# Reading roadmap: Generic thinness in finitely generated subgroups of SL_n(Z) The proposed argument compares ordinary and inverse-symmetric matrix norm balls. Cartan-coordinate volume estimates and lattice sector counts supply singular-value gaps and transverse partial flags; exterior-power ping-pong then gives freeness. Zariski density and infinite index give thinness. Separate arguments address dimension two, Hausdorff dimension of limit sets and Lyapunov exponents. Source SHA-256: `8adb9681e2cd5e4bdcb292162a32796b00ae7b757a5a900fbd580d0bc939e34e`. Indexed 81,049 characters. Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates. ## Fix the sampling model Distinguish ordinary operator-norm balls, balls bounding both the matrix and its inverse, and random walks. Read the separate conclusions with those measures fixed. Claims: thinsln, sl2, aounthm ## The two-dimensional branch Follow traces, circle geometry, the Schottky construction and the limit-set dimension argument. Claims: disjointcircles, sl2schottky, haussl2 ## Cartan coordinates and lattice counts Compare the ordinary-ball gap distribution with middle singular-value gaps in symmetrized balls; read the well-roundedness section alongside these calculations. Claims: toptwoth, middletwo, wellrounded1, gapwellrounded ## Exterior powers to thinness Follow contraction, angular incidence and ping-pong into generic freeness, then distinguish the Zariski-density and infinite-index inputs. Claims: epscontractextpower, contracts, prox, freesln, thinsln ## The random-product conclusion Read the matrix-product and reduced-word-length estimates as inputs to the separate Lyapunov-exponent assertion. Claims: singleop, twoops, freelen, thm:line:734 ## Internal claim index ### Thm 1.1 — `thinsln` Source lines 124–130. Let $G=\mathrm{SL}_n(\ZZ)$ where $n\geq 2$, and let $B_X'(G)$ and $\mu_X'$ be as above. Then we have \lim_{X\rightarrow\infty}\mu'_X(\{(g_1,g_2)\in (B_X'(G))^2\; |\; \langle g_1,g_2\rangle \mbox{ is thin}\})=1 *Extractive statement; proof gist not independently reconstructed.* - `freesln` (proof_dependency; curated_source_check): proving Proposition~\ref{freesln}. Since free subgroups of $\textrm{SL}_n(\mathbb Z)$ are infinite index in $\textrm{SL}_n(\mathbb Z)$, and since the generic finitely generated subgroup of $\textrm{SL}_n(\mathbb Z)$ is Zariski dense in $\SLn(\mathbb C)$ by \cite{R2}, Theorem~\ref{thinsln} follows. ### Thm 2.1 — `sl2` Source lines 147–153. Let $G=\mathrm{SL}_2(\ZZ)$, and let $\Gamma(g)$ and $\mu_X$ be as above. Then we have \lim_{X\rightarrow\infty}\mu_X(\{g=(g_1,g_2)\in G^2\; |\; \Gamma(g) \mbox{ is thin}\})=1 *Extractive statement; proof gist not independently reconstructed.* ### Lemma 2.2 — `disjointcircles` Source lines 186–192. For any pair $(g_1,g_2)\in G^2$ such that $|tr(g_i)|>2$ for $i=1,2$, let $\alpha_i$ and $\beta_i$ denote the fixed points of $g_i$. Let $$d_{\min}(g_1,g_2):= \min(d(\alpha_1,\beta_2),d(\alpha_1,\alpha_2),d(\beta_1,\beta_2),d(\beta_1,\alpha_2)),$$ where $d(\cdot,\cdot)$ denotes hyperbolic distance. Then for any $r>0$ we have \lim_{X\rightarrow\infty}\mu_X(\{(g_1,g_2)\in G^2\; | \; |tr(g_i)|>2, \,d_{\min}(g_1,g_2)>r\})=1. *Extractive statement; proof gist not independently reconstructed.* Proof opening: From (\ref{tracelarge}) we have that the probability that the traces of $g_i$ are different and greater than $2$ tends to $1$. We may therefore restrict to nonconjugate hyperbolic pairs $(g_1,g_2)$ in proving Lemma~\ref{disjointcircles}. Specifically, it suffices to show that \lim_{X\rightarrow\infty}\mu_X(\{(g_1,g_2)\in G^2\; | \; tr(g_1)\not=tr(g_2),\, d_{\min}(g_1,g_2)>r\})=1 Note now that to every pair of nonconjugate elements $g_1,g_2\in G$ one can associate a unique pair of distinct closed geodesics $L_1,L_2$ on $G\backslash\mathbb H$ fixed by $g_1$ and $g_2$ respectively. Furthermore, the length $\ell(L_i)$ is $$\ell(L_i)=\left((|tr(g_i)|+\sqrt{tr^2(g_i)-4})/2\right)^2$$ for $i=1,2$. Candidate references: `tracelarge` (candidate_proof_dependency); `wantshow2` (candidate_proof_dependency); `external:DRS` (external_reference); `external:EM` (external_reference); `external:sardav` (external_reference). ### Lemma 2.3 — `sl2schottky` Source lines 210–216. Let $G=\mathrm{SL}_2(\ZZ)$, and let $\Gamma(g)$ and $\mu_X$ be as before. Then we have \lim_{X\rightarrow\infty} \mu_X(\{g\in G^k \mbox{ s.t. } \Gamma(g) \mbox{ is Schottky}\})=1 *Extractive statement; proof gist not independently reconstructed.* ### Lemma 2.4 — `haussl2` Source lines 228–234. Let $g_i$, $\Gamma=\Gamma(g)$ and $\mu_X$ be as above, and let $\delta(\Gamma)$ denote the Hausdorff dimension of the limit set of $\Gamma$. Then for every $\epsilon>0$ we have \lim_{X\rightarrow\infty}\mu_X(\{g=(g_1,g_2) \mbox{ s.t. } 0<\delta(\Gamma)<\epsilon\})=1 *Extractive statement; proof gist not independently reconstructed.* Proof opening: Let $C_1,C_2,C_3,C_{4}$ be as above, and let $I_i$ denote the $i$th coordinate of I=(g_1,g_2,g_1^{-1},g_2^{-1}) By Lemma~\ref{sl2schottky}, the circles $C_i$ are mutually external and disjoint, and the generators $g_i$ of $\Gamma$ map the exterior of $C_i$ onto the interior of $C_{i+2}$ while their inverses do the opposite. We recall the following setup from \cite{B}. Let $K=3$, let $a(j)$ denote the center of $C_j$, and let $r(j)$ denote the radius of $C_j$. Define $\Sigma(m)$ to be the set of all sequences $(i_1,\dots,i_m)$ where i_1,\dots,i_m\in \{1,2,3, 4\} \mbox{ and } i_1\not=i_{2}\pm 2, \dots, i_{m-1}\not= i_m\pm 2 Define by I(i_1,\dots,i_m)= I_{i_1}\cdots I_{i_{m-1}}(C_{i_m}) where $ Candidate references: `sl2schottky` (candidate_proof_dependency); `hausupper` (candidate_proof_dependency); `hauslambda` (candidate_proof_dependency); `external:B` (external_reference); `external:R2` (external_reference); `external:KL` (external_reference). ### Defn 3.1 — `defn:line:308` Source lines 308–311. Two elements $g_1,g_2\in\SLn(\mathbb R)$ are called a ping-pong pair if both $g_1$ and $g_2$ are $(r,\epsilon)$-very proximal with respect to some $r>2\epsilon>0$, and if the attracting points of $g_i$ and $g_i^{-1}$ are at least distance $r$ apart from the repulsive hyperplanes of $g_j$ and $g_j^{-1}$ in $\mathbb P^{n-1}(\mathbb R)$, where $i\not=j$. *Extractive statement; proof gist not independently reconstructed.* ### Thm 3.2 — `gap` Source lines 318–323. Proposition~3.1 \cite{BG} Let $\epsilon<1/4$ and let $\gamma\in\mathrm{SL}_n(\mathbb R)$. Let $a_1(\gamma)$ and $a_2(\gamma)$ be the largest and second-largest singular values of $\gamma$, respectively (i.e. largest and second-largest eigenvalues of $\gamma^t\gamma$). If $\frac{a_2(\gamma)}{a_1(\gamma)} \leq \epsilon^2$, then $\gamma$ is $\epsilon$-contracting. More precisely, writing $\gamma = k_{\gamma}a_{\gamma}k_{\gamma}'$ , one can take $H_{\gamma}$ to be the projective hyperplane spanned by $\{k_{\gamma}'^{-1}(e_i)\}_{i=2}^n$, and $v_{\gamma}= k_{\gamma}(e_1)$, wh *Extractive statement; proof gist not independently reconstructed.* Candidate references: `external:BG` (external_reference). ### Thm 3.3 — `aounthm` Source lines 327–331. Given two long random products $w_1, w_2$ of generators of a Zariski dense subgroup $\Gamma$ of $SL_n(\mathbb{Z}),$ $w_1$ and $w_2$ generate a free subgroup. *Extractive statement; proof gist not independently reconstructed.* Proof opening: By the results of Guivarc'h and Raugi \cite{GR} (see also Goldsheid-Margulis \cite{goldmar}), the assumption of Zariski density implies that the Lyapunov exponents of $\Gamma$ with the given generating set are distinct, and so for the ratio of the top singular value to the second biggest grows exponentially fast as a function of $n.$ Since the words are also known to be equi-distributed in sectors (see, e.g. \cite{bougerol}), the result follows. Candidate references: `external:GR` (external_reference); `external:goldmar` (external_reference); `external:bougerol` (external_reference). ### Thm 3.4 — `toptwoth` Source lines 346–350. For $n\geq 3$ fixed, let $G=\mathrm{SL}_n(\mathbb R)$, and let $\mu$ be Haar measure on $\mathrm{SL}_n(\mathbb R)$. For $g\in G$, denote by $a_1(g),a_2(g),\dots,a_n(g)$ the nonzero entries of the diagonal matrix $a_g$ in the $KAK$ decomposition of $g$ in (\ref{KAK}) with $a_1(g)\geq a_2(g)\geq\cdots\geq a_n(g)>0$. Fix $\eta>4$. Then $$0<\frac{\lim_{X\rightarrow\infty} \, \mu(\{g\in G\, | \, ||g||\leq X, a_1(g)/a_2(g)\geq\eta^2\})}{\lim_{X\rightarrow\infty} \, \mu(\{g\in G\, | \, ||g||\leq X\})}<\frac{1}{\eta^{2(n^2-3n+2)}}.$$ *Extractive statement; proof gist not independently reconstructed.* Proof opening: Recall that for any $g\in \SLn(\mathbb R)$, we have that $g=k_ga_gk_g'$, where $a_g\in A=\{{\mbox{diag}}(e^{j_1},\dots,e^{j_n})\; |\; j_i\in\mathbb R, j_i\leq j_{i+1},\sum_i j_i=0\}$ and $k_g,k_g'\in SO_n(\mathbb R)$. Let $H\in A$ be the $n$-tuple $(j_1,\dots,j_n)$. Consider the Haar measure on $A$ which is \left(\prod_{\lambda\in\Sigma^+}\sinh(\lambda(H))\right)dH where $\Sigma^+=\{j_i-j_k\; |\; 1\leq k0$. Fix $\eta>4$. Then, if $n$ is even, we have $$\lim_{X\rightarrow\infty} \, \mu'_X\left(\left\{g\in G\, \Big| \, \frac{a_k(g)}{a_{k+1}(g)}\geq\eta^2 \mbox{ and } \frac{a_k(g^{-1})}{a_{k+1}(g^{-1})}\geq\eta^2\right\}\right)=1,$$ where $k=n/2$. If $n$ is odd, we have that $$\lim_{X\rightarrow\infty} *Extractive statement; proof gist not independently reconstructed.* Candidate references: `KAK` (internal_reference). ### Lemma 3.7 — `epscontractextpower` Source lines 536–542. Let $\epsilon>0$ and let $\gamma\in\mathrm{SL}_n(\mathbb R)$. Let $a_i(\gamma)$ denote the $i$-th largest singular value of $\gamma$. If $\frac{a_{k+1}(\gamma)}{a_{k}(\gamma)} \leq \epsilon^2$, then $\gamma$ is $\epsilon$-contracting when acting on $\mathbb P(\bigwedge^{k}(\mathbb R^n))$. More precisely, let $B:=\{e_{i_1}\wedge\cdots\wedge e_{i_k}\;|\; 1\leq i_12$, and let $\mu'_X$ be as before. For $g\in G$, let $g=k_ga_gk_g'$ be a Cartan decomposition of $g$ and let $k=n/2$ if $n$ is even, and $k=(n-1)/2$ if $n$ is odd. Denote by $H_g$ the projective hyperplane in $\mathbb P(\bigwedge^k(\mathbb R^n))$ spanned by $\{k_{g}'^{-1}(\mathbf{v})\;|\; \mathbf v\in B\backslash e_1\wedge e_2\wedge\cdots\wedge e_k\}$ where $B$ is as in Lemma~\ref{epscontractextpower}, and let $v_g=k_{g}(e_1\wedge e_2\wedge\cdots\wedge e_k)$. Then, as $\epsilon'$ tends to $0$, we have $$\lim *Extractive statement; proof gist not independently reconstructed.* Candidate references: `epscontractextpower` (internal_reference). ### Thm 4.1 — `emtheorem` Source lines 606–610. Theorem~1.4, Eskin-Mcmullen \cite{EM} Let $G$ be a connected semisimple Lie group with finite center and let $H 0$ there exists a neighborhood $U$ of identity in $SL_n(\mathbb R)$ such that (1-\epsilon')m\left(\bigcup_{g\in U} gKA_iK\right)< m(KA_iK)< (1+\epsilon')m\left(\bigcap_{g\in U} gKA_iK\right). for all $i$. *Extractive statement; proof gist not independently reconstructed.* ### Lemma 4.3 — `wellrounded1` Source lines 640–643. Let $C_T'$ be defined as above. Then the sequence $\{KC_{T}'K\}$ of regions (whose volume tends to infinity as $T\rightarrow\infty$) is well-rounded. *Extractive statement; proof gist not independently reconstructed.* Proof opening: We first show that for any $\epsilon>0$ there is a neighborhood $\mathcal N_{1,\epsilon}$ of identity in $SL_n(\mathbb R)$ such that $$\bigcup_{g\in \mathcal N_{1,\epsilon}} gKC_T'K\subset KC_{(1+\epsilon)T}'K.$$ Let $\mathcal O_{\sqrt[n-1]{1+\epsilon}}$ be the neighborhood of identity of radius $\sqrt[n-1]{1+\epsilon}$. Then $\mathcal O_{\sqrt[n-1]{1+\epsilon}}$ contains $U_1V$ for some neighborhood $U_1$ of identity in $K$ and some neighborhood $V$ of identity in $A$. By the strong wavefront lemma (Theorem~2.1 in \cite{GO}) there exists a neighborhood $\mathcal N_{1,\epsilon}$ of identity in $SL_n(\mathbb R)$ such that $$\mathcal N_{1,\epsilon}\gamma\subset k_1U_1aVk_2=k_1U_1Vak_2$$ for al Candidate references: `modified` (candidate_proof_dependency); `wellrounded` (candidate_proof_dependency); `external:GO` (external_reference). ### Defn 4.4 — `wellroundeddefn2` Source lines 667–672. The sequence $\{B_n\}$ of sets is well-rounded if for any $\epsilon>0$ there exists an open neighborhood $U$ of the identity in $SL_n(\mathbb R)$ such that $$\frac{m(U\partial B_n)}{m(B_n)}<\epsilon$$ for all $n$. *Extractive statement; proof gist not independently reconstructed.* ### Lemma 4.5 — `gapwellrounded` Source lines 673–676. Let $\{KC_{T,\beta}K\}$ and $\{KC_{T,\beta}'K\}$ be as above. Then there exists some $N\in \mathbb N$ such that the sequence $\{KC_{T,\beta}'K\}_{T>N}$ is well-rounded. *Extractive statement; proof gist not independently reconstructed.* ### Lemma 5.1 — `singleop` Source lines 712–716. Let $A \in M^{n\times n},$ with $\lambda_1(A) \gg \lambda_2(A),$ and let $w$ be a vector. Then, $$A^k w \sim \langle w, v_1(A)\rangle \langle v_1(A), u_1(A)\rangle^{k-1} \lambda_1^k u_1(A).$$ *Extractive statement; proof gist not independently reconstructed.* Proof opening: Simple computation. ### Lemma 5.2 — `twoops` Source lines 719–725. Let $A, B \in \GLn(\mathbb Z)$ be such that $\lambda_1(A), \lambda_1(B), \lambda_1(A^{-1}), \lambda_1(B^{-1}) \geq \lambda \gg 1.$ Assume, further, that all the inner products $\langle u_1(A), v_1(A)\rangle, \langle u_1(A), v_1(B)\rangle, \langle u_1(A),\\ v_1(B), \langle(u_1(B), v_1(B)\rangle$ are bigger than $\epsilon$ in absolute value. Let $w(A, B)$ be a reduced word of length $k$ in $A, B, A^{-1}, B^{-1}.$ Then \| w(A, B) u_1(A)\| \geq (\epsilon \lambda)^k. *Extractive statement; proof gist not independently reconstructed.* Proof opening: Simple computation using Lemma \ref{singleop}, and the observation that $u_1(A^{-1}) = v_1(A),$ etc. Candidate references: `singleop` (candidate_proof_dependency). ### Fact 5.3 — `freelen` Source lines 729–733. Let $w(x, y)$ be a random product of $x, y, x^{-1}, y^{-1},$ of length $n.$ Then, with probability tending to one as $n$ goes to infinity, the reduced length of $w$ is at least $3\ell(w)/4.$ *Extractive statement; proof gist not independently reconstructed.* ### Thm 5.4 — `thm:line:734` Source lines 734–737. Let $A, B$ be picked uniformly at random from the symmetrized ball of radius $R.$ Then, the Lyapunov exponent of the group generated by $A, B$ goes to infinity with $R,$ with high probability. *Extractive statement; proof gist not independently reconstructed.* Proof opening: By Lemma \ref{middletwo}, we know that the the gap between the top and second singular values of $A, B$ (when acting on the appropriate exterior power) goes to infinity with $R.$ On the other hand, by equidistribution in sectors, the vectors $u_1(A), v_1(A), u_1(B), v_1(b)$ are uniformly distributed in the orthogonal group, so the absolute values of their inner products are bounded below by $\epsilon$ with probability $p(\epsilon)$ asymptotically independent of $R.$ The result now follows from Fact \ref{freelen} and Lemma \ref{twoops}. - `middletwo` (proof_dependency; curated_source_check): R,$ with high probability. \end{thm} \begin{proof} By Lemma \ref{middletwo}, we know that the the gap between the top and second singular values of $A, B$ - `freelen` (proof_dependency; curated_source_check): ically independent of $R.$ The result now follows from Fact \ref{freelen} and Lemma \ref{twoops}. \end{proof} \section{Directions for further study} \lab - `twoops` (proof_dependency; curated_source_check): .$ The result now follows from Fact \ref{freelen} and Lemma \ref{twoops}. \end{proof} \section{Directions for further study} \label{future} Here, we enu Candidate references: `middletwo` (candidate_proof_dependency); `freelen` (candidate_proof_dependency); `twoops` (candidate_proof_dependency). ## External inputs and references ### [1] Aoun — `external:Aoun` Source lines 789–794. Richard Aoun et~al. \newblock Random subgroups of linear groups are free. \newblock {\em Duke Mathematical Journal}, 160(1):117--173, 2011. Candidate references: `section:1` (external_reference). ### [2] B — `external:B` Source lines 794–800. AF~Beardon. \newblock The {H}ausdorff dimension of singular sets of properly discontinuous groups. \newblock {\em American Journal of Mathematics}, pages 722--736, 1966. Candidate references: `haussl2` (external_reference). ### [3] bougerol — `external:bougerol` Source lines 800–806. Philippe Bougerol and Jean Lacroix. \newblock {\em Products of random matrices with applications to {S}chr{\"o}dinger operators}. \newblock Birkh{\"a}user Boston, 1985. Candidate references: `aounthm` (external_reference). ### [4] bgs — `external:bgs` Source lines 806–811. Jean Bourgain, Alex Gamburd, and Peter Sarnak. \newblock Affine linear sieve, expanders, and sum-product. \newblock {\em Inventiones mathematicae}, 179(3):559--644, 2010. Candidate references: `section:1` (external_reference). ### [5] BT — `external:BT` Source lines 811–816. Christopher Brav and Hugh Thomas. \newblock Thin monodromy in {S}p(4). \newblock {\em Compos. Math.}, 150(3):333--343, 2014. Candidate references: `section:1` (external_reference). ### [6] BG — `external:BG` Source lines 816–821. Emmanuel Breuillard and Tsachik Gelander. \newblock On dense free subgroups of {L}ie groups. \newblock {\em Journal of Algebra}, 261(2):448--467, 2003. Candidate references: `section:1` (external_reference); `sln` (external_reference); `gap` (external_reference); `epscontractextpower` (external_reference). ### [7] D — `external:D` Source lines 821–826. Jean Delsarte. \newblock Sur le gitter fuchsien. \newblock {\em CR Acad. Sci. Paris}, 214(147-179):1, 1942. Candidate references: `sl2sec` (external_reference). ### [8] DRS — `external:DRS` Source lines 826–831. W.~Duke, Z.~Rudnick, and P.~Sarnak. \newblock Density of integer points on affine homogeneous varieties. \newblock {\em Duke Math. J.}, 71(1):143--179, 1993. Candidate references: `sl2sec` (external_reference); `disjointcircles` (external_reference); `toptwoth` (external_reference). ### [9] EM — `external:EM` Source lines 831–836. Alex Eskin and Curt McMullen. \newblock Mixing, counting, and equidistribution in {L}ie groups. \newblock {\em Duke Math. J.}, 71(1):181--209, 1993. Candidate references: `section:1` (external_reference); `disjointcircles` (external_reference); `sln` (external_reference); `wellroundedsec` (external_reference); `emtheorem` (external_reference). ### [10] Fuchsbull — `external:Fuchsbull` Source lines 836–841. Elena Fuchs. \newblock Counting problems in {A}pollonian packings. \newblock {\em Bull. Amer. Math. Soc. (N.S.)}, 50(2):229--266, 2013. Candidate references: `section:1` (external_reference). ### [11] FMS — `external:FMS` Source lines 841–847. Elena Fuchs, Chen Meiri, and Peter Sarnak. \newblock Hyperbolic monodromy groups for the hypergeometric equation and {C}artan involutions. \newblock {\em J. Eur. Math. Soc. (JEMS)}, 16(8):1617--1671, 2014. Candidate references: `section:1` (external_reference). ### [12] furstenberg1960products — `external:furstenberg1960products` Source lines 847–852. Harry Furstenberg and Harry Kesten. \newblock Products of random matrices. \newblock {\em The Annals of Mathematical Statistics}, pages 457--469, 1960. Candidate references: `lyapsec` (external_reference). ### [13] goldmar — `external:goldmar` Source lines 852–857. Ilya Goldsheid and Gregori~A. Margulis. \newblock Lyapunov exponents of a product of random matrices. \newblock {\em Uspekhi Mat. Nauk}, 269(5):11--71, 1989. Candidate references: `aounthm` (external_reference). ### [14] GO — `external:GO` Source lines 857–863. Alexander Gorodnik and Hee Oh. \newblock Orbits of discrete subgroups on a symmetric space and the {F}urstenberg boundary. \newblock {\em Duke Math. J.}, 139(3):483--525, 2007. Candidate references: `section:1` (external_reference); `sln` (external_reference); `wellroundedsec` (external_reference); `wellrounded1` (external_reference). ### [15] GR — `external:GR` Source lines 863–870. Yves Guivarc'h and Albert Raugi. \newblock Propri\'et\'es de contraction d'un semi-groupe de matrices inversibles. {C}oefficients de {L}iapunoff d'un produit de matrices al\'eatoires ind\'ependantes. \newblock {\em Israel J. Math.}, 65(2):165--196, 1989. Candidate references: `aounthm` (external_reference). ### [16] KL — `external:KL` Source lines 870–875. William~M. Kantor and Alexander Lubotzky. \newblock The probability of generating a finite classical group. \newblock {\em Geom. Dedicata}, 36(1):67--87, 1990. Candidate references: `haussl2` (external_reference). ### [17] knapp — `external:knapp` Source lines 875–881. Anthony Knapp. \newblock {\em Representation theory of semisimple groups: An overview based on examples}, volume~36. \newblock Princeton University Press, Princeton, NJ, 2001. Candidate references: `toptwoth` (external_reference). ### [18] Kontbull — `external:Kontbull` Source lines 881–887. Alex Kontorovich. \newblock From {A}pollonius to {Z}aremba: local-global phenomena in thin orbits. \newblock {\em Bull. Amer. Math. Soc. (N.S.)}, 50(2):187--228, 2013. Candidate references: `section:1` (external_reference). ### [19] R1 — `external:R1` Source lines 887–893. Igor Rivin. \newblock Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms. \newblock {\em Duke Math. J.}, 142(2):353--379, 2008. Candidate references: `section:1` (external_reference). ### [20] R3 — `external:R3` Source lines 893–898. Igor Rivin. \newblock Walks on graphs and lattices---effective bounds and applications. \newblock {\em Forum Math.}, 21(4):673--685, 2009. Candidate references: `section:1` (external_reference). ### [21] R2 — `external:R2` Source lines 898–903. Igor Rivin. \newblock Zariski density and genericity. \newblock {\em Int. Math. Res. Not.}, 19:3649--3657, 2010. Candidate references: `section:1` (external_reference); `haussl2` (external_reference); `sln` (external_reference). ### [22] salsarnak — `external:salsarnak` Source lines 903–908. Alireza Salehi~Golsefidy and Peter Sarnak. \newblock The {A}ffine sieve. \newblock {\em J. Amer. Math. Soc.}, 26(4):1085--1105, 2013. Candidate references: `section:1` (external_reference). ### [23] SalehiVarju — `external:SalehiVarju` Source lines 908–913. Alireza Salehi~Golsefidy and Peter Varju. \newblock Expansion in perfect groups. \newblock {\em Geometric and Functional Analysis}, 22(6):1832--1891, 2012. Candidate references: `section:1` (external_reference). ### [24] sardav — `external:sardav` Source lines 913–919. Peter Sarnak. \newblock Letter to {J}. davis about reciprocal geodesics. \newblock \url{http://web.math.princeton.edu/sarnak/SarnakDavisLtr05.pdf}, 2005. Candidate references: `disjointcircles` (external_reference). ### [25] SV — `external:SV` Source lines 919–924. Sandip Singh and T.~N. Venkataramana. \newblock Arithmeticity of certain symplectic hypergeometric groups. \newblock {\em Duke Math. J.}, 163(3):591--617, 2014. Candidate references: `section:1` (external_reference). ## Coverage and limits {"mode": "native_latex_index", "claims": 23, "sections": 6, "external_references": 25, "source_bibliography_entries": 25, "unresolved_bibliography": 0, "unresolved": 0} - Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer. - Numbers reflect simple source counters; labels and line anchors are authoritative. - Extractive statements and proof openings are marked; they are not reconstructed proof gists. - Macros, included files and custom counters require an expanded source or annotated fallback. - Primary panel source is arXiv 1506.01735v1, 4 June 2015, 28 PDF pages; IMRN publication is 30 pages and its revised Section 2 is not assumed identical. Journal PDF download unsuccessful; indexed publisher excerpts are separate edition evidence. - Fact environments are included as claim units without changing their printed labels. TeX decoded from recorded latin-1 encoding into UTF-8; original bytes retained. No mathematical source corrections were made. - Figure pixels and proofs of external references are not supplied in native TeX packets. - The dated adjudication adds a partial set of coordinator-audited proof edges. Missing dependencies and unchecked nodes remain unresolved. A sink is not necessarily a headline theorem.