\begin{bibdiv}
\begin{biblist}

\bib{Aigner}{book}{
  author={Aigner, Martin},
  title={Markov's theorem and 100 years of the uniqueness conjecture},
  subtitle={A mathematical journey from irrational numbers to perfect matchings},
  publisher={Springer},
  address={Cham},
  date={2013},
}

\bib{Baragar}{article}{
  author={Baragar, Arthur},
  title={On the unicity conjecture for Markoff numbers},
  journal={Canad. Math. Bull.},
  volume={39},
  date={1996},
  number={1},
  pages={3--9},
}

\bib{Borosh}{article}{
  author={Borosh, Itshak},
  title={More numerical evidence on the uniqueness of Markov numbers},
  journal={BIT},
  volume={15},
  date={1975},
  pages={351--357},
}

\bib{BGS}{article}{
  author={Bourgain, Jean},
  author={Gamburd, Alexander},
  author={Sarnak, Peter},
  title={Markoff triples and strong approximation},
  journal={C. R. Math. Acad. Sci. Paris},
  volume={354},
  date={2016},
  number={2},
  pages={131--135},
}

\bib{BGS2}{article}{
  author={Bourgain, Jean},
  author={Gamburd, Alexander},
  author={Sarnak, Peter},
  title={Strong approximation and Diophantine properties of Markoff triples},
  journal={J. Amer. Math. Soc.},
  volume={39},
  date={2026},
  number={1},
  pages={177--204},
}

\bib{BNRW}{article}{
  author={Burrin, Claire},
  author={Nevo, Amos},
  author={R\"uhr, Rene},
  author={Weiss, Barak},
  title={Effective counting for discrete lattice orbits in the plane via Eisenstein series},
  journal={Enseign. Math.},
  volume={66},
  date={2020},
  number={3-4},
  pages={259--304},
  note={arXiv:1905.01493. The theorem and section numbers cited in the text refer to the authors' version dated 23 May 2020},
}

\bib{Button}{article}{
  author={Button, Jack O.},
  title={The uniqueness of the prime Markoff numbers},
  journal={J. London Math. Soc. (2)},
  volume={58},
  date={1998},
  number={1},
  pages={9--17},
}

\bib{Button2001}{article}{
  author={Button, Jack O.},
  title={Markoff numbers, principal ideals and continued fraction expansions},
  journal={J. Number Theory},
  volume={87},
  date={2001},
  number={1},
  pages={77--95},
}

\bib{Cassels}{book}{
  author={Cassels, J. W. S.},
  title={An introduction to Diophantine approximation},
  series={Cambridge Tracts in Mathematics and Mathematical Physics},
  volume={45},
  publisher={Cambridge University Press},
  date={1957},
}

\bib{CasselsLF}{book}{
  author={Cassels, J. W. S.},
  title={Local fields},
  series={London Mathematical Society Student Texts},
  volume={3},
  publisher={Cambridge University Press},
  date={1986},
}

\bib{Chen}{article}{
  author={Chen, William Y.},
  title={Nonabelian level structures, Nielsen equivalence, and Markoff triples},
  journal={Ann. of Math. (2)},
  volume={199},
  date={2024},
  number={1},
  pages={301--443},
}

\bib{Cohn1955}{article}{
  author={Cohn, Harvey},
  title={Approach to Markoff's minimal forms through modular functions},
  journal={Ann. of Math. (2)},
  volume={61},
  date={1955},
  pages={1--12},
}

\bib{Cohn1971}{article}{
  author={Cohn, Harvey},
  title={Representation of Markoff's binary quadratic forms by geodesics on a perforated torus},
  journal={Acta Arith.},
  volume={18},
  date={1971},
  pages={125--136},
}

\bib{DLN}{article}{
  author={Doan, Nhat Minh},
  author={Li, Xiaobin},
  author={Nguyen, Van},
  title={Stable-norm boundary geometry on once-punctured tori},
  date={2026},
  note={arXiv:2605.14574v3},
}

\bib{Frobenius}{article}{
  author={Frobenius, Georg},
  title={\"Uber die Markoffschen Zahlen},
  journal={Sitzungsber. K\"onigl. Preuss. Akad. Wiss. Berlin},
  date={1913},
  pages={458--487},
}

\bib{Goldman}{article}{
  author={Goldman, William M.},
  title={Trace coordinates on Fricke spaces of some simple hyperbolic surfaces},
  conference={
    title={Handbook of Teichm\"uller theory. Vol. II},
  },
  book={
    series={IRMA Lect. Math. Theor. Phys.},
    volume={13},
    publisher={Eur. Math. Soc.},
    address={Z\"urich},
  },
  date={2009},
  pages={611--684},
}

\bib{HardyWright}{book}{
  author={Hardy, G. H.},
  author={Wright, E. M.},
  title={An introduction to the theory of numbers},
  edition={6},
  publisher={Oxford University Press},
  address={Oxford},
  date={2008},
}

\bib{LangTan}{article}{
  author={Lang, Mong Lung},
  author={Tan, Ser Peow},
  title={A simple proof of the Markoff conjecture for prime powers},
  journal={Geom. Dedicata},
  volume={129},
  date={2007},
  pages={15--22},
}

\bib{MKS}{book}{
  author={Magnus, Wilhelm},
  author={Karrass, Abraham},
  author={Solitar, Donald},
  title={Combinatorial group theory},
  edition={2},
  publisher={Dover},
  address={New York},
  date={1976},
}

\bib{Markoff1879}{article}{
  author={Markoff, A.},
  title={Sur les formes quadratiques binaires ind\'efinies},
  journal={Math. Ann.},
  volume={15},
  date={1879},
  pages={381--406},
}

\bib{Markoff1880}{article}{
  author={Markoff, A.},
  title={Sur les formes quadratiques binaires ind\'efinies. II},
  journal={Math. Ann.},
  volume={17},
  date={1880},
  pages={379--399},
}

\bib{MR}{article}{
  author={McShane, Greg},
  author={Rivin, Igor},
  title={A norm on homology of surfaces and counting simple geodesics},
  journal={Internat. Math. Res. Notices},
  date={1995},
  number={2},
  pages={61--69},
}

\bib{MRcr}{article}{
  author={McShane, Greg},
  author={Rivin, Igor},
  title={Simple curves on hyperbolic tori},
  journal={C. R. Acad. Sci. Paris S\'er. I Math.},
  volume={320},
  date={1995},
  number={12},
  pages={1523--1528},
}

\bib{Poonen}{article}{
  author={Poonen, Bjorn},
  title={$p$-adic interpolation of iterates},
  journal={Bull. Lond. Math. Soc.},
  volume={46},
  date={2014},
  number={3},
  pages={525--527},
}

\bib{Schmutz}{article}{
  author={Schmutz, Paul},
  title={Systoles of arithmetic surfaces and the Markoff spectrum},
  journal={Math. Ann.},
  volume={305},
  date={1996},
  pages={191--203},
}

\bib{Series}{article}{
  author={Series, Caroline},
  title={The geometry of Markoff numbers},
  journal={Math. Intelligencer},
  volume={7},
  date={1985},
  number={3},
  pages={20--29},
}

\bib{SorrentinoVeselov}{article}{
  author={Sorrentino, Alfonso},
  author={Veselov, Alexander P.},
  title={Markov numbers, Mather's $\beta$ function and stable norm},
  journal={Nonlinearity},
  volume={32},
  date={2019},
  number={6},
  pages={2147--2156},
}

\bib{Srinivasan}{article}{
  author={Srinivasan, Anitha},
  title={Markoff numbers and ambiguous classes},
  journal={J. Th\'eor. Nombres Bordeaux},
  volume={21},
  date={2009},
  number={3},
  pages={757--770},
}

\bib{Strassmann}{article}{
  author={Strassmann, Reinhold},
  title={\"Uber den Wertevorrat von Potenzreihen im Gebiet der $p$-adischen Zahlen},
  journal={J. Reine Angew. Math.},
  volume={159},
  date={1928},
  pages={13--28},
}

\bib{Zagier}{article}{
  author={Zagier, Don},
  title={On the number of Markoff numbers below a given bound},
  journal={Math. Comp.},
  volume={39},
  date={1982},
  number={160},
  pages={709--723},
}

\bib{Zhang}{article}{
  author={Zhang, Ying},
  title={Congruence and uniqueness of certain Markov numbers},
  journal={Acta Arith.},
  volume={128},
  date={2007},
  number={3},
  pages={295--301},
}

\end{biblist}
\end{bibdiv}