Proof. This follows from the fact that, for any sequence \((h_n)_{n\in \mathbb{N}}\) in \(\mathrm{GL}_n(\mathbb{R})\) with \(\lim_n h_n=h_0\), then \(h_n\) is also biproximal for large \(n\), \(\lim_n x_{h_n}^{\pm}=x_{h_0}^{\pm}\), \(\lim_n V_{h_n}^{\pm}=V_{h_0}^{\pm}\) and \(\lim_n V_{h_n}^{+}\cap V_{h_n}^{-}=V_{h_0}^{+}\cap V_{h_0}^{-}\). ◻

Proof. The existence of \(m_0>1\) such that, for all \(m\geq m_0\), the subgroup \(\langle {\mathsf a}^m,{\mathsf b}^m\rangle\) is free and projective Anosov follows from [16]. We now check that, after increasing \(m_0\) and shrinking the neighborhood \(\Omega_m\), if necessary, the estimate in ([prop-item2]) also holds.

By the hypotheses on \({\mathsf a}\) and \({\mathsf b}\), after conjugating in \(\operatorname{SL}_n(\mathbb{R})\), we may assume that \[{\mathsf a}= \begin{pmatrix} \nu_1 & &\\ & A &\\ & & \nu_n \end{pmatrix}, \qquad {\mathsf b}= g \begin{pmatrix} \mu_1 & &\\ & B &\\ & & \mu_n \end{pmatrix} g^{-1},\] where \(A,B\in \operatorname{GL}_{n-2}(\mathbb{R})\) and \(g\in \operatorname{SL}_n(\mathbb{R})\). Here \(\nu_1\) and \(\nu_n\) are the eigenvalues of \({\mathsf a}\) of maximum and minimum moduli, respectively, and \(\mu_1\) and \(\mu_n\) are the corresponding eigenvalues of \({\mathsf b}\). With this normalization, \[(x_{\mathsf a}^+,V_{\mathsf a}^+) = \big([e_1],\langle e_1,\ldots,e_{n-1}\rangle\big), \qquad (x_{\mathsf a}^-,V_{\mathsf a}^-) = \big([e_n],\langle e_2,\ldots,e_n\rangle\big),\] \[(x_{\mathsf b}^+,V_{\mathsf b}^+) = \big([ge_1],\langle ge_1,\ldots,ge_{n-1}\rangle\big), \qquad (x_{\mathsf b}^-,V_{\mathsf b}^-) = \big([ge_n],\langle ge_2,\ldots,ge_n\rangle\big).\]

The antipodality assumption implies that there exists \(0<\theta<10^{-2}\) such that all unit vectors in the directions of \(g^{\pm1}e_1\) and \(g^{\pm1}e_n\) have first and \(n\)-th coordinates of modulus at least \(\theta\).

For \(g\in \operatorname{SL}_n(\mathbb{R})\), set \[C_g:=2\|g\|\,\|g^{-1}\|\ge1.\] Then both \(g\) and \(g^{-1}\) act \(C_g\)-Lipschitzly on \(\mathbb{P}(\mathbb{R}^n)\): for unit vectors \(v_1,v_2\), \[d_{\mathbb{P}}([gv_1],[gv_2]) \le \left\| \frac{gv_1}{\|gv_1\|} - \frac{gv_2}{\|gv_2\|} \right\| \le C_g\|v_1-v_2\|.\]

Fix \(m>1\) large and choose \(\varepsilon=\varepsilon(m)>0\) satisfying \[\label{bound-epsilon} 0<\varepsilon< \min\left\{ \|A^{\pm m}\|,\, \|B^{\pm m}\|,\, (10^2C_g)^{-4},\, \theta^{10} \right\}.\tag{5}\]

For \(M\in \operatorname{SL}_{n-2}(\mathbb{R})\), \(\xi_1,\xi_n\in\mathbb{R}^\times\), and \(\varepsilon>0\), set \[\mathcal{B}(M,\xi_1,\xi_n;\varepsilon) := \left\{ h \begin{pmatrix} t_1 & &\\ & M' &\\ & & t_n \end{pmatrix} h^{-1} \;\middle|\; \substack{ |t_1-\xi_1|<\varepsilon,\; \|(M')^{\pm1}-M^{\pm1}\|<\varepsilon\\ \|h-I_n\|<\varepsilon,\; |t_n^{-1}-\xi_n^{-1}|<\varepsilon } \right\}.\]

Then define \[\begin{align} \Omega({\mathsf a}^m,\varepsilon) &\mathrel{\vcenter{:}}= \mathcal{B}(A^m,\nu_1^m,\nu_n^m;\varepsilon);\\ \Omega({\mathsf b}^m,\varepsilon) &\mathrel{\vcenter{:}}= g\,\mathcal{B}(B^m,\mu_1^m,\mu_n^m;\varepsilon)\,g^{-1}. \end{align}\] For the rest of the proof, if \(h\in \operatorname{SL}_n(\mathbb{R})\), we denote by \[\ell_1(h)\ge \ell_2(h)\] the two largest moduli of the eigenvalues of \(h\). We now record several estimates.

(i) Any two matrices \(C\in \Omega({\mathsf a}^m,\varepsilon)\) and \(D\in \Omega({\mathsf b}^m,\varepsilon)\) are biproximal. Moreover, their two largest eigenvalue moduli satisfy \[\label{obs1-ineq1} \min\Bigg\{ \frac{\ell_1(C^{\pm1})}{\ell_2(C^{\pm1})}, \frac{\ell_1(D^{\pm1})}{\ell_2(D^{\pm1})} \Bigg\} \geq E(m),\tag{6}\] where \[\label{obs1-ineq2} E(m)\mathrel{\vcenter{:}}= \frac{1}{4} \min\Bigg\{ \frac{|\nu_1|^m}{\|A^m\|},\; \frac{1}{|\nu_n|^m\|A^{-m}\|},\; \frac{|\mu_1|^m}{\|B^m\|},\; \frac{1}{|\mu_n|^m\|B^{-m}\|} \Bigg\}.\tag{7}\] In addition, by the choice of \(\varepsilon>0\) and the definition of the projective metric \(d_{\mathbb{P}}\), the attracting and repelling fixed points and hyperplanes of \(C\) and \(D\) satisfy

\[\begin{align} \operatorname{dist}(x_C^\pm,x_{\mathsf a}^\pm) &\le \max_{\|h-I_n\|\le\varepsilon}\max_{i=1,n} d_{\mathbb{P}}([he_i],[e_i]) \le 10\varepsilon \le \sqrt\varepsilon,\\ \operatorname{dist}(x_D^\pm,x_{\mathsf b}^\pm) &\le \max_{\|h-I_n\|\le\varepsilon}\max_{i=1,n} d_{\mathbb{P}}([ghe_i],[ge_i]) \le 10C_g\varepsilon \le \sqrt\varepsilon,\\ \operatorname{dist}(\mathbb{P}(V_C^\pm),\mathbb{P}(V_{\mathsf a}^\pm)) &\le \max_{\|h-I_n\|\le\varepsilon}\max_{i=1,n} d_{\mathbb{P}}([h^{-t}e_i],[e_i]) \le 10\varepsilon \le \sqrt\varepsilon,\\ \operatorname{dist}(\mathbb{P}(V_D^\pm),\mathbb{P}(V_{\mathsf b}^\pm)) &\le \max_{\|h-I_n\|\le\varepsilon}\max_{i=1,n} d_{\mathbb{P}}([(gh)^{-t}e_i],[g^{-t}e_i]) \le 10C_g\varepsilon \le \sqrt\varepsilon, \end{align}\] where \((\cdot)^{-t}\) denotes the inverse transpose.

(ii) Recall that every unit vector representing one of the four lines \[[ge_1],\;[ge_n],\;[g^{-1}e_1],\;[g^{-1}e_n]\] has first and \(n\)-th coordinates of modulus at least \(\theta\). Define \[\mathcal{C}_1 \mathrel{\vcenter{:}}= B_{5\sqrt{\varepsilon}}(x_{\mathsf a}^+) \cup B_{5\sqrt{\varepsilon}}(x_{\mathsf a}^-), \qquad \mathcal{C}_2 \mathrel{\vcenter{:}}= B_{5\sqrt{\varepsilon}}(x_{\mathsf b}^+) \cup B_{5\sqrt{\varepsilon}}(x_{\mathsf b}^-).\]

Since \(\lim_{m\to\infty} E(m)^{1/m}>1\), after increasing \(m_0\) we may assume that, for every \(m\ge m_0\), the following hold:

1. \[\min\left\{ \ell_1({\mathsf a}^{\pm1})^{m\varepsilon}, \ell_1({\mathsf b}^{\pm1})^{m\varepsilon} \right\} \geq 2^{40}\|g\|^{10}\cdot\|g^{-1}\|^{10}\theta^{-10};\]

2. for any \(C\in \Omega({\mathsf a}^m,\varepsilon)\), any \(D\in \Omega({\mathsf b}^m,\varepsilon)\), and any \(p\in \mathbb{Z}^\ast\), the ratios \[\frac{\ell_1(C^{\pm1})}{\ell_2(C^{\pm1})}, \qquad \frac{\ell_1(D^{\pm1})}{\ell_2(D^{\pm1})}\] are sufficiently large so that \[C^p \left( \mathbb{P}(\mathbb{R}^n) - \mathcal{N}_{\theta/2}\big(\mathbb{P}(V_C^+\cup V_C^-)\big) \right) \subset B_{\sqrt{\varepsilon}}(x_{C}^{\pm}) \subset \mathcal{C}_1;\] \[D^p \left( \mathbb{P}(\mathbb{R}^n) - \mathcal{N}_{\theta/2}\big(\mathbb{P}(V_D^+\cup V_D^-)\big) \right) \subset B_{\sqrt{\varepsilon}}(x_{D}^{\pm})\subset \mathcal{C}_2.\]

For the rest of the proof, fix \(m\geq m_0\) and \(\varepsilon>0\) satisfying 5 . Then \[\mathrm{dist}(\mathcal{C}_1,\mathbb{P}(V_D^\pm)) \geq \mathrm{dist}(x_{\mathsf a}^\pm,\mathbb{P}(V_{\mathsf b}^+\cup V_{\mathsf b}^-))-6\sqrt{\varepsilon} \geq {\theta}/{2},\] and similarly, \[\mathrm{dist}(\mathcal{C}_2,\mathbb{P}(V_C^\pm))\geq {\theta}/{2}.\] Consequently, for any \(p\in \mathbb{Z}^\ast\), \[\label{pp-incl} C^p\mathcal{C}_2\subset \mathcal{C}_1, \qquad D^p\mathcal{C}_1\subset \mathcal{C}_2.\tag{8}\]

Fix \(h\in \operatorname{SL}_n(\mathbb{R})\) with \(\|h-I_n\|<\varepsilon\). For any \([u_2]\in \mathcal{C}_2\), the first and \(n\)-th coordinates of the unit vector \(\frac{h^{-1}u_2}{\|h^{-1}u_2\|}\) have moduli at least \(\theta/10\). Similarly, for any \([u_1]\in \mathcal{C}_1\), the first and \(n\)-th coordinates of the unit vector \(\frac{h^{-1}g^{-1}u_1}{\|h^{-1}g^{-1}u_1\|}\) have moduli at least \(\theta/10\). For any \(C\in \Omega({\mathsf a}^m,\varepsilon)\), any \(D\in \Omega({\mathsf b}^m,\varepsilon)\) and any \(p\in \mathbb{Z}^\ast\), \[\begin{align} \frac{1}{4}\ell_1(C^p) &\leq ||C^p||\leq 4\ell_1(C^p); \\ \frac{\ell_1(D^p)}{4||g||\cdot||g^{-1}||} &\leq ||D^p||\leq 4||g||\cdot||g^{-1}||\ell_1(D^p). \end{align}\]

Thus, for any \([u_i]\in \mathcal{C}_i\), \(i=1,2\) and \(p\in \mathbb{Z}^{\ast}\), we have \[\begin{align} \begin{aligned}\label{pp-ineq1} \|C^pu_2\| &\geq \frac{\theta\,\ell_1(C^p)\|u_2\|}{10^2} \geq \frac{\theta}{10^3}\|C^p\|\cdot \|u_2\|;\\ \|D^pu_1\| &\geq \frac{\theta\,\ell_1(D^p)\|u_1\|}{10^2\|g\|\cdot\|g^{-1}\|} \geq \frac{\theta\,\|D^p\|\cdot \|u_1\|}{(10^2\|g\|\cdot\|g^{-1}\|)^2}. \end{aligned} \end{align}\tag{9}\]

(iii) Set \[c:=(10^2\|g\|\cdot\|g^{-1}\|)^{-2}\theta.\] By the previous choices, for any \[C,C'\in \Omega({\mathsf a}^m,\varepsilon), \qquad D,D'\in \Omega({\mathsf b}^m,\varepsilon),\] and any \(p\in \mathbb{Z}^\ast\), we have \[\label{obs-iii} \frac{\|C^p\|}{\|{C'}^p\|^{1-\varepsilon}}\geq \frac{1}{c}, \qquad \frac{\|D^p\|}{\|{D'}^p\|^{1-\varepsilon}}\geq \frac{1}{c}.\tag{10}\]

Indeed, let \(\delta\in\{-1,1\}\) and \(p\in \mathbb{N}\). By the choice of \(\varepsilon\), \[\frac{\ell_1(C^{\delta p})}{\ell_1({C'}^{\delta p})} = \frac{\ell_1(C^\delta)^p}{\ell_1({C'}^\delta)^p} \geq \frac{1}{4^p}, \qquad \frac{\ell_1(D^{\delta p})}{\ell_1({D'}^{\delta p})} = \frac{\ell_1(D^\delta)^p}{\ell_1({D'}^\delta)^p} \geq \frac{1}{4^p}.\] Moreover, since \(\frac{1}{2}\ell_1({\mathsf a}^{\delta})^m \leq \ell_1(C^{\delta})\leq 2\ell_1({\mathsf a}^{\delta})^m\), \(\frac{1}{2}\ell_1({\mathsf b}^{\delta})^m \leq \ell_1(D^{\delta})\leq 2\ell_1({\mathsf b}^{\delta})^m\), we conclude that \[\frac{\ell_1({\mathsf a}^\delta)^{mp}}{2^{p+2}} \leq \|C^{\delta p}\| \leq 2^{p+2}\ell_1({\mathsf a}^\delta)^{mp},\] and \[\frac{\ell_1({\mathsf b}^\delta)^{mp}}{2^{p+2}\|g\|\cdot\|g^{-1}\|} \leq \|D^{\delta p}\| \leq 2^{p+2}\|g\|\cdot\|g^{-1}\|\ell_1({\mathsf b}^\delta)^{mp}.\] Using these bounds for the matrices \(C,D\) (and also for \(C',D'\)), we obtain \[\frac{\|C^{\delta p}\|}{\|{C'}^{\delta p}\|^{1-\varepsilon}} \geq \frac{\|{C'}^{\delta p}\|^\varepsilon}{4^{2p}} \geq \frac{\ell_1({\mathsf a}^\delta)^{mp\varepsilon}}{2^{10p}} \geq \frac{1}{c},\] and similarly \[\frac{\|D^{\delta p}\|}{\|{D'}^{\delta p}\|^{1-\varepsilon}} \geq \frac{\|{D'}^{\delta p}\|^\varepsilon}{4^{2p}||g||^2\cdot||g^{-1}||^2} \geq \frac{\ell_1({\mathsf b}^\delta)^{mp\varepsilon}}{2^{10p}\|g\|^3\cdot\|g^{-1}\|^3} \geq \frac{1}{c}.\]

We now prove the desired comparison estimate. Let \(F_2=\langle a\rangle\ast \langle b\rangle\) and let \(\rho_1,\rho_2:F_2\to \operatorname{SL}_n(\mathbb{R})\) be representations such that \[\rho_i(a)\in \Omega({\mathsf a}^m,\varepsilon), \qquad \rho_i(b)\in \Omega({\mathsf b}^m,\varepsilon), \qquad i=1,2.\] We show that \(\rho_1\) and \(\rho_2\) satisfy the estimate in ([prop-item2]), assuming \(0<2\varepsilon<1\). Since \(\ell_1(\cdot)\) is conjugacy invariant, it suffices to consider cyclically reduced words. For this, let \[h=\prod_{j=1}^{l} a^{p_j}b^{q_j}, \qquad p_j,q_j\neq 0.\] The other cyclically reduced forms are handled in the same way, after interchanging the roles of \(a\) and \(b\) if necessary. Using the ping-pong inclusions 8 and the estimates in 9 , we obtain \[\label{pp-ineq3} \|\rho_1(h)\| \geq \prod_{j=1}^{l} \big(c\|\rho_1(a)^{p_j}\|\big) \big(c\|\rho_1(b)^{q_j}\|\big).\tag{11}\]

Therefore, by 11 and the submultiplicativity of the operator norm, \[\frac{\|\rho_1(h)\|}{\|\rho_2(h)\|^{1-\varepsilon}} \geq \prod_{j=1}^{l} \left( c\frac{\|\rho_1(a)^{p_j}\|}{\|\rho_2(a)^{p_j}\|^{1-\varepsilon}} \right) \left( c\frac{\|\rho_1(b)^{q_j}\|}{\|\rho_2(b)^{q_j}\|^{1-\varepsilon}} \right).\] By 10 , each factor on the right-hand side is at least \(1\). Hence \[{\|\rho_1(h)\|}\ge {\|\rho_2(h)\|^{1-\varepsilon}}.\] Applying this inequality to \(h^r\), \(r\geq 1\), and using the fact that \[\lim_{r\to\infty}\|\rho_i(h^r)\|^{1/r} = \ell_1(\rho_i(h)),\] we get \[\ell_1(\rho_1(h)) \geq \ell_1(\rho_2(h))^{1-\varepsilon}.\] Since \(\rho_1\) and \(\rho_2\) were arbitrary, switching their roles gives \[\ell_1(\rho_1(h)) \leq \ell_1(\rho_2(h))^{1/(1-\varepsilon)} \leq \ell_1(\rho_2(h))^{1+2\varepsilon},\] provided \(\varepsilon\) is sufficiently small. Taking logarithms, we obtain \[(1-\varepsilon)\lambda_1(\rho_2(h)) \leq \lambda_1(\rho_1(h)) \leq (1+2\varepsilon)\lambda_1(\rho_2(h)).\] This proves the desired estimate. Finally, since \({\mathsf a}^m\) and \({\mathsf b}^m\) are biproximal, Lemma 1, together with the openness of the projective Anosov property [4], gives an open neighborhood \(\Omega_m\subset \mathrm{Hom}(F_2,\operatorname{SL}_n(\mathbb{R}))\) of the representation \(a\mapsto {\mathsf a}^m, b\mapsto {\mathsf b}^m,\) consisting entirely of projective Anosov representations and contained in \(\Omega({\mathsf a}^m,\varepsilon)\times \Omega({\mathsf b}^m,\varepsilon)\). This completes the proof. ◻

Proof. Fix a convex cocompact representation \(j:F_2\to \operatorname{SL}_2(\mathbb{R})\), or equivalently, a projective Anosov representation. Since \(j\) is projective Anosov, there exists \(c>0\) such that \[\lambda_1(j(\gamma))\ge c|\gamma|_\infty \qquad(\gamma\in F_2).\] On the other hand, since \(\chi\) is a homomorphism with \(|\chi(\eta)|\le |\eta|\) for all \(\eta\in F_2\), we have that, for all \(k\in \mathbb{N}\), \[k|\chi(\gamma)| = |\chi(\gamma^k)| \le |\gamma^k|.\] and hence \[|\chi(\gamma)|\le |\gamma|_\infty .\] It follows that \[\label{sup} \sup_{\gamma\in F_2-\{e\}} \frac{|\chi(\gamma)|}{\lambda_1(j(\gamma))} <\infty.\tag{12}\]

Let \[\tau_{2d}:\operatorname{SL}_2(\mathbb{R})\to \operatorname{SL}_{2d}(\mathbb{R})\] be the unique irreducible representation of dimension \(2d\), up to conjugation. Then \[\tau_{2d}\circ j:F_2\to \operatorname{SL}_{2d}(\mathbb{R})\] is Borel-Anosov. Moreover, for any \(\gamma\in F_2\), \[\lambda_d(\tau_{2d}(j(\gamma)))=\lambda_1(j(\gamma)).\]

For \(\kappa>0\), define a representation \(\rho_\kappa:F_2\to \operatorname{SL}_{2d+1}(\mathbb{R})\) by \[\label{rk} \rho_\kappa(\gamma) = \begin{pmatrix} e^{-\frac{\kappa{\chi}(\gamma)}{2d}} \tau_{2d}(j(\gamma)) &\\ & e^{\kappa{\chi}(\gamma)} \end{pmatrix}, \qquad \gamma\in F_2.\tag{13}\] By 12 , we may choose \(\kappa>0\) sufficiently small so that \[\label{choice} \kappa\frac{2d+1}{2d} \sup_{\gamma\in F_2\smallsetminus\{e\}} \frac{|{\chi}(\gamma)|}{\lambda_1(j(\gamma))} <1.\tag{14}\] By [17], this condition ensures that \(\rho_\kappa\) is Borel-Anosov in \(\operatorname{SL}_{2d+1}(\mathbb{R})\).

We claim that \[\label{middle} \lambda_{d+1}(\rho_\kappa(\gamma)) = \kappa{\chi}(\gamma) \quad\text{ for any \gamma\in F_2.}\tag{15}\] Indeed, let \(\gamma\ne e\), and set \(L:=\lambda_1(j(\gamma))>0\) and \(q:=\chi(\gamma)\). The Jordan coordinates of \(\tau_{2d}(j(\gamma))\) are \[(2d-1)L,\;(2d-3)L,\ldots, L,\;-L,\ldots, -(2d-3)L,\;-(2d-1)L.\] After multiplying this \(2d\)-dimensional block by \(e^{-\kappa q/(2d)}\), all these coordinates are shifted by \(-\kappa q/(2d)\). Hence the middle two coordinates of the twisted \(\tau_{2d}\)-block are \[L-\frac{\kappa q}{2d} \qquad\text{and}\qquad -L-\frac{\kappa q}{2d}.\] The remaining one-dimensional block has logarithmic eigenvalue modulus \[\kappa q.\] By the choice of \(\kappa\) as in 14 , we have \[\frac{2d+1}{2d}\kappa |q|<L.\] Equivalently, \[-L-\frac{\kappa q}{2d} < \kappa q < L-\frac{\kappa q}{2d}.\] Thus the coordinate \(\kappa q\) lies strictly between the \(d\)-th and \((d+1)\)-st coordinates of the twisted \(2d\)-dimensional block. Therefore, after arranging all \(2d+1\) logarithmic eigenvalue moduli in decreasing order, the middle coordinate is precisely \[\lambda_{d+1}(\rho_\kappa(\gamma))=\kappa\chi(\gamma).\] The case \(\gamma=e\) is immediate.

Recall that any infinite order element of a Borel-Anosov group is loxodromic. Since \(\rho_\kappa(a)\) and \(\rho_\kappa(b)\) are loxodromic with antipodal fixed flags in the full flag space \(\mathcal{F}(\mathbb{R}^n)\), 6 gives an integer \(m>0\) and an open neighborhood \(\Omega_m\subset \mathrm{Hom}(F_2,\operatorname{SL}_{2d+1}(\mathbb{R}))\) of the representation \(\rho_\kappa':F_2\to \operatorname{SL}_{2d+1}(\mathbb{R})\) defined by \[\rho_\kappa'(a)=\rho_\kappa(a^m), \qquad \rho_\kappa'(b)=\rho_\kappa(b^m),\] such that any representation in \(\Omega_m\) is Borel-Anosov and, for any \(\sigma\in \Omega_m\) and any \(w\in F_2\), \[\|\lambda(\sigma(w))-\lambda(\rho_\kappa'(w))\| \leq \frac{\kappa}{80d^2\,\lambda_1(j(a))} \|\lambda(\rho_\kappa'(w))\|.\] We shall use the elementary inequality \[\left\| \frac{x}{\|x\|}-\frac{y}{\|y\|} \right\| \le 2\frac{\|x-y\|}{\|y\|}\] for nonzero vectors \(x,y\). Applying this with \(x=\lambda(\sigma(w))\) and \(y=\lambda(\rho_\kappa'(w))\), and using the preceding estimate, we obtain that for any non-trivial \(w\in F_2\), \[\label{main1-ineq1} \left\| \frac{\lambda(\sigma(w))}{\|\lambda(\sigma(w))\|} - \frac{\lambda(\rho_\kappa'(w))}{\|\lambda(\rho_\kappa'(w))\|} \right\| \leq \frac{\kappa}{40d^2\,\lambda_1(j(a))}.\tag{16}\]

Fixing \(\sigma\in \Omega_m\), we claim \[\label{mL} \mathcal{L}_{\sigma({\mathcal{N}})} \subsetneq \mathcal{L}_{\sigma(F_2)}\tag{17}\]

We will first show: \[\label{hN} \inf_{h\in {\mathcal{N}}- \{e\}} \left\| \frac{\lambda(\sigma(h))}{\|\lambda(\sigma(h))\|} - \frac{\lambda(\sigma(a))}{\|\lambda(\sigma(a))\|} \right\| \ge \frac{\kappa}{5d^2\,\lambda_1(j(a))}.\tag{18}\] Let \(h=\prod_{j=1}^{l} a^{s_j}b^{t_j} \in {\mathcal{N}}\) be any non-trivial element. Then \(s_1+\cdots+s_l=0\) and hence, by the definition of \(\rho_\kappa'\), \[\lambda_{d+1}(\rho_\kappa'(h))=\lambda_{d+1} ( \prod_{j=1}^{l}\rho_{\kappa}(a^{ms_j}b^{m t_j}))=\kappa \,{\chi}( \prod_{j=1}^{l} a^{ms_j}b^{mt_j})=\kappa m\sum_{i=1}^l s_i=0.\] Therefore 16 implies: \[\sup_{h\in N\smallsetminus\{e\}} \frac{|\lambda_{d+1}(\sigma(h))|}{\|\lambda(\sigma(h))\|} \leq \frac{\kappa}{40d^2\,\lambda_1(j(a))}.\]

On the other hand, since \(\chi(a)=1\), we obtain from 15 and the choice of \(\kappa\): \[\lambda_{d+1}(\rho_\kappa(a))=\kappa \quad\text{and}\quad \frac{\kappa}{2d}\le \frac{\lambda_1(j(a))}{2d+1}.\] The largest absolute value of a Jordan coordinate of \(\tau_{2d}(j(a))\) is \((2d-1)\lambda_1(j(a))\). Since the \(2d\)-dimensional block in \(\rho_\kappa(a)\) is shifted by \(-\kappa/(2d)\), and the remaining one-dimensional block contributes \(\kappa\), we get that for any \(1\leq j\leq 2d+1\), \[|\lambda_{j}(\rho_k(a))|\leq |\lambda_{1}(\tau_{2d}(j(a)))|+\frac{\kappa}{2d} \le \left( (2d-1) +\tfrac{1}{(2d+1)}\right)\lambda_1(j(a)) \leq 2d\lambda_1(j(a)).\] Hence \[||\lambda(\rho_{\kappa}(a))||\leq 2d\sqrt{2d+1}\lambda_1(\rho_{\kappa}(a))\leq{ 4 d^2 }\lambda_1(j(a)).\] Since \(\rho'_\kappa(a)=\rho_\kappa(a^m)\), we conclude that \[\frac{\lambda_{d+1}(\rho_\kappa'(a))}{\|\lambda(\rho_\kappa'(a))\|} = \frac{m\lambda_{d+1}(\rho_\kappa(a))}{\|m\lambda(\rho_\kappa(a))\|} \geq \frac{\kappa}{4d^2\,\lambda_1(j(a))}.\] Again applying 16 , we get \[\frac{\lambda_{d+1}(\sigma(a))}{\|\lambda(\sigma(a))\|} \ge \frac{\lambda_{d+1}(\rho_\kappa'(a))}{\|\lambda(\rho_\kappa'(a))\|} - \frac{\kappa}{40d^2\,\lambda_1(j(a))} \ge \frac{9\kappa}{40d^2\,\lambda_1(j(a))}.\] It follows that, for every \(h\in\mathcal{N}-\{e\}\), \[\begin{align} \left\| \frac{\lambda(\sigma(a))}{\|\lambda(\sigma(a))\|} - \frac{\lambda(\sigma(h))}{\|\lambda(\sigma(h))\|} \right\| &\ge \left| \frac{\lambda_{d+1}(\sigma(a))}{\|\lambda(\sigma(a))\|} - \frac{\lambda_{d+1}(\sigma(h))}{\|\lambda(\sigma(h))\|} \right| \\ &\ge \frac{9\kappa}{40d^2\,\lambda_1(j(a))} - \frac{\kappa}{40d^2\,\lambda_1(j(a))} \\ &= \frac{\kappa}{5d^2\,\lambda_1(j(a))}. \end{align}\] This proves 18 . It follows that the direction of \(\lambda(\sigma(a))\) is not contained in the set of accumulation directions of the Jordan projections of elements of \(\sigma(\mathcal{N})\). By Theorem 9 below, 18 implies 17 .

Since every representation in \(\Omega_m\) is Borel-Anosov and \(F_2\) has no torsion, it is faithful. Therefore \[\sigma(F_2)/\sigma(N)\simeq F_2/N\simeq \mathbb{Z}.\] Finally, Zariski dense representations form an open dense subset of the set \(\mathrm{Hom}(F_2,\operatorname{SL}_{2d+1}(\mathbb{R}))\) [18], and it is clearly non-empty for \(F_2\). After shrinking \(\Omega_m\), we may therefore choose a non-empty open subset \(\Omega\subset \Omega_m\) consisting entirely of Zariski dense representations. Since \(\sigma({\mathcal{N}})\) is normal in \(\sigma (F_2)\) for any \(\sigma\in \Omega\), its Zariski closure is a normal and infinite subgroup of \(\operatorname{SL}_{2d+1}(\mathbb{R})\). Since \(\operatorname{SL}_{2d+1}(\mathbb{R})\) is a simple algebraic group, its Zariski closure is all of \(\operatorname{SL}_{2d+1}(\mathbb{R})\). ◻

Proof. Let \(\gamma\in\Gamma\) be loxodromic. Since \(N\) is Zariski dense, we may choose a loxodromic element \(h\in N\) in general position with respect to \(\gamma\). In particular, the attracting and repelling flags of \(h\) are transverse to those of \(\gamma\). For \(m\geq 1\), set \[w_m=[\gamma^m,h]=\gamma^m h\gamma^{-m}h^{-1}.\] Then \(w_m\in N\). By the standard Schottky product estimate for loxodromic elements in general position [3], we have \[\mu(w_m) = \mu(\gamma^m)+\mu(\gamma^{-m})+O(1),\] where the error is independent of \(m\). Dividing by \(m\), and using \[\frac{1}{m}\mu(\gamma^m)\to \lambda(\gamma), \qquad \frac{1}{m}\mu(\gamma^{-m})\to \lambda(\gamma^{-1}) = \operatorname{i}(\lambda(\gamma)),\] we obtain \[\frac{1}{m}\mu(w_m) \to \lambda(\gamma)+\operatorname{i}(\lambda(\gamma)).\] Since \(w_m\in N\), the limit belongs to \(\mathcal{L}_N\). By 9, the claim follows. ◻

Proof. By Theorem 1, we can choose \(v\in \mathcal{L}_{\rho(F_2)}- \mathcal{L}_{\rho(\mathcal{N})}\). Since \(\rho(\mathcal{N})\) is Zariski dense, \(\mathcal{L}_{\rho(\mathcal{N})}\) is a closed convex cone with non-empty interior. By the separating hyperplane theorem, there exists a linear form \(\psi \in \mathfrak a^*\) which is positive on \(\mathcal{L}_{\rho(\mathcal{N})}-\{0\}\) but satisfies \(\psi(v)<0\). Then \[\delta_{\rho(\mathcal{N}),\psi}<\infty \qquad\text{whereas}\qquad \delta_{\rho(F_2),\psi}=\infty.\] ◻

Proof. Indeed, since \(\Gamma\) is Borel-Anosov, the orbit map of \(\Gamma\) into the symmetric space is a quasi-isometric embedding. Equivalently, there exist constants \(A\ge1\) and \(B\ge0\) such that \[A^{-1}|\gamma|-B \le \|\mu(\gamma)\| \le A|\gamma|+B \qquad(\gamma\in\Gamma).\] On the other hand, since \(\varphi\) is positive on \(\mathcal{L}_\Gamma-\{0\}\), there exists \(c_\varphi>0\) such that \[\varphi(v)\ge c_\varphi\|v\| \qquad(v\in\mathcal{L}_\Gamma).\] Since \(\mu(\Gamma)\) stays within a bounded distance of \(\mathcal{L}_\Gamma\) [3], there exists \(B'_\varphi\ge0\) such that \[\varphi(\mu(\gamma)) \ge c_\varphi\|\mu(\gamma)\|-B'_\varphi \qquad(\gamma\in\Gamma).\] The reverse inequality \(\varphi(\mu(\gamma))\le \|\varphi\|\,\|\mu(\gamma)\|\) is immediate. This proves the lemma. ◻

Proof. For fixed \(x,g\in\Gamma\), we have \[\sup_{n\in N} \|\mu(x^{-1}ng)-\mu(n)\|<\infty.\] Since \(a>\delta_{N,\varphi}\), it follows that the measures \(\mu_x^C\) are finite.

We first prove the upper bound in 26 . Let \(c\in C\cap\mathcal{O}_R(x,y)\), and choose \(u\in[x,c]\) with \(d_w(u,y)\le R\). By 25 , applied to the word geodesic \([x,c]\), we have \[d_\varphi(x,c)\ge d_\varphi(x,u)+d_\varphi(u,c)-D.\] Since \(d_w(u,y)\le R\), both \(d_\varphi(u,y)\) and \(d_\varphi(y,u)\) are bounded by a constant depending only on \(R\). Using Proposition 12, we get \[d_\varphi(x,u)\ge d_\varphi(x,y)-O_R(1) \quad\text{and}\quad d_\varphi(u,c)\ge d_\varphi(y,c)-O_R(1).\] Therefore \[d_\varphi(x,c)\ge d_\varphi(x,y)+d_\varphi(y,c)-O_R(1).\] It follows that \[e^{-a d_\varphi(x,c)} \le A e^{-a d_\varphi(x,y)}e^{-a d_\varphi(y,c)}\] for some \(A=A(R)>0\). Summing over \(c\in C\cap\mathcal{O}_R(x,y)\) gives the upper bound.

We prove the lower bound by contradiction. Suppose it fails. Then there are \(R_i\to\infty\), cosets \(C_i=g_i N\), and \(x_i,y_i\in\Gamma\) such that \[\label{eq:shadow-contra} e^{a d_\varphi(x_i,y_i)} \frac{\mu_{x_i}^{C_i}(\mathcal{O}_{R_i}(x_i,y_i))}{|\mu_{y_i}^{C_i}|} \to 0.\tag{27}\] Left-translating by \(x_i^{-1}\), and using normality of \(N\), we may assume that \(x_i=e\). Write \(h_i=y_i\). Since \(d_\varphi\) is quasi-isometric to the word metric, \(h_i\to\infty\); otherwise, for large \(i\), the shadow \(\mathcal{O}_{R_i}(e,h_i)\) would be all of \(\overline{\Gamma}\), contradicting 27 . Passing to a subsequence, we may assume that the sequence \(h_i^{-1}\) converges to some \(\xi\in\partial\Gamma\).

Define probability measures on \(\overline{\Gamma}\): \[\nu_i \mathrel{\vcenter{:}}= \frac{1}{|\mu_{h_i}^{C_i}|}(h_i^{-1})_*\mu_{h_i}^{C_i}.\] After passing to a subsequence, \(\nu_i\to\nu\) weakly. We claim that \(\nu=\delta_\xi\). Let \(V\subset\overline{\Gamma}\) be a compact subset such that \(V\cap\{\xi\}=\varnothing\). Since \(h_i^{-1}\to\xi\), word hyperbolicity gives \[V\subset \mathcal{O}_{R_i}(h_i^{-1},e) \quad\text{for all large i.}\] Equivalently, \(h_iV\subset \mathcal{O}_{R_i}(e,h_i)\). For \(z\in h_iV\), Proposition 12, applied to the triple \((e,h_i,z)\), gives \[d_\varphi(h_i,z)\ge d_\varphi(e,z)-d_\varphi(e,h_i)-D.\]

Therefore \[\begin{align} \nu_i(V) &= \frac{\mu_{h_i}^{C_i}(h_iV)}{|\mu_{h_i}^{C_i}|} \le e^{aD}e^{a d_\varphi(e,h_i)} \frac{\mu_e^{C_i}(\mathcal{O}_{R_i}(e,h_i))}{|\mu_{h_i}^{C_i}|}. \end{align}\] By 27 , the right-hand side tends to \(0\). Hence \(\nu_i(V)\to0\), and so \(\nu(V)=0\). This implies that \(\nu=\delta_\xi\).

We claim that for any \(n\in N\), there exists \(C_n\ge1\), independent of \(i\), such that for any non-negative continuous function \(f\) on \(\overline{\Gamma}\), \[\label{cn} C_n^{-1}\nu_i(f)\le (n_*\nu_i)(f)\le C_n\nu_i(f).\tag{28}\] Indeed, the support of \(\nu_i\) is \(h_i^{-1}C_i\), which is left \(N\)-invariant by normality.

Moreover, Proposition 12 gives a uniform comparison of the weights on the support. For \(z\in\Gamma\), we get \[d_\varphi(e,n^{-1}z) \le d_\varphi(e,n^{-1})+d_\varphi(n^{-1},n^{-1}z)+D = d_\varphi(e,n^{-1})+d_\varphi(e,z)+D,\] and \[d_\varphi(e,z) \le d_\varphi(e,n)+d_\varphi(e,n^{-1}z)+D.\] Hence \[|d_\varphi(e,n^{-1}z)-d_\varphi(e,z)| \le \max\{d_\varphi(e,n),d_\varphi(e,n^{-1})\}+D.\] Thus, for a constant \(C_n\ge1\), independent of \(i\), \[C_n^{-1}e^{-a d_\varphi(e,z)} \le e^{-a d_\varphi(e,n^{-1}z)} \le C_n e^{-a d_\varphi(e,z)}.\] Since \(h_i^{-1}C_i\) is left \(N\)-invariant, changing variables \(w=nz\) in the sum defining \(\nu_i\) gives, for any non-negative continuous function \(f\) on \(\overline{\Gamma}\), \[C_n^{-1}\int f\,d\nu_i \le \int f(nz)\,d\nu_i(z) \le C_n\int f\,d\nu_i .\] This proves 28 . Passing to the weak limit gives \[C_n^{-1}\nu(f) \le n_*\nu (f) \le C_n \nu (f)\quad \text{ for all n\in N.}\] Since \(\nu=\delta_\xi\), it follows that any \(n\in N\) fixes \(\xi\). This contradicts the non-elementarity of \(N\). Hence this proves the lower bound. ◻

Proof. Let \(R,A\) be the constants from Lemma 5. By 4, we may choose a negative integer \(m_0\) such that \(d_\varphi(x,y)\ge m_0\) for all \(x,y\in\Gamma\), and set \[S_m(x)\mathrel{\vcenter{:}}=\{y\in\Gamma:m\le d_\varphi(x,y)<m+1\}, \qquad m\ge m_0.\]

We first note that the family \[\{\mathcal{O}_R(x,y):y\in S_m(x)\}\] has uniformly bounded multiplicity, independently of \(x\) and \(m\). Indeed, suppose \[z\in\mathcal{O}_R(x,y)\cap\mathcal{O}_R(x,y') \qquad\text{with }y,y'\in S_m(x).\] Choose points \(u,u'\) on a word geodesic \([x,z]\) such that \(d_w(u,y)\le R\) and \(d_w(u',y')\le R\). Since \(d_\varphi\) is quasi-isometric to \(d_w\), all \(d_\varphi\)-distances between points at \(d_w\)-distance at most \(R\) are uniformly bounded. Hence, by Proposition 12, \[d_\varphi(x,u)=d_\varphi(x,y)+O_R(1) \quad\text{ and}\quad d_\varphi(x,u')=d_\varphi(x,y')+O_R(1).\] Since \(y,y'\in S_m(x)\), it follows that \[d_\varphi(x,u)=d_\varphi(x,u')+O_R(1).\] After interchanging \(u\) and \(u'\) if necessary, assume that \(u\) lies between \(x\) and \(u'\) on the geodesic \([x,z]\). Applying 25 to the geodesic segment \([x,u']\), we obtain \[d_\varphi(u,u') \le d_\varphi(x,u')-d_\varphi(x,u)+D = O_R(1).\] By 4, this implies \(d_w(u,u')=O_R(1)\). Therefore \(d_w(y,y')=O_R(1)\). Properness of the word metric gives a uniform multiplicity bound; call it \(M\).

Using the lower bound in 26 for the coset \(C=N\), we get \[U(y)e^{-a d_\varphi(x,y)} \le A\,\mu_x^N(\mathcal{O}_R(x,y)).\] Therefore, for any \(m\ge m_0\), \[\sum_{y\in S_m(x)} U(y)e^{-a d_\varphi(x,y)} \le A\sum_{y\in S_m(x)}\mu_x^N(\mathcal{O}_R(x,y)) \le AMU(x).\] Since \(d_\varphi(x,y)\ge m\) for \(y\in S_m(x)\), we obtain \[\sum_{y\in S_m(x)} U(y)e^{-s d_\varphi(x,y)} \le AMe^{-(s-a)m}U(x).\] Summing over \(m\ge m_0\) proves the lemma. ◻

Proof. Fix \(s>\delta_{N,\varphi}\), and choose \(a\) such that \[\delta_{N,\varphi}<a<s.\] Let \(U\) be the function defined in Lemma 6 for this choice of \(a\). First observe that \(U\) descends to a function on \(\Gamma/N\). Indeed, for \(n_0\in N\), \[\begin{align} U(\gamma n_0) &= \sum_{n\in N}e^{-a d_\varphi(\gamma n_0,n)} = \sum_{n\in N}e^{-a d_\varphi(e,n_0^{-1}\gamma^{-1}n)} \\ &= \sum_{n\in N} e^{-a d_\varphi(e,\gamma^{-1}(\gamma n_0^{-1}\gamma^{-1})n)} = \sum_{n'\in N}e^{-a d_\varphi(e,\gamma^{-1}n')} = U(\gamma), \end{align}\] where normality of \(N\) was used in the third equality. Hence we may write \(U(\gamma N)\mathrel{\vcenter{:}}= U(\gamma)\).

For \(\gamma_0\in\Gamma\), define a function \(b_{\gamma_0}\) on \(\Gamma/N\): \[b_{\gamma_0}(\gamma N) \mathrel{\vcenter{:}}= \log U(\gamma\gamma_0)-\log U(\gamma),\] which is well-defined by the normality of \(N\). Moreover, \(b_{\gamma_0}\) is bounded. To see this, let \(\gamma\in \Gamma\) and \(n\in N\). By Proposition 12, \[d_\varphi(\gamma\gamma_0,n) \ge d_\varphi(\gamma,n)-d_\varphi(e,\gamma_0)-D,\] and hence, summing over \(n\in N\) gives \[U(\gamma\gamma_0) \le e^{a(d_\varphi(e,\gamma_0)+D)}U(\gamma).\] The reverse inequality follows similarly, using \(\gamma=\gamma\gamma_0\gamma_0^{-1}\). Thus \[|b_{\gamma_0}(\gamma N)| \le a\max\{d_\varphi(e,\gamma_0),d_\varphi(e,\gamma_0^{-1})\}+O(1),\] proving the boundedness of \(b_{\gamma_0}\). So \(b_{\gamma_0}\in \ell^\infty(\Gamma/N)\).

Using the mean \(\mathfrak m:\ell^\infty(\Gamma/N)\to\mathbb{R}\) given by 29 , define \(\chi:\Gamma\to \mathbb{R}\) by \[\chi(\gamma_0)\mathrel{\vcenter{:}}=\mathfrak m(b_{\gamma_0}) \quad\text{ for \gamma_0\in \Gamma}.\] The identity \(b_{\gamma_0\gamma_1}(\gamma N) = b_{\gamma_1}(\gamma\gamma_0N)+b_{\gamma_0}(\gamma N)\) and the right \(\Gamma\)-invariance of \(\mathfrak m\) imply \[\chi(\gamma_0\gamma_1)=\chi(\gamma_1)+\chi(\gamma_0) \quad\text{for all \gamma_0, \gamma_1\in \Gamma.}\] Therefore \(\chi:\Gamma\to\mathbb{R}\) is a character. Moreover, \(\chi|_N=0\) since \(b_{\gamma_0}=0\) for any \(\gamma_0\in N\).

By Jensen’s inequality for the positive normalized mean \(\mathfrak m\), \[e^{\chi(\gamma_0)} = e^{\mathfrak m(b_{\gamma_0})} \le \mathfrak m(e^{b_{\gamma_0}}) = \mathfrak m\left( \gamma N\mapsto \frac{U(\gamma\gamma_0)}{U(\gamma)} \right).\] Let \(F\subset\Gamma\) be finite and let \(s>a\). For every \(\gamma,\gamma_0\in\Gamma\), we have \[d_\varphi(\gamma,\gamma\gamma_0) = \varphi(\mu(\gamma^{-1}\gamma\gamma_0)) = \varphi(\mu(\gamma_0)) = d_\varphi(e,\gamma_0).\] Hence, by Jensen’s inequality, for each \(\gamma_0\in F\), \[e^{\chi(\gamma_0)}e^{-s d_\varphi(e,\gamma_0)} \le \mathfrak m\left( \gamma N\mapsto \frac{U(\gamma\gamma_0)}{U(\gamma)} e^{-s d_\varphi(\gamma,\gamma\gamma_0)} \right).\] Summing over \(\gamma_0\in F\), and using the positivity and linearity of \(\mathfrak m\), we obtain \[\begin{align} \sum_{\gamma_0\in F} e^{\chi(\gamma_0)}e^{-s d_\varphi(e,\gamma_0)} &\le \mathfrak m\left( \gamma N\mapsto \frac{1}{U(\gamma)} \sum_{\gamma_0\in F} U(\gamma\gamma_0)e^{-s d_\varphi(\gamma,\gamma\gamma_0)} \right). \end{align}\] By [eq:conv-estimate], applied with \(x=\gamma\), we have \[\sum_{\gamma_0\in F} U(\gamma\gamma_0)e^{-s d_\varphi(\gamma,\gamma\gamma_0)} \le \sum_{y\in\Gamma} U(y)e^{-s d_\varphi(\gamma,y)} \le b_sU(\gamma).\] Therefore the function inside the mean is bounded above by \(b_s\), and so \[\sum_{\gamma_0\in F} e^{\chi(\gamma_0)}e^{-s d_\varphi(e,\gamma_0)} \le b_s\quad\text{ and hence } \quad \sum_{\gamma\in\Gamma} e^{\chi(\gamma)}e^{-s d_\varphi(e,\gamma)}<\infty.\]

Changing variables \(\gamma\mapsto\gamma^{-1}\) and using \(\chi(\gamma^{-1})=-\chi(\gamma)\), gives \[\label{eq:opposite-twisted-series} \sum_{\gamma\in\Gamma} e^{-\chi(\gamma)}e^{-s d_\varphi(e,\gamma^{-1})} = \sum_{\gamma\in\Gamma} e^{-\chi(\gamma)}e^{-s d_{\varphi\circ\operatorname{i}}(e,\gamma)} <\infty.\tag{31}\] Combining these estimates and applying the arithmetic-geometric mean inequality, we get \[\begin{align} \infty &> \sum_{\gamma\in\Gamma} \left( e^{\chi(\gamma)}e^{-s d_\varphi(e,\gamma)} + e^{-\chi(\gamma)}e^{-s d_{\varphi\circ\operatorname{i}}(e,\gamma)} \right) \\ &\ge 2\sum_{\gamma\in\Gamma} \exp\left( -\frac{s}{2} \bigl(d_\varphi(e,\gamma)+d_{\varphi\circ\operatorname{i}}(e,\gamma)\bigr) \right) = 2\sum_{\gamma\in\Gamma} e^{-s d_{\bar\varphi}(e,\gamma)}. \end{align}\] This proves 30 . ◻

Proof. By 2, \(\mathcal{L}_N^{\operatorname{i}}=\mathcal{L}_\Gamma^{\operatorname{i}}\). Since \(\psi_{N}=-\infty\) outside \(\mathcal{L}_N\), it suffices to show that \(\psi_N(v)=\psi_\Gamma(v)\) for all \(v\in \mathcal{L}_N\) and \(\operatorname{i}(v)=v\). First consider the case where \(\operatorname{int}\mathcal{L}_\Gamma\). Let \(\varphi\in\mathfrak a^*\) be a linear form tangent to \(\psi_{\Gamma}\) at \(v\), that is, \(\varphi\ge \psi_{\Gamma}\) and \(\varphi(v)=\psi_{\Gamma}(v)\). Such a form exists since \(v\in \operatorname{int}\mathcal{L}_\Gamma\) and \(\psi_{\Gamma}\) is concave (cf. [8]). Since \(\psi_{\Gamma}\) is invariant under \(\operatorname{i}\) and \(\operatorname{i}(v)=v\), the linear form \(\varphi\circ\operatorname{i}\) is also tangent to \(\psi_{\Gamma}\) at \(v\). Hence replacing \(\varphi\) with \(\bar{\varphi}\), we may assume that \(\varphi=\varphi\circ\operatorname{i}\). Moreover, since \(\Gamma\) is Borel-Anosov, a tangent form to \(\psi_{\Gamma}\) at an interior point of \(\mathcal{L}_\Gamma\) is positive on \(\mathcal{L}_\Gamma-\{0\}\). Because \(\varphi\) is tangent to \(\psi_{\Gamma}\), we have \(\delta_{\Gamma,\varphi}=1\). Since \(\varphi\) is symmetric, Theorem 10 gives \[\delta_{N,\varphi}=\delta_{\Gamma,\varphi}=1.\]

Since \(\varphi >0\) on \(\mathcal{L}_N-\{0\}\) and \(\delta_{N,\varphi}=1\), it follows that \(\varphi\) is tangent to \(\psi_{N}\) at some \(u\in \mathcal{L}_N\) (see [21]). Now \[\varphi(u)=\psi_{N}(u)\le \psi_{\Gamma}(u)\le \varphi(u),\] and hence \(\psi_{\Gamma}(u)=\varphi(u)\). We claim that \(u\) lies on the ray \(\mathbb{R}_{>0}v\). Suppose not. For \(0<t<1\), set \[w_t=(1-t)v+tu.\] Since \(v\in\operatorname{int}\mathcal{L}_\Gamma\) and \(u\in\mathcal{L}_\Gamma\), we have \(w_t\in\operatorname{int}\mathcal{L}_\Gamma\). By concavity of \(\psi_{\Gamma}\) and the inequalities \(\psi_{\Gamma}\le\varphi\), we get \[\psi_{\Gamma}(w_t) \ge (1-t)\psi_{\Gamma}(v)+t\psi_{\Gamma}(u) = (1-t)\varphi(v)+t\varphi(u) = \varphi(w_t) \ge \psi_{\Gamma}(w_t).\] Therefore equality holds throughout. Hence \(\psi_{\Gamma}\) is affine on the segment joining \(v\) to \(u\), whose interior lies in \(\operatorname{int}\mathcal{L}_\Gamma\). Since \(\psi_{\Gamma}\) is strictly concave on \(\operatorname{int}\mathcal{L}_\Gamma\), this is possible only if \(u\) and \(v\) lie on the same ray. Thus \(u=tv\) for some \(t>0\). By homogeneity, we get \(\psi_{N}(v)=\psi_{\Gamma}(v)\), as desired.

We now extend this to the case when \(v\) lies in the boundary of \(\mathcal{L}_\Gamma\). Since \(\mathcal{L}_N\) has non-empty interior and \(\operatorname{i}\)-invariant, we can choose \(w\in \operatorname{int}\mathcal{L}_N\) with \(\operatorname{i}(w)=w\). For \(0<t\le 1\), define \(v_t=(1-t)v+tw\). Then for all \(0<t<1\), \(\operatorname{i}(v_t)=v_t\) and \(v_t\in \operatorname{int}\mathcal{L}_N\subset \operatorname{int}\mathcal{L}_\Gamma\), and hence \(\psi_N(v_t)=\psi_\Gamma(v_t)\). We use the elementary fact that a finite concave upper-semicontinuous function on a compact interval is continuous. Applying this to the restrictions of \(\psi_N\) and \(\psi_\Gamma\) to the line segment \(\{v_t: 0\le t\le 1\}\), we get \(\psi_N(v)=\psi_\Gamma(v)\). ◻

Proof. Let \(v_N\in\mathcal{L}_N\) be the maximal growth direction of \(N\). By Quint’s theorem [22], the linear form \[\varphi_{N}(u)\mathrel{\vcenter{:}}=\langle u,v_N\rangle\] satisfies \[\delta_{N,\varphi_{N}}=\delta_N.\] Since \(\Gamma\) is Borel-Anosov, we have \(\mathcal{L}_\Gamma-\{0\}\subset \operatorname{int}\mathfrak a^+\) and hence \(v_N\in\operatorname{int}\mathfrak a^+\). Since \(\mathfrak a^+\) is acute with respect to the inner product \(\langle \cdot, \cdot\rangle\), \(\varphi_{N}\) is positive on \(\mathcal{L}_{\Gamma} -\{0\}\). . Since both \(\psi_N\) and \(v_N\) are invariant under \(\operatorname{i}\), the uniqueness of the maximal growth direction implies \(\operatorname{i}(v_N)=v_N\) and hence \(\varphi_N\) is symmetric. Therefore Theorem 10 gives \[\label{pre}\delta_{\Gamma,\varphi_{N}}=\delta_{N,\varphi_{N}}=\delta_N .\tag{34}\] On the other hand, since \(\|v_N\|=1\), the Cauchy–Schwarz inequality gives \[\varphi_{N}(\mu(\gamma))=\langle \mu(\gamma),v_N\rangle \le \|\mu(\gamma)\| \quad\text{ for all }\gamma\in\Gamma.\] Hence \[\{\gamma\in\Gamma:\|\mu(\gamma)\|<T\} \subset \{\gamma\in\Gamma:\varphi_{N}(\mu(\gamma))<T\},\] and therefore \[\delta_\Gamma\le \delta_{\Gamma,\varphi_{N}}.\] Combining this with 34 gives \(\delta_\Gamma\le \delta_N\). Hence \(\delta_N=\delta_\Gamma\). ◻

Proof. Let \(S=\{a^{\pm1},b^{\pm1}\}\) be the free generating set of \(F_2\), and let \[\Sigma=\{(x_i)_{i\ge 0}\in S^{\mathbb{N}}: x_{i+1}\ne x_i^{-1}\}\] be the one-sided subshift coding reduced words. We denote by \(\sigma\) the shift map. By Quint’s Schottky coding for orbital counting [25], there exists a positive Hölder function \(L:\Sigma\to\mathbb{R}\) with the following property: there is a constant \(C>0\) such that, for every reduced word \(\gamma=s_0s_1\cdots s_{n-1}\in F_2\) and every \(x\in\Sigma\) beginning with the block \(s_0s_1\cdots s_{n-1}\), we have \[\left| \sum_{k=0}^{n-1} L(\sigma^k x)-\ell(\gamma) \right|\le C.\]

Since \(\chi_0:F_2\to\mathbb{R}\) is a homomorphism, it is represented by the locally constant function \[\mathcal{X}(x)=\chi_0(x_0),\qquad x=(x_i)_{i\ge0}\in\Sigma.\] Thus, if \(x\) begins with the reduced word \(\gamma=s_0\cdots s_{n-1}\), then \[\sum_{k=0}^{n-1}\mathcal{X}(\sigma^k x)=\chi_0(\gamma).\] Therefore, for every \(t\in\mathbb{R}\), \[\left| \sum_{k=0}^{n-1}(L+t\mathcal{X})(\sigma^k x) - \bigl(\ell(\gamma)+t\chi_0(\gamma)\bigr) \right| \le C.\]

Since \(L\) is positive and continuous on the compact space \(\Sigma\), there is \(c_0>0\) such that \(L\ge c_0\). Hence, for all sufficiently small \(|t|\), the function \[L_t:=L+t\mathcal{X}\] is still positive. In particular, \(\ell+t\chi_0\) is proper on \(F_2\), and the bounded error above does not change the exponential growth exponent. Thus \[H(t):=\delta_{F_2,\ell+t\chi_0}\] is characterized by the pressure equation \[p(-H(t)L_t)=0,\] where \(p\) denotes topological pressure for \((\Sigma,\sigma)\).

The map \((s,t)\mapsto p(-sL_t)\) is real analytic. Moreover, \[\frac{\partial}{\partial s}p(-sL_t) = -\int L_t\,dm_{s,t}<0,\] where \(m_{s,t}\) is the equilibrium state of \(-sL_t\). Hence the implicit function theorem implies that \(H(t)\) is real analytic for \(|t|\) small.

We next observe that \(H\) is even. Since \(\ell(\gamma^{-1})=\ell(\gamma)\) and \(\chi_0(\gamma^{-1})=-\chi_0(\gamma)\), the change of variables \(\gamma\mapsto\gamma^{-1}\) gives \[\#\{\gamma\in F_2:\ell(\gamma)+t\chi_0(\gamma)<T\} = \#\{\gamma\in F_2:\ell(\gamma)-t\chi_0(\gamma)<T\}.\] Therefore \(H(t)=H(-t)\) and hence \(H'(0)=0.\)

Let \(h=H(0)=\delta_{F_2,\ell}\), and let \(m\) be the equilibrium state for the potential \(-hL\). Differentiating \(p\bigl(-H(t)(L+t\mathcal{X})\bigr)=0\) at \(t=0\), and using the pressure derivative formula, gives \[0 = \int \bigl(-H'(0)L-h\mathcal{X}\bigr)\,dm.\] Since \(H'(0)=0\), we obtain \(\int \mathcal{X}\,dm=0\). Differentiating a second time and using the standard pressure Hessian formula gives \[0 = -H''(0)\int L\,dm + h^2\operatorname{Var}_m(\mathcal{X}),\] where \(\operatorname{Var}_m(\mathcal{X})\) is the asymptotic variance of \(\mathcal{X}\) with respect to the equilibrium state \(m\). Hence \[H''(0)\int L\,dm = h^2\operatorname{Var}_m(\mathcal{X}).\] Since \(L>0\), we have \(\int L\,dm>0\). It remains to show that \(\operatorname{Var}_m(\mathcal{X})>0\).

By the Livsic degeneracy criterion for a topologically mixing subshift of finite type, \(\operatorname{Var}_m(\mathcal{X})=0\) if and only if \(\mathcal{X}\) is cohomologous to a constant. Suppose, for contradiction, that \[\mathcal{X}=c+u-u\circ\sigma\] for some continuous function \(u\) and some constant \(c\). Then for every periodic word \(w\) of period \(n\), \[\chi_0(w)=\sum_{k=0}^{n-1}\mathcal{X}(\sigma^k x)=cn.\] Applying this to \(w^{-1}\), which has the same period \(n\), gives \(\chi_0(w)=-cn.\) Thus \(\chi_0(w)=0\) for every cyclically reduced word \(w\). This is impossible because \(\chi_0\) is a non-trivial homomorphism on \(F_2\). Therefore \[\operatorname{Var}_m(\mathcal{X})>0\quad \text{ and hence}\quad H''(0)>0.\]

Since \(H\) is real analytic, even, and satisfies \(H''(0)>0\), the point \(t=0\) is a strict local minimum of \(H\). Hence, for all sufficiently small \(t\ne0\), \[\delta_{F_2,\ell+t\chi_0}> \delta_{F_2,\ell}.\] This proves the lemma. ◻

Proof. By Lemma 8, applied to \(j\) and \({\chi}\), there exists \(\eta>0\) such that \[\label{eq:pressure-gap-all-positive-forms} \delta_{F_2,\ell+t{\chi}} > \delta_{F_2,\ell} \quad\text{for all 0<|t|<\eta.}\tag{38}\] Choose \(\kappa>0\) sufficiently small so that \(\kappa/2<\eta\) and, for all \(\gamma\in F_2-\{e\}\), \[\label{eq:kappa-choice-all-positive-forms} \ell(\gamma)>\frac{3\kappa}{2}|{\chi}(\gamma)|.\tag{39}\] This is possible because \(j\) is convex cocompact, so \(\ell(\gamma)\) grows linearly in the word length, while \(|{\chi}(\gamma)|\) is bounded above linearly in the word length. The choice of \(\kappa\) ensures that, for every non-trivial \(\gamma\in F_2\), the Cartan projection of \(\rho_{\kappa}(\gamma)\) is \[\mu(\rho_{\kappa}(\gamma)) = \operatorname{diag}\left( \ell(\gamma)-\frac{\kappa}{2}{\chi}(\gamma),\, \kappa{\chi}(\gamma),\, -\ell(\gamma)-\frac{\kappa}{2}{\chi}(\gamma) \right).\]

Hence, for all \(\gamma\in F_2-\{e\}\), \[\varphi(\mu(\rho_{\kappa}(\gamma))) = (s_1+s_2) \left( \ell(\gamma) + t_{\varphi} {\chi}(\gamma) \right)\] where \(t_{\varphi}:= \frac{\kappa(s_2-s_1)}{2(s_1+s_2)}\). Thus we obtain \[\delta_{\Gamma_{\kappa},\varphi} = \tfrac{1}{s_1+s_2}\, \delta_{F_2,\ell+t_{\varphi}{\chi}}.\]

On the other hand, \({\chi}=0\) on \(\mathcal{N}\). Hence, for every \(h\in\mathcal{N}-\{e\}\), \(\varphi(\mu(\rho_{\kappa}(h))) = (s_1+s_2)\ell(h).\) Therefore \[\delta_{N_{\kappa},\varphi} = \tfrac{1}{s_1+s_2}\,\delta_{\mathcal{N},\ell}.\] Since \(j(\mathcal{N})\) is cocyclic in the convex cocompact group \(j(F_2)<\operatorname{SL}_2(\mathbb{R})\), Roblin’s rank-one theorem [1] gives \(\delta_{\mathcal{N},\ell} = \delta_{F_2,\ell}.\) Thus \[\delta_{N_{\kappa},\varphi} = \tfrac{1}{s_1+s_2}\,\delta_{F_2,\ell}.\]

Since \(|s_2-s_1|\le s_1+s_2\), we have \(|t_{\varphi}|\le \kappa/2<\eta\). If \(s_1\ne s_2\), then \(t_{\varphi}\ne0\), and hence 38 gives \(\delta_{F_2,\ell+t_{\varphi}{\chi}} > \delta_{F_2,\ell}.\) Therefore \(\delta_{\Gamma_{\kappa},\varphi} > \delta_{N_{\kappa},\varphi}.\) If \(s_1=s_2\), then \(t_{\varphi}=0\), so the two displayed formulae give \(\delta_{\Gamma_{\kappa},\varphi} = \delta_{N_{\kappa},\varphi}.\) This completes the proof. ◻

Proof. Since \(\mathsf c\) is loxodromic, both \(\lambda(\mathsf c)\) and \(\lambda(\mathsf c^{-1})\) lie in \(\operatorname{int}\mathfrak a^+\). Compactness of \(\mathcal{U}\subset(\mathfrak a^+)^\vee-\{0\}\), together with \[n^{-1}\mu(\mathsf c^n)\to\lambda(\mathsf c), \qquad n^{-1}\mu(\mathsf c^{-n})\to\lambda(\mathsf c^{-1}),\] gives 41 . For 42 , apply the usual Schottky ping-pong estimate in the two fundamental representations \(\wedge^j\mathbb{R}^3\), \(j=1,2\) (see [3]). Taking logarithms and then applying \(\varphi\) gives the stated lower bound, uniformly for \(\varphi\in\mathcal{U}\). The upper estimate follows directly from submultiplicativity in the same two fundamental representations. ◻

Proof. Since \(\widehat\rho_{\kappa,r}|_{F_2}=\rho_\kappa\), we have \(\delta_{\Gamma_{\kappa,r},\varphi} \ge \delta_{\rho_\kappa(F_2),\varphi}\) for all \(r\) and all \(\varphi\in\mathcal{U}\).

Fix \(\varepsilon>0\), and put \[s_\varphi:=\delta_{\rho_\kappa(F_2),\varphi}+\varepsilon.\] Since \(\rho_\kappa(F_2)\) is Borel-Anosov and \(\mathcal{U}\subset(\mathfrak a^+)^\vee-\{0\}\) is compact, the pressure formalism for Schottky groups implies that \(s_\varphi\) is bounded above and bounded away from zero on \(\mathcal{U}\). Moreover, \[S_F(\varphi) := \sum_{g\in F_2} e^{-s_\varphi\varphi(\mu(\rho_\kappa(g)))}\] is uniformly bounded for \(\varphi\in\mathcal{U}\). By 41 , \[S_{\mathsf c}(r,\varphi) := \sum_{q\in\mathbb{Z}-\{0\}} e^{-s_\varphi\varphi(\mu(\mathsf c^{rq}))} \to0 \quad\text{uniformly for \varphi\in\mathcal{U}}.\]

Now let \(\gamma=g_0c^{q_1}g_1\cdots c^{q_k}g_k\) be reduced. By 42 , \[\begin{align} &\sum_{\gamma\in F_3} e^{-s_\varphi\varphi(\mu(\widehat\rho_{\kappa,r}(\gamma)))} \le e^{s_\varphi A_{\mathcal{U}}}S_F(\varphi) \sum_{k\ge0} \left( e^{s_\varphi A_{\mathcal{U}}} S_F(\varphi)S_{\mathsf c}(r,\varphi) \right)^k . \end{align}\] Since \(s_\varphi\) and \(S_F(\varphi)\) are uniformly bounded on \(\mathcal{U}\), while \(S_{\mathsf c}(r,\varphi)\to0\) uniformly, the geometric series converges for all sufficiently large \(r\), uniformly in \(\varphi\in\mathcal{U}\). Hence \[\delta_{\Gamma_{\kappa,r},\varphi} \le \delta_{\rho_\kappa(F_2),\varphi}+\varepsilon\] for all \(\varphi\in\mathcal{U}\) and all sufficiently large \(r\). Combining this with the lower bound and letting \(\varepsilon\to0\) proves the proposition. ◻

Proof. Apply Lemma 9 with \(\mathcal{U}=\{\varphi\}\), and write the resulting constant as \(A_\varphi\). Let \(w=w_0c^{q_1}w_1\cdots c^{q_k}w_k\in\mathcal{N}'\) with \(w_i\in F_2\) and \(q_i\in\mathbb{Z}-\{0\}\). Since \(w\in\mathcal{N}'\), the sum of the \(a\)-exponents of the \(F_2\)-blocks is zero: \(\sum_{i=0}^{k}\mathcal{X}(w_i)=0\).

For \(u\in F_2-\{e\}\), by the definition of \(\rho_{\kappa}\) and the choice of \(\kappa\), we have \[\mu(\rho_{\kappa}(u)) = \operatorname{diag}\left( \ell(u)-\frac{\kappa}{2}{\mathcal{X}}(u),\, \kappa{\mathcal{X}}(u),\, -\ell(u)-\frac{\kappa}{2}{\mathcal{X}}(u) \right).\] Hence \[\omega_1(\mu(\rho_{\kappa}(u))) = \ell(u)-\frac{\kappa}{2}{\mathcal{X}}(u)\text{ and } \omega_2(\mu(\rho_{\kappa}(u))) = \ell(u)+\frac{\kappa}{2}{\mathcal{X}}(u).\] Therefore \[\varphi(\mu(\rho_{\kappa}(u))) = \varphi(\mu(\rho_{0}(u))) + \frac{\kappa}{2}(s_2-s_1){\mathcal{X}}(u).\] Applying this to the \(F_2\)-blocks \(w_i\), and using \(\sum_i{\mathcal{X}}(w_i)=0\), gives \[\label{eq:F2-block-cancellation-general} \sum_{i=0}^{k} \varphi(\mu(\rho_{\kappa}(w_i))) = \sum_{i=0}^{k} \varphi(\mu(\rho_{0}(w_i))).\tag{43}\]

Let \(r'=\lfloor r/2\rfloor\). By 41 , after increasing \(r\), we may assume that, for every \(q\ne0\), \[\label{eq:C-comparison-varphi} \varphi(\mu(\mathsf c^{rq})) \ge \varphi(\mu(\mathsf c^{r'q}))+A_\varphi .\tag{44}\] Combining 42 , 43 , and 44 , we get \[\begin{align} \varphi(\mu(\widehat\rho_{\kappa,r}(w))) \ge& \sum_{i=0}^{k} \varphi(\mu(\rho_0(w_i))) + \sum_{i=1}^{k} \varphi(\mu(\mathsf c^{r'q_i})) - A_\varphi \\\ge & \varphi(\mu(\widehat\rho_{0,r'}(w)))-A_\varphi \qquad(w\in\mathcal{N}') \end{align}\]

It follows that, for every \(s>0\), \[\begin{align} \sum_{w\in\mathcal{N}'} e^{-s\varphi(\mu(\widehat\rho_{\kappa,r}(w)))} &\le e^{sA_\varphi} \sum_{w\in\mathcal{N}'} e^{-s\varphi(\mu(\widehat\rho_{0,r'}(w)))} \le e^{sA_\varphi} \sum_{\gamma\in F_3} e^{-s\varphi(\mu(\widehat\rho_{0,r'}(\gamma)))} . \end{align}\] Therefore \[\delta_{N'_{\kappa,r},\varphi} \le \delta_{\widehat\rho_{0,r'}(F_3),\varphi}.\] Using Proposition 18 for \(\widehat\rho_{0,r'}\), we obtain \[\limsup_{r\to\infty} \delta_{N'_{\kappa,r},\varphi} \le \delta_{\Gamma_0,\varphi}.\] ◻

Proof. Write \(\varphi=s_1\omega_1+s_2\omega_2\) with \(s_1,s_2\ge0\) and \(s_1+s_2>0\). By Theorem 17, together with Lemma 8, we have \[\delta_{\Gamma_{\kappa},\varphi} > \delta_{\Gamma_0,\varphi} \quad \text{ for all sufficiently small \kappa>0}.\] Indeed, if \(t_\varphi := \frac{\kappa(s_2-s_1)}{2(s_1+s_2)}\ne 0\) is sufficiently small, then \[\delta_{\Gamma_{\kappa},\varphi} = \tfrac{1}{s_1+s_2}\, \delta_{F_2,\ell+t_\varphi{\chi}} > \tfrac{1}{s_1+s_2}\, \delta_{F_2,\ell} = \delta_{\Gamma_0,\varphi}.\] By Proposition 18, we also have \[\lim_{r\to \infty} \delta_{\Gamma_{\kappa,r},\varphi} = \delta_{\Gamma_{\kappa},\varphi}.\] On the other hand, by Lemma 10, \[\limsup_{r\to\infty} \delta_{N'_{\kappa,r},\varphi} \le \delta_{\Gamma_0,\varphi}.\] Combining the last three displays gives \[\limsup_{r\to\infty} \delta_{N'_{\kappa,r},\varphi} < \lim_{r\to\infty} \delta_{\Gamma_{\kappa,r},\varphi}.\] Hence, for all sufficiently large \(r\), \[\delta_{N'_{\kappa,r},\varphi} < \delta_{\Gamma_{\kappa,r},\varphi}.\] Fix such an \(r\), and set \[\Gamma:=\Gamma_{\kappa,r}, \qquad N:=N'_{\kappa,r}.\] For \(r\) sufficiently large, \(\Gamma\) is Borel-Anosov and Zariski dense by the choice of \(\mathsf c\). Since the representation is Borel-Anosov, it is faithful. Hence \[\Gamma/N\simeq F_3/\mathcal{N}'\simeq \mathbb{Z},\] so \(N\) is a cocyclic normal subgroup of \(\Gamma\). Finally, \(N\) is Zariski dense. This proves the theorem. ◻

Proof. Choose a compact neighborhood \(\mathcal{U}\) of \(\varphi_0\) in \(\operatorname{int}((\mathfrak a^+)^\vee)\). By Proposition 18, we have \[\delta_{\Gamma_r,\varphi} \to \delta_{\Gamma_\infty,\varphi} \quad\text{uniformly for \varphi\in\mathcal{U}}.\] After shrinking \(\mathcal{U}\) if necessary, the limiting exponent is bounded away from zero on \(\mathcal{U}\). For each sufficiently large \(r\), the uniqueness of the maximizer noted above implies that \(\varphi\mapsto\delta_{\Gamma_r,\varphi}\) is differentiable on \(\mathcal{U}\). For the limiting model group, writing \(\varphi=s_1\omega_1+s_2\omega_2\) the explicit model formula gives \[\delta_{\Gamma_\infty,\varphi} = \frac{1}{s_1+s_2}\, \delta_{F_2,\ell+ \frac{\kappa(s_2-s_1)}{2(s_1+s_2)}\chi}.\] By the pressure argument used in the proof of 8, the right-hand side is real analytic for \(\varphi\) near \(\varphi_0\). Hence \[\varphi\mapsto\delta_{\Gamma_\infty,\varphi}\] is \(C^1\) on \(\mathcal{U}\), after shrinking \(\mathcal{U}\) if necessary. We now use the following standard consequence of convexity: if differentiable convex functions converge locally uniformly to a \(C^1\) convex function, then their differentials converge locally uniformly on compact subsets. Since the functions \(\varphi\mapsto \delta_{\Gamma_r,\varphi}\) are convex and converge uniformly on \(\mathcal{U}\) to the \(C^1\) function \(\varphi\mapsto \delta_{\Gamma_\infty,\varphi}\), we obtain \[D_\varphi\delta_{\Gamma_r,\varphi} \to D_\varphi\delta_{\Gamma_\infty,\varphi} \qquad\text{uniformly for }\varphi\in\mathcal{U}.\]

For every \(\eta\in\mathfrak a^*\), we get \[\label{dvv} D_\varphi\delta_{\Gamma_r,\varphi}(\eta) = -\delta_{\Gamma_r,\varphi}\,\eta(v_{r,\varphi}).\tag{45}\] Indeed, set \(F(\varphi)=\delta_{\Gamma_r,\varphi}\). By 23 , for small \(t\), \[F(\varphi+t\eta) \ge \frac{\psi_{\Gamma_r}(v_{r,\varphi})}{(\varphi+t\eta)(v)} = \frac{F(\varphi)}{1+t\eta(v_{r,\varphi})}.\] Dividing by \(t>0\) and letting \(t\to0^+\) gives \(D_\varphi F(\eta)\ge -F(\varphi)\eta(v_{r,\varphi}).\) Applying the same argument with \(-\eta\) in place of \(\eta\) gives the reverse inequality, yielding 45 .

The same formula holds for the model group: \[D_\varphi\delta_{\Gamma_\infty,\varphi}(\eta) = -\delta_{\Gamma_\infty,\varphi}\,\eta(v_{\infty,\varphi}).\] Since \(\delta_{\Gamma_r,\varphi}\to\delta_{\Gamma_\infty,\varphi}>0\) uniformly on \(\mathcal{U}\), the convergence of differentials implies that for every \(\eta\in\mathfrak a^*\), \[\eta(v_{r,\varphi})\to \eta(v_{\infty,\varphi}) \qquad\text{uniformly for }\varphi\in\mathcal{U}.\] Choosing a basis of \(\mathfrak a^*\), we conclude that \(v_{r,\varphi}\to v_{\infty,\varphi}\) uniformly for \(\varphi\in\mathcal{U}\). ◻

Proof. Let \(U\) be a conic neighborhood of the barycentric ray whose closure is contained in the interior of the convex cone spanned by \(\lambda({\mathsf c})\) and \(\lambda({\mathsf c}^{-1})\). Since \(c,c^{-1}\in\mathcal{N}'\), the limit cone \(\mathcal{L}_{N_r'}\) contains the convex cone spanned by \(\lambda({\mathsf c})\) and \(\lambda({\mathsf c}^{-1})\). Therefore \(U\subset \operatorname{int}\mathcal{L}_{N_r'}.\)

It remains to show that \(v_{r,\varphi}\in U\) for \(\varphi\) near \(\varphi_0\) and \(r\) large. Since \(\varphi_0\) and \(\psi_{\Gamma_\infty}\) are both \(\operatorname{i}\)-invariant, it follows from the uniqueness that \(v_{\infty,\varphi_0}\) lies on the barycentric ray. The map \[\varphi\mapsto v_{\infty,\varphi}\] is continuous near \(\varphi_0\). Indeed, since \(\varphi\mapsto\delta_{\Gamma_\infty,\varphi}\) is real analytic near \(\varphi_0\), and since \(v_{\infty, \varphi}\) satisfies \[D_\varphi\delta_{\Gamma_\infty,\varphi}(\eta) = -\delta_{\Gamma_\infty,\varphi}\,\eta(v_{\infty,\varphi}) \qquad(\eta\in\mathfrak a^*),\] the vector \(v_{\infty, \varphi}\) depends continuously on \(\varphi\). Therefore, after shrinking \(\mathcal{U}\), we have \[v_{\infty,\varphi}\in U \qquad(\varphi\in\mathcal{U}).\] By the uniform convergence \(v_{r,\varphi}\to v_{\infty,\varphi}\) (11), after increasing \(r_0\) we obtain \(v_{r,\varphi}\in U\) for all \(r\ge r_0\) and \(\varphi\in\mathcal{U}\). This proves the lemma. ◻

Proof. Let \(\mathcal{C}_{\mathsf c}\) denote the convex cone spanned by two rays \(\lambda(\mathsf c)\) and \(\lambda(\mathsf c^{-1})\). Since \(\mathcal{L}_\Delta\) is \(\operatorname{i}\)-invariant, so is the smallest convex cone containing it. Since \(\operatorname{i}(\lambda(\mathsf c))=\lambda(\mathsf c^{-1})\), the first hypothesis implies that \(\lambda(\mathsf c^{-1})\) lies outside the smallest convex cone containing \(\mathcal{L}_\Delta\). Therefore we have \(\mathcal{L}_\Delta-\{0\}\subset \operatorname{int}\mathcal{C}_{\mathsf c}\).

We first arrange a ping-pong configuration. Since \(\Gamma\) is Borel-Anosov, its boundary map to the full flag variety \(\mathcal{F}\) is antipodal. Since \(\Delta\) is a free factor of the word-hyperbolic group \(\Gamma\), it is quasiconvex in \(\Gamma\). Hence \(\Delta\) is again Borel-Anosov, with limit set \(\Lambda_\Delta\subset \mathcal{F}\).

Let \(\xi_{\mathsf c}\in \mathcal{F}\) and \(\xi_{\mathsf c^{-1}}\in \mathcal{F}\) denote the attracting fixed points of \(\mathsf c\) and \(\mathsf c^{-1}\) respectively. Choose compact neighborhoods \(Y\) of \(\Lambda_\Delta\) and \(X\) of \(\{\xi_{\mathsf c} ,\xi_{\mathsf c^{-1}}\}\cup \bigcup_{n\ne0}\mathsf c^n\Lambda_\Delta\) so that every flag in \(X\) is antipodal to every flag in \(Y\), and so that for all \(n\in \mathbb{Z}-\{0\}\), \[\mathsf c^nY\subset\operatorname{int}X\] By the convergence dynamics of the Anosov subgroup \(\Delta\), all but finitely many elements \(g\in\Delta\) satisfy \[gX\subset\operatorname{int}Y.\] Since \(\Delta\) is finitely generated and linear, it is residually finite. Passing to a finite-index subgroup \(\Delta'<\Delta\) which avoids this finite exceptional set, we obtain \[gX\subset\operatorname{int}Y \qquad \text{for all g\in\Delta'-\{e\}.}\] Thus \(\Delta'\) and \(\langle\mathsf c\rangle\) play ping-pong with the sets \(X\) and \(Y\).

By [28] (see also [3]), there exists a constant \(C_0>0\) such that every reduced word \[w=g_1\mathsf c^{n_1}\cdots g_k\mathsf c^{n_k}, \qquad g_i\in\Delta'-\{e\},\quad n_i\in\mathbb{Z}-\{0\},\] satisfies \[\left\| \mu(w) - \sum_{i=1}^k\mu(g_i) - \sum_{i=1}^k\mu(\mathsf c^{n_i}) \right\| \le C_0k.\] Since \[\|\mu(\mathsf c^n)-\lambda(\mathsf c^n)\|=O(1) \qquad(n\in\mathbb{Z}),\] we may enlarge \(C_0\) and write \[\left\| \mu(w) - \sum_{i=1}^k\mu(g_i) - \sum_{i=1}^k |n_i|\lambda(\mathsf c^{\operatorname{sgn}n_i}) \right\| \le C_0k.\]

If \(w\) is cyclically reduced, applying this estimate to \(w^m\), dividing by \(m\), and letting \(m\to\infty\), gives \[\label{give} \left\| \lambda(w) - \sum_{i=1}^k\mu(g_i) - \sum_{i=1}^k |n_i|\lambda(\mathsf c^{\operatorname{sgn}n_i}) \right\| \le C_0k.\tag{47}\]

Indeed, when \(w\) is cyclically reduced, the word \(w^m\) is obtained by concatenating \(m\) copies of \(w\) without cancellation, and \(\lambda(w)=\lim_{m\to\infty}\frac{1}{m}\mu(w^m)\).

If \(w\) is not cyclically reduced, then \(w\) is conjugate either to a cyclically reduced word or to an element of one of the free factors. Since the Jordan projection is invariant under conjugation, replacing \(w\) by a conjugate does not change \(\lambda(w)\). Thus the preceding estimate applies after conjugating \(w\) to cyclically reduced form; if \(w\) is conjugate into \(\Delta'\) or into \(\langle \mathsf c\rangle\), the desired cone containment follows directly from \(\mathcal{L}_{\Delta'}\subset \mathcal{C}_{\mathsf c}\) or \(\lambda(\mathsf c^{\pm1})\in\mathcal{C}_{\mathsf c}\).

We now show that every Jordan projection \(\lambda(w)\) lies in \(\mathcal{C}_{\mathsf c}\). For \(v\in\operatorname{int}\mathfrak a^+\), set \[R(v):=\frac{\alpha_1(v)}{\alpha_2(v)}.\] Let \[R_+:= \max\{R(\lambda(\mathsf c)),R(\lambda(\mathsf c^{-1}))\}, \qquad R_-:= \min\{R(\lambda(\mathsf c)),R(\lambda(\mathsf c^{-1}))\}.\] Then \(\mathcal{C}_{\mathsf c}\) is precisely the closed cone of vectors \(v\in\mathfrak a^+\) satisfying \[R_-\le R(v)\le R_+.\]

Since \[\mathcal{L}_\Delta-\{0\} \subset \operatorname{int}\mathcal{C}_{\mathsf c},\] the ratios \(R(v)\), for \(v\in\mathcal{L}_\Delta-\{0\}\), are uniformly bounded away from \(R_+\) and \(R_-\). Moreover, since \(\mathcal{L}_\Delta\) is the asymptotic cone of \(\mu(\Delta)\), after passing to a deeper finite-index subgroup \(\Delta'\), if necessary, we may therefore assume that, for every non-trivial \(g\in\Delta'\), \[\label{muc} \alpha_2(\mu(g))>C_0 \quad\text{ and }\quad \frac{\alpha_1(\mu(g))+C_0}{\alpha_2(\mu(g))-C_0} < R_+.\tag{48}\]

So \[D_w:= \sum_{i=1}^k\bigl(\alpha_2(\mu(g_i))-C_0\bigr) + \sum_{i=1}^k |n_i| \alpha_2\bigl(\lambda(\mathsf c^{\operatorname{sgn} n_i})\bigr) >0.\] Since \({\alpha_1(\lambda(\mathsf c^{\pm1}))}\le R_+ {\alpha_2(\lambda(\mathsf c^{\pm1}))}\) and \({\alpha_1(\mu(g))+C_0}< R_+ {\alpha_2(\mu(g))-C_0}\) for all \(g\in \Delta'\), the estimate for \(\lambda(w)\) in 47 gives \[\alpha_1(\lambda(w))\le R_+D_w, \qquad \alpha_2(\lambda(w))\ge D_w.\] Hence \[R(\lambda(w))\le R_+.\]

Applying the same argument to \(w^{-1}\), and using \(\lambda(w^{-1})=\operatorname{i}(\lambda(w))\) gives the lower bound \[R(\lambda(w))\ge R_-.\] Therefore \[\lambda(w)\in\mathcal{C}_{\mathsf c} \qquad \text{for all w\in\Delta'*\langle\mathsf c\rangle}.\]

By hypothesis, \(\Delta'*\langle\mathsf c\rangle\) is Zariski dense in \(\operatorname{SL}_3(\mathbb{R})\). Hence, its limit cone is the smallest closed convex cone containing the Jordan projections of its elements. Since all these Jordan projections lie in \(\mathcal{C}_{\mathsf c}\), we get \[\mathcal{L}_{\Delta'*\langle\mathsf c\rangle} \subset \mathcal{C}_{\mathsf c}.\] The reverse inclusion holds because \(\mathsf c\) and \(\mathsf c^{-1}\) belong to \(\Delta'*\langle\mathsf c\rangle\). This proves the lemma. ◻

Proof. Since \(\langle w\rangle\) is a quasiconvex subgroup of \(\Gamma\), it acts cocompactly on \[\mathcal{X}_w := \big(\Gamma\cup \partial\Gamma\big) \smallsetminus \{w^+,w^-\},\] where \(w^+\) and \(w^-\) are the attracting and repelling fixed points of \(w\) in \(\partial\Gamma\). Choose a compact fundamental domain \(\mathcal{F}_1\subset \mathcal{X}_w\) for this action, so that \(\mathcal{X}_w=\langle w\rangle \mathcal{F}_1\). In particular, the points of \(\mathcal{F}_1\) are uniformly separated from \(w^+\) and \(w^-\) in the visual metric. Let \(T_1:=\Gamma\cap \mathcal{F}_1\). For each \(t\in T_1\), write \(t^{-1}=w^{m(t)}\overline{t}\) for some \(m(t)\in\mathbb{Z}\) and some \(\overline{t}\in T_1\), and set \(T:=\{\overline{t}:t\in T_1\}\). Then any element of \(T\), as well as its inverse up to left multiplication by a power of \(w\), remains uniformly away from \(\{w^+,w^-\}\) in the visual metric. Choose from \(T\) one representative for each double coset of \(\langle w\rangle\) in \(\Gamma - \langle w\rangle\), and denote the resulting set by \(Q\). By construction, for any \(h\in Q\), the points represented by \(h\) and \(h^{-1}\) are uniformly away from \(w^+\) and \(w^-\). Hence the Gromov products \((h^{\mp1} \cdot w^r)\), \(h\in Q,\;r\in\mathbb{Z}\), are uniformly bounded. This proves the first estimate. The same compactness argument shows that since for any \(h\in Q\) non-trivial, \(h\{w^{+},w^{-}\}\) is uniformly away from \(\{w^{+},w^{-}\}\) in \(\Gamma \cup \partial \Gamma\), the Gromov products \(\big((w^s h^{\pm1})^{-1}\cdot w^r\big)\), \(h\in Q,\;r,s\in\mathbb{Z}\), are uniformly bounded. Equivalently, \(\big||w^s h^{\pm 1}w^r|-|w^s h^{\pm 1}|-|w^r|\big|\) is uniformly bounded. Enlarging the constant if necessary completes the proof. ◻

Proof. Let \(Q\subset\Gamma\) be the set of double coset representatives provided by Lemma 14. Since \(Q\) is a set of representatives for \(\langle w\rangle\backslash \big(\Gamma\smallsetminus \langle w\rangle\big)/ \langle w\rangle\) any element of \(\Gamma\smallsetminus\langle w\rangle\) can be written uniquely in the form \[w^r f w^s, \qquad r,s\in\mathbb{Z},\quad f\in Q.\]

By Lemma 14, there exists \(C>0\) such that, for any \(r,s\in\mathbb{Z}\) and any \(f\in Q\), \[\big||w^r f w^s|-|f|-|w^r|-|w^s|\big|\leq C.\] Applying Proposition 11, we obtain a constant \(L>0\) such that \[\label{exponent-ineq4} \sup_{r,s\in\mathbb{Z}} \sup_{f\in Q} \big\| \mu(w^r f w^s)-\mu(f)-\mu(w^r)-\mu(w^s) \big\| \leq L.\tag{51}\]

Since \(\langle w\rangle\) is quasi-isometrically embedded, there exists \(C_1>0\) such that, for all \(T\geq 1\), \[\#\{r\in\mathbb{Z}:\|\mu(w^r)\|\leq T\}\leq C_1T.\] Using 51 , we obtain, after enlarging constants if necessary, \[\begin{gather} \#\{g\in\Gamma:\|\mu(g)\|\leq T\} \leq \#\{r\in\mathbb{Z}:\|\mu(w^r)\|\leq T\} \\ \quad+ \#\Big\{ (r,s,f)\in\mathbb{Z}^2\times Q: \|\mu(w^r)\|+\|\mu(w^s)\|+\|\mu(f)\|\leq T+L \Big\} \\ \leq C_2T^2 \#\{f\in Q:\|\mu(f)\|\leq T+L\}+C_2T . \end{gather}\]

Absorbing the fixed additive error \(L\) and the lower-order term into the polynomial factor proves 50 . ◻

Proof of Theorem 22. Since \(\Gamma\) is torsion-free hyperbolic and \(N\) is non-virtually cyclic, there exist a primitive element \(w\in N\) (e.g. see the proof of [29]).

Let \(Q\) be the set of double coset representatives of \(\langle w\rangle\) given by Lemma 15. Since \(\Gamma\) is torsion-free hyperbolic and \(w\) is primitive, the centralizer of \(w\) is \(\langle w\rangle\). Hence the map \(Q\to \Gamma\) given by \(f\mapsto fw f^{-1}\) is injective. Moreover, by subadditivity of the Cartan projection and by \(\|\mu(f^{-1})\|=\|\mu(f)\|\), we have \[\|\mu(fw f^{-1})\| \leq 2\|\mu(f)\|+\|\mu(w)\|.\] Hence, for any \(T\geq 1\), \[\#\{h\in N:\|\mu(h)\|\leq 2T+\|\mu(w)\|\} \geq \#\{f\in Q:\|\mu(f)\|\leq T\}.\] Using 50 , we obtain \[\#\{h\in N:\|\mu(h)\|\leq 2T+\|\mu(w)\|\} \geq {c}{T^{-2}} \#\{g\in\Gamma:\|\mu(g)\|\leq T\}.\] Taking logarithms, dividing by \(2T+\|\mu(w)\|\), and passing to the upper limit gives \(\delta_N\geq \delta_\Gamma/2\). ◻