus fixes the point $x^\bullet =(1,0)$ and acts by $-\operatorname {id}$ on $T_{x^\bullet }{{{\mathbb H}}^2_{{\mathbb C}}}$ . In particular, it maps $v^\bullet $ to $-v^\bullet $ . By equivariance,

$$ \begin{align*} \text{for all }g\in G,\text{we have}\quad\iota\,{\widetilde{\varphi}}(g)={\widetilde{\varphi}}(gS). \end{align*} $$

The element S centralizes M and normalizes $\Phi ^{\mathbb R}\,$ ; more precisely,

$$ \begin{align*} \text{for all }t\in {\mathbb R}, \text{we have}\quad S\Phi^tS^{-1}=\Phi^{-t}. \end{align*} $$

Since S is the projective image of the matrix of the Hermitian form $q=-z_0\overline {z_2} -z_2\overline {z_0}+|z_1|^2$ , we have $^*g\,S\,g=S$ for every $g\in G$ , hence

$$ \begin{align*} \text{for all }g\in G,\text{we have}\quad ^*g^{-1}=S\,g\,S^{-1}. \end{align*} $$

For all $x\in \Gamma {\backslash } G$ and $s\in {\mathbb R}$ , we again have $\Theta (x\Phi ^s) =\Theta (x)\Phi ^{-s}$ and $\iota \circ \varphi =\varphi \circ \Theta $ .

The (closed) horoball in ${{{\mathbb H}}^2_{{\mathbb C}}}$ centred at $\infty $ whose boundary $\partial {\cal H}_\infty $ contains $x^\bullet $ is

$$ \begin{align*} {\cal H}_\infty=\{(w_0,w)\in {{{\mathbb H}}^2_{{\mathbb C}}}:2\,{\operatorname{Re}}\; w_0 -|w|^2\geq 2\}. \end{align*} $$

The Heisenberg group H acts simply transitively on $\partial {\cal H}_\infty $ and on ${\partial ^1_{\pm }} {\cal H}_\infty $ , which contains $\pm v^\bullet $ . Thus again with $\Phi ^{\geq t_0}=\{\Phi ^t:t\geq t_0\}$ , equation (30) is still satisfied. By for instance [Reference Parkkonen and PaulinPaP6, p. 90], the stabilizer $\Gamma \!_{{\cal H}_\infty }$ in $\Gamma $ of the horoball ${\cal H}_\infty $ , as well as that of ${\partial ^1_{\pm }} {\cal H}_\infty $ , is equal to $\Gamma _H$ . The $\Gamma $ -equivariant families of horoballs

$$ \begin{align*} {\cal D}^-={\cal D}^+=(\gamma\cdot{\cal H}_\infty)_{\gamma\in\Gamma} \end{align*} $$

are again locally finite, since $\infty $ is a bounded parabolic fixed point of $\Gamma $ .

For every $\gamma \in \Gamma $ having a representative (whose choice does not change the following claims) in $\operatorname {SU}_q$ with first column $\bigg (\!\begin {smallmatrix} a \\ \alpha \\ c \end {smallmatrix}\!\bigg )\in {\cal M}_{3,1}({\mathcal {O}}_K)$ , we have $\gamma \notin \Gamma \!_{{\cal H}_\infty }$ if and only if $c\neq 0$ (see, for instance, [Reference Parkkonen and PaulinPaP1, Eq. (42)]) and then,

1. (i) since $\infty =[1:0:0]$ , the point at infinity $\gamma \cdot \infty $ is equal to $({a}/{c},{\alpha }/{c})$ ;

2. (ii) since H acts simply transitively on $\partial _\infty {{{\mathbb H}}^2_{{\mathbb C}}}-\{\infty \}$ , there exists a unique $r_\gamma \in H$ such that $r_\gamma \cdot 0=\gamma \cdot \infty $ , and we have $r_\gamma ={\mathfrak n}_-({a}/{c},{\alpha }/{c})$ ;

3. (iii) by [Reference Parkkonen and PaulinPaP1, Lemma 6.3], we have $d({\cal H}_\infty ,\gamma \cdot {\cal H}_\infty )=\ln |c|=\tfrac 12\ln ({\operatorname {\texttt{n}}}(c))$ .

Therefore, by [Reference Parkkonen and PaulinPaP1, Proposition 6.5(2)] with ${\cal I}={\mathcal {O}}_K$ , the map $\gamma \mapsto r_\gamma $ induces, for all $t,t_0\in {\mathbb R}$ , a bijection from $\{[\gamma ]\in \Gamma \!_{{\cal H}_\infty }{\backslash }(\Gamma -\Gamma \!_{{\cal H}_\infty })/\Gamma \!_{{\cal H}_\infty }: d({\cal H}_\infty ,\gamma \cdot {\cal H}_\infty )\leq t-t_0\}$ to ${\mathcal {F}}_{t-t_0}$ .

Again using the simple transitivity of the action of H on ${\partial ^1_{\pm }}{\cal H}_\infty $ , we have a $\Gamma _H$ -equivariant homeomorphism ${\widetilde {\psi }}: {\partial ^1_{+}}{\cal H}_\infty {\rightarrow } H$ which associates to $v\in {\partial ^1_{+}}{\cal H}_\infty $ the unique element ${\widetilde {\psi }}(v)\in H$ such that ${\widetilde {\psi }}(v)\cdot (v^\bullet )=v$ .

For every $\gamma \in \Gamma -\Gamma \!_{{\cal H}_\infty }$ , with $\rho _\gamma $ the element of ${\partial ^1_{+}}{\cal H}_\infty $ whose point at infinity is $\gamma \cdot \infty $ , the map ${\widetilde {\psi }}$ induces a homeomorphism $\psi : \Gamma _H{\backslash }{\partial ^1_{+}} {\cal H}_\infty {\rightarrow }\Gamma _H{\backslash } H$ such that

$$ \begin{align*} \psi_*(\Delta_{\Gamma\rho_\gamma})=\Delta_{\Gamma_H r_\gamma}. \end{align*} $$

In the remainder of the proof of Corollary 4.3, we use the same normalization of the Patterson–Sullivan measures $(\mu _x)_{x\in {{{\mathbb H}}^2_{{\mathbb C}}}}$ as in [Reference Parkkonen and PaulinPaP6, §4]. We denote by

$$\begin{align*}\delta_{x,y}=\left\{\begin{array}{@{}l} 1\quad\mathrm{if }\ x=y\\0\quad\mathrm{ otherwise}\end{array} \right.\end{align*}$$

the Kronecker symbol.

Lemma 4.4. We have

$$\begin{align*} \|\sigma ^\mp _{{\cal D}^\pm }\|= \frac{(1+2\,\delta _{D_K,-3})\,|D_K|}{4\,|{\mathcal {O}}_K^\times |}. \end{align*}$$

Proof. By [Reference Parkkonen and PaulinPaP6, Lemma 12(iv)] with $n=2$ , we have $\|\sigma ^\mp _{{\cal D}^\pm }\|=8\operatorname {Vol}(\Gamma \!_{{\cal H}_\infty }{\backslash } {\cal H}_\infty )$ , where $\operatorname {Vol}$ is the Riemannian volume. Denoting as in [Reference Parkkonen and PaulinPaP6, §3], for every $s\in {\mathbb R}$ ,

$$ \begin{align*} {\cal H}_s=\{(w_0,w)\in {{{\mathbb H}}^2_{{\mathbb C}}}:2\,{\operatorname{Re}}\; w_0 -|w|^2\geq s\}, \end{align*} $$

we have ${\cal H}_\infty ={\cal H}_2$ and the horoballs ${\cal H}_s$ all have the same stabilizer $\Gamma \!_{{\cal H}_s}=\Gamma \!_{{\cal H}_\infty }$ for $s\in {\mathbb R}$ . By the comment following [Reference Parkkonen and PaulinPaP6, Eq. (11)], we have $\operatorname {Vol}(\Gamma \!_{{\cal H}_\infty } {\backslash } {\cal H}_\infty )=\tfrac 14\,\operatorname {Vol}(\Gamma \!_{{\cal H}_1}{\backslash }{\cal H}_1)$ . The result then follows from [Reference Parkkonen and PaulinPaP6, Lemma 16] which says that

$$ \begin{align*} \operatorname{Vol}(\Gamma\!_{{\cal H}_1} {\backslash} {\cal H}_1)=\frac{(1+2\delta_{D_K,-3})\,|D_K|} {8\,|{\mathcal{O}}_K^\times|}.\\[-46pt] \end{align*} $$

Since we normalized $\mu _{\Gamma _H{\backslash } H}$ to be a probability measure, it follows from Lemma 4.4 that for $x\in \Gamma _H{\backslash } H$ ,

$$ \begin{align*} \psi_*(\sigma^+_{{\cal D}^-})=(\varphi^{-1})_*(\sigma^+_{{\cal D}^-})= \frac{(1+2\,\delta_{D_K,-3})\,|D_K|}{4\,|{\mathcal{O}}_K^\times|}\; \mu_{\Gamma_H{\backslash} H}. \end{align*} $$

By [Reference Parkkonen and PaulinPaP6, Lemma 12(iii)] with $n=2$ and by the Holzapfel–Stover volume formula (see [Reference Parkkonen and PaulinPaP6, Lemma 17] for the appropriate normalization of the volume form), we have

$$ \begin{align*} \|m_{\mathrm{BM}}\|=\frac{\pi^2}{2}\;\operatorname{Vol}(M)= \frac{\pi\,(1+2\,\delta_{D_K,-3})\,|D_K|^{5/2}\,\zeta_K(3)}{96\;\zeta(3)}. \end{align*} $$

By [Reference Parkkonen and PaulinPaP6, Eq. (21)] and the comment following it, the index of $H\cap \Gamma $ in $\Gamma _H$ is equal to ${|{\mathcal {O}}_K^\times |}/({1+2\,\delta _{D_K,-3}})$ . The map from

$$\begin{align*}\left\{(a,\alpha,c)\in{\mathcal{O}}_K \times {\mathcal{O}}_K \times{\mathcal{O}}_K: \begin{array}{c}\langle a,\alpha,c\rangle ={\mathcal{O}}_K\\ {\operatorname{\texttt{tr}}}(a\,\overline{c})={\operatorname{\texttt{n}}}(\alpha),c\neq 0\end{array}\!\right\} \end{align*}$$

to H defined by $(a,\alpha ,c)\mapsto {\mathfrak n}_-({a}/{c}, {\alpha }/{c})$ is $|{\mathcal {O}}_K^\times |$ -to-one onto its image. Hence, using the (lifted linear) action of ${\mathfrak n}_-(w_0,w)\in H\cap \Gamma $ on $(a,\alpha ,c)\in {\mathcal {O}}_K \times {\mathcal {O}}_K \times {\mathcal {O}}_K$ defined by

$$ \begin{align*} {\mathfrak n}_-(w_0,w)\cdot(a,\alpha,c)= (a+\overline{w}\,\alpha +w_0\,c,\alpha+\omega \,c,c), \end{align*} $$

by [Reference Parkkonen and PaulinPaP6, Theorem 4], for every $t_0\in {\mathbb R}$ , we have, as $t{\rightarrow }+\infty $ ,

$$ \begin{align*} {\operatorname{Card}}\;{\mathcal{F}}_{t-t_0}&=\frac{1+2\,\delta_{D_K,-3}}{|{\mathcal{O}}_K^\times|^2} \\&\qquad\times {\operatorname{Card}}\!\left(\!(H\kern1.2pt{\cap}\kern1.2pt \Gamma){\backslash}\!\kern-1pt\left\{\kern-2pt(a,\alpha,c)\kern1.2pt{\in}\kern1.2pt{\mathcal{O}}_K \kern1.2pt{\times}\kern1.2pt {\mathcal{O}}_K \kern1.2pt{\times}\kern1.2pt{\mathcal{O}}_K: \begin{array}{c}\langle a,\alpha,c\rangle ={\mathcal{O}}_K\\ {\operatorname{\texttt{tr}}}(a\,\overline{c})={\operatorname{\texttt{n}}}(\alpha)\\0<{\operatorname{\texttt{n}}}(c)\leq e^{2\,t-2\,t_0} \end{array}\!\!\right\}\kern-2pt\right) \\ &\quad\sim \frac{3\,(1+2\,\delta_{D_K,-3})\,\zeta(3)} {2\,\pi\,|{\mathcal{O}}_K^\times|^2\,\sqrt{|D_K|}\;\zeta_K(3)}\;e^{4\,t-4\,t_0}. \end{align*} $$

Since $H\cap \Gamma $ has finite index in $\Gamma _H=\Gamma \!_{{\cal H}_\infty }$ and acts freely on $\partial {\cal H}_\infty $ , there are only finitely many elliptic elements in $\Gamma $ up to conjugation by $\Gamma \cap H$ whose fixed point set contains $\infty =[1:0:0]$ as a point at infinity. There are only finitely many $\Gamma \!_{{\cal H}_\infty }$ -orbits of images of ${\cal H}_\infty $ by $\Gamma $ meeting ${\cal H}_\infty $ . Hence, there again exists a finite subset F of the set of double cosets $\Gamma \!_{{\cal H}_\infty }{\backslash }\Gamma / \Gamma \!_{{\cal H}_\infty }$ such that for every $[\gamma ]\in \Gamma \!_{{\cal H}_\infty }{\backslash }\Gamma / \Gamma \!_{{\cal H}_\infty }-F$ , we have

$$ \begin{align*} d({\cal H}_\infty,\gamma\cdot{\cal H}_\infty)>0 \quad \mathrm{and}\quad m_\gamma=1. \end{align*} $$

We have similarly, for all $\gamma \in \Gamma -\Gamma \!_{{\cal H}_\infty }$ and $t\in {\mathbb R}$ ,

$$ \begin{align*} (\varphi^{-1})_*(\Delta_{\Gamma{\texttt{g}^{t}}\rho_\gamma})= \Delta_{\Gamma r_\gamma \Phi^{t}M} \end{align*} $$

and by Lemma 4.4, for all $y\in \Gamma _H{\backslash } H$ and $s\in {\mathbb R}$ with $s\geq t_0$ ,

$$ \begin{align*} d((\varphi^{-1})_*(\mu^{0+}_{{\cal D}^+,t_0})) (\Theta(y\,\Phi^{-s})) &=\|\sigma^-_{{\cal D}^+}\|\; d\mu_{\Gamma_H{\backslash} H}(y)\,e^{-4\,s}\,ds\\ & =\frac{(1+2\,\delta_{D_K,-3})\,|D_K|}{4\,|{\mathcal{O}}_K^\times|}\; d\mu_{\Gamma_H{\backslash} H}(y)\,e^{-4\,s}\,ds. \end{align*} $$

The end of the proof of Corollary 4.3 now proceeds like that of Corollary 4.1.

Proof. We mostly indicate the differences with the proof of Corollary 4.3. We refer to [Reference Kim and ParkerKiP, Reference MostowMos, Reference PhilippePhi] as well as [Reference Parkkonen and PaulinPaP8, §3] for background on quaternionic hyperbolic geometry. We follow the conventions of the latter reference concerning the normalization of the sectional curvature and the choice of the quaternionic Hermitian form with Witt signature $(1,2)$ .

We now consider $X={\textbf {H}^2_{{\mathbb H}}}$ the Siegel domain model of the quaternionic hyperbolic plane, that is, the quaternionic manifold

$$ \begin{align*} \{(w_0,w)\in{\mathbb H}^2: {\operatorname{\texttt{tr}}} (w_0) -{\operatorname{\texttt{n}}}(w)>0\}, \end{align*} $$

endowed with the Riemannian metric

(34) $$ \begin{align} ds^2_{\,{\textbf{H}^2_{{\mathbb H}}}}=\frac{1}{({\operatorname{\texttt{tr}}} w_0 -{\operatorname{\texttt{n}}}(w))^2} ({\operatorname{\texttt{n}}}(dw_0-\overline{dw}\, w)+ ({\operatorname{\texttt{tr}}}(w_0) -{\operatorname{\texttt{n}}}(w))\,{\operatorname{\texttt{n}}}(dw)). \end{align} $$

This metric is again normalized so that its sectional curvatures are in $[-4,-1]$ . The boundary at infinity of ${\textbf {H}^2_{{\mathbb H}}}$ is

$$ \begin{align*} \partial_\infty{\textbf{H}^2_{{\mathbb H}}}=\{(w_0,w)\in {\mathbb H}^2 \;:{\operatorname{\texttt{tr}}}( w_0) -{\operatorname{\texttt{n}}}(w)=0\}\cup\{\infty\}. \end{align*} $$

Using right-homogeneous coordinates, we identify ${\textbf {H}^2_{{\mathbb H}}}\cup \partial _\infty {\textbf {H}^2_{{\mathbb H}}}$ with its image in the right projective plane ${\mathbb P}^2_r({\mathbb H})$ over ${\mathbb H}$ by the map $(w_0,w)\mapsto [w_0:w:1]$ and $\infty \mapsto [1:0:0]$ . We denote by $\cdot $ the left projective action of G on ${\textbf {H}^2_{{\mathbb H}}}\cup \partial _\infty {\textbf {H}^2_{{\mathbb H}}}$ , as well as its derived action on $T^1{\textbf {H}^2_{{\mathbb H}}}$ .

The critical exponent of the (non-uniform arithmetic) lattice $\Gamma $ of G is now (see, for instance, [Reference Corlette and IozziCI, Theorem 4.4(i)])

$$ \begin{align*} \delta_\Gamma=10. \end{align*} $$

We again fix $v^\bullet =((1,0),(-2,0))\in T^1{\textbf {H}^2_{{\mathbb H}}}$ , which is indeed a unit tangent vector with footpoint $x^\bullet =(1,0)$ by equation (34). The stabilizer of $v^\bullet $ in G is again equal to M and is hence centralized by $\Phi ^{\mathbb R}$ . The G-equivariant orbital map ${\widetilde {\varphi }}: g\mapsto g\cdot v^\bullet $ now defines a homeomorphism $\varphi : \Gamma {\backslash } G/M {\rightarrow } \Gamma {\backslash } T^1{\textbf {H}^2_{{\mathbb H}}}$ .

For every $t\in {\mathbb R}$ , the element $\Phi ^t$ acts on ${\textbf {H}^2_{{\mathbb H}}}$ by the map $(w_0,w)\mapsto (e^{-2t}w_0,e^{-t}w)$ . The geodesic line $\ell $ in ${\textbf {H}^2_{{\mathbb H}}}$ such that $\ell (0)=x^\bullet $ and $\ell '(0)= v^\bullet $ is $t\mapsto (e^{-2\,t},0)$ . Hence, as in the complex case (see the proof of Corollary 4.3),

$$ \begin{align*} \text{for all }t\in{\mathbb R},\text{and }g\in G,\text{we have}\quad {\texttt{g}^{t}}{\widetilde{\varphi}}(g)= {\widetilde{\varphi}}(g\Phi^t). \end{align*} $$

The order-two element $S=\bigg [\!\begin {smallmatrix} \ \ 0 & 0&-1\\\ \ 0&1&\ \ 0\\-1 & 0 & \ \ 0 \end {smallmatrix}\!\bigg ]$ still belongs to $\Gamma $ , it centralizes M and normalizes $\Phi ^{\mathbb R}$ , and it acts by the map $(w_0,w)\mapsto (w_0^{-1},-w\,w_0^{-1})$ on ${\textbf {H}^2_{{\mathbb H}}}$ . Since S is the projective image of the matrix of the quaternionic Hermitian form $q=-{\operatorname {\texttt{tr}}}(\overline {z_0}\,z_2) +{\operatorname {\texttt{n}}}(z_1)$ ,

$$ \begin{align*} \text{for all }g\in G,\text{we have}\quad ^*g^{-1}=S\,g\,S^{-1}. \end{align*} $$

As in the complex case, for all $g\in G$ , $t\in {\mathbb R}$ and $x\in \Gamma {\backslash } G/M$ , we have

$$ \begin{align*} \iota\,{\widetilde{\varphi}}(g)={\widetilde{\varphi}}(gS),\quad S\Phi^tS^{-1}=\Phi^{-t}, \quad \iota\circ \varphi=\varphi\circ\Theta \quad\mathrm{and}\quad \Theta(x\Phi^t)=\Theta(x)\Phi^{-t}. \end{align*} $$

The (closed) horoball in ${\textbf {H}^2_{{\mathbb H}}}$ centred at $\infty $ whose boundary $\partial {\cal H}_\infty $ contains $x^\bullet $ is

$$ \begin{align*} {\cal H}_\infty=\{(w_0,w)\in {\textbf{H}^2_{{\mathbb H}}}:{\operatorname{\texttt{tr}}}( w_0) -{\operatorname{\texttt{n}}}(w)\geq 2\}. \end{align*} $$

The quaternionic Heisenberg group H again acts simply transitively on $\partial {\cal H}_\infty $ , and on ${\partial ^1_{\pm }} {\cal H}_\infty $ which contains $\pm v^\bullet $ . Thus again with $\Phi ^{\geq t_0}=\{\Phi ^t:t\geq t_0\}$ , equation (30) is still satisfied. By for instance the end of §3 in [Reference Parkkonen and PaulinPaP8], the stabilizer $\Gamma \!_{{\cal H}_\infty }$ in $\Gamma $ of the horoball ${\cal H}_\infty $ , as well as that of ${\partial ^1_{\pm }} {\cal H}_\infty $ , is equal to $\Gamma _H$ . The $\Gamma $ -equivariant families of horoballs

$$ \begin{align*} {\cal D}^+={\cal D}^-=(\gamma\cdot{\cal H}_\infty)_{\gamma\in\Gamma} \end{align*} $$

are again locally finite, since $\infty $ is again a bounded parabolic fixed point of $\Gamma $ .

For every $\gamma \in \Gamma $ having a representative in $U_q$ with first column $\bigg (\!\begin {smallmatrix} a \\ \alpha \\ c \end {smallmatrix}\!\bigg )\in {\cal M}_{3,1}({\mathcal {O}})$ , we have $\gamma \notin \Gamma \!_{{\cal H}_\infty }$ if and only if $c\neq 0$ (see, for instance, [Reference Kim and ParkerKiP], [Reference Parkkonen and PaulinPaP8, Eq. (3 $.$ 3)]) and then

1. (i) since $\infty =[1:0:0]$ , the point at infinity $\gamma \cdot \infty $ is equal to $(a\,c^{-1},\alpha \,c^{-1})$ ;

2. (ii) since H acts simply transitively on $\partial _\infty {{{\mathbb H}}^2_{{\mathbb C}}}-\{\infty \}$ , there exists a unique $r_\gamma \in H$ such that $r_\gamma \cdot 0=\gamma \cdot \infty $ , and we have $r_\gamma = {\mathfrak n}_-(a\,c^{-1},\alpha \,c^{-1})$ ;

3. (iii) with ${\cal H}_s =\{(w_0,w)\in {\textbf {H}^2_{{\mathbb H}}}: {\operatorname {\texttt{tr}}}(w_0)-{\operatorname {\texttt{n}}}(w)=s\}$ for $s>0$ , by [Reference Parkkonen and PaulinPaP8, Lemma 6 $.$ 5] where we take $s=2$ so that ${\cal H}_2={\cal H}_\infty $ , we have $d({\cal H}_\infty , \gamma \cdot {\cal H}_\infty )= \tfrac 12\ln ({\operatorname {\texttt{n}}}(c))$ .

Therefore, by [Reference Parkkonen and PaulinPaP8, Proposition 4 $.$ 2(ii)] with ${\mathfrak m} ={\mathcal {O}}$ , the map $\gamma \mapsto r_\gamma $ induces, for all $t,t_0 \in {\mathbb R}$ , a bijection from $\{[\gamma ]\in \Gamma \!_{{\cal H}_\infty }{\backslash }(\Gamma -\Gamma \!_{{\cal H}_\infty })/ \Gamma \!_{{\cal H}_\infty }: d({\cal H}_\infty ,\gamma \cdot {\cal H}_\infty )\leq t-t_0\}$ to ${\mathcal {F}}_{t-t_0}$ .

As in the complex case, we have homeomorphisms $\psi : \Gamma _H{\backslash }{\partial ^1_{+}}{\cal H}_\infty {\rightarrow }\Gamma _H{\backslash } H$ such that

$$ \begin{align*} \psi_*(\Delta_{\Gamma\rho_\gamma})=\Delta_{\Gamma_H r_\gamma}. \end{align*} $$

In the remainder of the proof of Corollary 4.5, we use the same normalization of the Patterson–Sullivan measures $(\mu _x)_{x\in {\textbf {H}^2_{{\mathbb H}}}}$ as in [Reference Parkkonen and PaulinPaP8, §7].

Lemma 4.6. We have

$$ \begin{align*} \|\sigma ^\mp _{{\cal D}^\pm }\|=\frac{D_A^2}{64\;|{\mathcal {O}}^\times |^2} .\end{align*} $$

Proof. By [Reference Parkkonen and PaulinPaP8, Lemma 7 $.$ 2(iv)] with $n=2$ , we have $\|\sigma ^\mp _{{\cal D}^\pm }\|=80\operatorname {Vol}(\Gamma \!_{{\cal H}_\infty }{\backslash } {\cal H}_\infty )$ , where $\operatorname {Vol}$ is the Riemannian volume. By [Reference Parkkonen and PaulinPaP8, Lemma 7 $.$ 1] and the arguments in its proofs, and by [Reference Parkkonen and PaulinPaP8, Eq. (8.4)] for the last equality, we have

$$ \begin{align*} \operatorname{Vol}(\Gamma\!_{{\cal H}_\infty} {\backslash} {\cal H}_\infty)&=\frac{1}{10}\operatorname{Vol}(\Gamma\!_{{\cal H}_\infty} {\backslash} \partial{\cal H}_\infty)=\frac{1}{10}\,\frac{1}{2^5}\,\operatorname{Vol}(\Gamma\!_{{\cal H}_1} {\backslash} \partial{\cal H}_1))=\frac{1}{2^5}\operatorname{Vol}(\Gamma\!_{{\cal H}_1} {\backslash}{\cal H}_1)\\&= \frac{1}{2^5}\frac{D_A^2}{160\;|{\mathcal{O}}^\times|^2}. \end{align*} $$

The result follows.

Since we normalized $\mu _{\Gamma _H{\backslash } H}$ to be a probability measure, it follows from Lemma 4.6 that for $x\in \Gamma _H{\backslash } H$ ,

$$ \begin{align*} \psi_*(\sigma^+_{{\cal D}^-})=(\varphi^{-1})_*(\sigma^+_{{\cal D}^-})= \frac{D_A^2}{64\;|{\mathcal{O}}^\times|^2}\;\mu_{\Gamma_H{\backslash} H}. \end{align*} $$

Let $m_A=24$ if $D_A$ is even, and $m_A=1$ otherwise. By respectively Lemma 7.2(iii) with $n=2$ and Theorem 1.4 in [Reference Parkkonen and PaulinPaP8], we have, with p ranging over primes,