@incollection{BGW2016,
  author    = {Paul Baum and Erik Guentner and Rufus Willett},
  title     = {{Exactness and the Kadison--Kaplansky Conjecture}},
  booktitle = {{Operator Algebras and Their Applications: A Tribute to Richard V. Kadison}},
  editor    = {Robert S. Doran and Efton Park},
  series    = {Contemporary Mathematics},
  volume    = {671},
  publisher = {American Mathematical Society},
  address   = {Providence, RI},
  year      = {2016},
  pages     = {1--33},
  doi       = {10.1090/conm/671/13501},
  note      = {\href{https://doi.org/10.1090/conm/671/13501}{\nolinkurl{doi:10.1090/conm/671/13501}}},
  url       = {https://math.hawaii.edu/~erik/papers/kadison.pdf}
}

@article{HLS2002,
  author    = {Nigel Higson and Vincent Lafforgue and Georges Skandalis},
  title     = {{Counterexamples to the Baum--Connes Conjecture}},
  journal   = {Geometric and Functional Analysis},
  volume    = {12},
  number    = {2},
  year      = {2002},
  pages     = {330--354},
  doi       = {10.1007/s00039-002-8249-5},
  note      = {\href{https://doi.org/10.1007/s00039-002-8249-5}{\nolinkurl{doi:10.1007/s00039-002-8249-5}}},
  url       = {https://vlafforg.perso.math.cnrs.fr/files/counterex.pdf}
}

@article{Austin2013,
  author    = {Tim Austin},
  title     = {{Rational Group Ring Elements with Kernels Having Irrational Dimension}},
  journal   = {Proceedings of the London Mathematical Society},
  series    = {3},
  volume    = {107},
  number    = {6},
  year      = {2013},
  pages     = {1424--1448},
  doi       = {10.1112/plms/pdt029},
  note      = {\href{https://doi.org/10.1112/plms/pdt029}{\nolinkurl{doi:10.1112/plms/pdt029}}},
  eprint    = {0909.2360},
  archivePrefix = {arXiv},
  primaryClass = {math.GR},
  url       = {https://arxiv.org/abs/0909.2360}
}

@incollection{GJV2019,
  author    = {G{\'o}mez Aparicio, Maria Paula and Julg, Pierre and Valette, Alain},
  title     = {{The Baum--Connes Conjecture: An Extended Survey}},
  booktitle = {{Advances in Noncommutative Geometry: On the Occasion of Alain Connes' 70th Birthday}},
  editor    = {Ali Chamseddine and Caterina Consani and Nigel Higson and Masoud Khalkhali and Henri Moscovici and Guoliang Yu},
  publisher = {Springer},
  address   = {Cham},
  year      = {2019},
  pages     = {127--244},
  doi       = {10.1007/978-3-030-29597-4_3},
  note      = {\href{https://doi.org/10.1007/978-3-030-29597-4_3}{\nolinkurl{doi:10.1007/978-3-030-29597-4_3}}},
  eprint    = {1905.10081},
  archivePrefix = {arXiv},
  primaryClass = {math.KT},
  url       = {https://arxiv.org/abs/1905.10081}
}

@book{ReedSimon1980,
  author    = {Michael Reed and Barry Simon},
  title     = {{Methods of Modern Mathematical Physics. I: Functional Analysis}},
  edition   = {Revised and enlarged},
  publisher = {Academic Press},
  year      = {1980},
  isbn      = {978-0-12-585050-6},
  url       = {https://shop.elsevier.com/books/i-functional-analysis/reed/978-0-08-057048-8}
}

@book{Folland1999,
  author    = {Gerald B. Folland},
  title     = {{Real Analysis: Modern Techniques and Their Applications}},
  edition   = {Second},
  publisher = {John Wiley \& Sons},
  year      = {1999},
  isbn      = {978-0-471-31716-6},
  url       = {https://www.wiley-vch.de/en/areas-interest/mathematics-statistics/real-analysis-978-0-471-31716-6}
}

@book{LyndonSchupp1977,
  author    = {Roger C. Lyndon and Paul E. Schupp},
  title     = {{Combinatorial Group Theory}},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {89},
  publisher = {Springer-Verlag},
  address   = {Berlin--Heidelberg--New York},
  year      = {1977},
  isbn      = {3-540-07642-5},
  url       = {https://link.springer.com/content/pdf/bfm:978-3-642-61896-3/1}
}

@book{AlonSpencer2016,
  author    = {Noga Alon and Joel H. Spencer},
  title     = {{The Probabilistic Method}},
  edition   = {Fourth},
  series    = {Wiley Series in Discrete Mathematics and Optimization},
  publisher = {John Wiley \& Sons},
  year      = {2016},
  isbn      = {978-1-119-06195-3},
  url       = {https://www.wiley-vch.de/en/areas-interest/mathematics-statistics/the-probabilistic-method-978-1-119-06195-3}
}

@incollection{ErdosLovasz1975,
  author    = {Paul Erd{\H o}s and L{\'a}szl{\'o} Lov{\'a}sz},
  title     = {{Problems and Results on 3-Chromatic Hypergraphs and Some Related Questions}},
  booktitle = {{Infinite and Finite Sets, Vol. II}},
  series    = {Colloquia Mathematica Societatis J{\'a}nos Bolyai},
  volume    = {10},
  publisher = {North-Holland},
  address   = {Amsterdam},
  year      = {1975},
  pages     = {609--627},
  url       = {https://www.renyi.hu/~p_erdos/1975-34.pdf}
}

@article{DR1998,
  author    = {Kenneth J. Dykema and Mikael R{\o}rdam},
  title     = {{Projections in Free Product {$C^*$}-Algebras}},
  journal   = {Geometric and Functional Analysis},
  volume    = {8},
  number    = {1},
  year      = {1998},
  pages     = {1--16},
  doi       = {10.1007/s000390050046},
  note      = {\href{https://doi.org/10.1007/s000390050046}{\nolinkurl{doi:10.1007/s000390050046}}},
  url       = {https://web.math.ku.dk/~rordam/manus/gafa.pdf}
}

@article{DR2000,
  author    = {Kenneth J. Dykema and Mikael R{\o}rdam},
  title     = {{Erratum to Projections in Free Product {$C^*$}-Algebras}},
  journal   = {Geometric and Functional Analysis},
  volume    = {10},
  year      = {2000},
  pages     = {975},
  doi       = {10.1007/PL00001644},
  note      = {\href{https://doi.org/10.1007/PL00001644}{\nolinkurl{doi:10.1007/PL00001644}}},
  url       = {https://link.springer.com/article/10.1007/PL00001644}
}


@article{Kadison2008,
 author={Richard V. Kadison},
 title={Irving Kaplansky's Role in Mid-Twentieth Century Functional Analysis},
 journal={Notices of the American Mathematical Society},
 volume={55},number={2},year={2008},pages={216--225}
}
@article{Luck2002,
 author={Wolfgang L{\"u}ck},
 title={The Relation between the {Baum--Connes} Conjecture and the Trace Conjecture},
 journal={Inventiones Mathematicae},volume={149},year={2002},pages={123--152},
 doi={10.1007/s002220200215},
 note={\href{https://doi.org/10.1007/s002220200215}{\nolinkurl{doi:10.1007/s002220200215}}}
}
@article{HK2001,
 author={Nigel Higson and Gennadi Kasparov},
 title={{E}-Theory and {KK}-Theory for Groups Which Act Properly and Isometrically on {Hilbert} Space},
 journal={Inventiones Mathematicae},volume={144},year={2001},pages={23--74},
 doi={10.1007/s002220000118},
 note={\href{https://doi.org/10.1007/s002220000118}{\nolinkurl{doi:10.1007/s002220000118}}}
}
@article{MY2002,
 author={Igor Mineyev and Guoliang Yu},
 title={The {Baum--Connes} Conjecture for Hyperbolic Groups},
 journal={Inventiones Mathematicae},volume={149},year={2002},pages={97--122},
 doi={10.1007/s002220200214},
 note={\href{https://doi.org/10.1007/s002220200214}{\nolinkurl{doi:10.1007/s002220200214}}}
}
@article{Lafforgue2012,
 author={Vincent Lafforgue},
 title={La conjecture de {Baum--Connes} {\`a} coefficients pour les groupes hyperboliques},
 journal={Journal of Noncommutative Geometry},volume={6},number={1},year={2012},pages={1--197},
 doi={10.4171/JNCG/89},
 note={\href{https://doi.org/10.4171/JNCG/89}{\nolinkurl{doi:10.4171/JNCG/89}}}
}
@article{DRII2000,
 author={Kenneth J. Dykema and Mikael R{\o}rdam},
 title={Projections in Free Product {$C^*$}-Algebras, {II}},
 journal={Mathematische Zeitschrift},volume={234},year={2000},pages={103--113},
 doi={10.1007/s002090050505},
 note={\href{https://doi.org/10.1007/s002090050505}{\nolinkurl{doi:10.1007/s002090050505}}}
}
@misc{FKOP2026,
  author={Felipe Flores and Mario Klisse and {\'O} Cobhthaigh, M{\'i}che{\'a}l and Matteo Pagliero},
 title={Selfless Reduced Free Products and Graph Products of {$C^*$}-Algebras},
 year={2026},
 note={arXiv:2510.24675v3, \href{https://arxiv.org/abs/2510.24675v3}{August 5, 2026}}
}
@misc{Robert2025,
 author={Leonel Robert},
 title={Selfless {$C^*$}-Algebras},
 year={2025},
 note={arXiv:2309.14188v3, \href{https://arxiv.org/abs/2309.14188v3}{June 6, 2025}}
}
@misc{Sauers2026,
 author={{Sauers}},
 title={A Matrix Projection of Non-Integer Trace over {$G$} Gives a Projection of Fractional Trace in {$C^*_r(G*\mathbb Z)$}, So {Kadison--Kaplansky} Is Equivalent to the Torsion-Free Trace Conjecture},
 year={2026},
 note={Public research note, September 13, 2026.
 \href{https://github.com/SauersML/group-approximation/blob/5080b7d3bb4ddd1de98d28c7f8904f48a87be463/research/fractional-matrix-trace-gives-projection-in-g-star-z.md}{Archived public text}}
}

@article{Gromov2003,
 author={Mikhail Gromov},title={Random Walk in Random Groups},
 journal={Geometric and Functional Analysis},volume={13},year={2003},pages={73--146},
 doi={10.1007/s000390300002},note={\href{https://doi.org/10.1007/s000390300002}{\nolinkurl{doi:10.1007/s000390300002}}}
}
@article{Osajda2020,
 author={Damian Osajda},title={Small Cancellation Labellings of Some Infinite Graphs and Applications},
 journal={Acta Mathematica},volume={225},year={2020},pages={159--191},
 doi={10.4310/ACTA.2020.v225.n1.a3},note={\href{https://doi.org/10.4310/ACTA.2020.v225.n1.a3}{\nolinkurl{doi:10.4310/ACTA.2020.v225.n1.a3}}}
}
@article{Ollivier2006,
 author={Yann Ollivier},title={On a Small Cancellation Theorem of {Gromov}},
 journal={Bulletin of the Belgian Mathematical Society -- Simon Stevin},volume={13},year={2006},pages={75--89},
 note={Revised author version, \href{https://arxiv.org/abs/math/0310022v2}{arXiv:math/0310022v2}, September 25, 2024}
}
@article{Gruber2015,
 author={Dominik Gruber},title={Groups with Graphical {$C(6)$} and {$C(7)$} Small Cancellation Presentations},
 journal={Transactions of the American Mathematical Society},volume={367},year={2015},pages={2051--2078},
 note={\href{https://arxiv.org/abs/1210.0178}{arXiv:1210.0178}}
}
@article{Gross1974,
 author={Jonathan L. Gross},title={Voltage Graphs},journal={Discrete Mathematics},volume={9},number={3},year={1974},pages={239--246},
 doi={10.1016/0012-365X(74)90006-5},note={\href{https://doi.org/10.1016/0012-365X(74)90006-5}{\nolinkurl{doi:10.1016/0012-365X(74)90006-5}}}
}
@article{GT1977,
 author={Jonathan L. Gross and Thomas W. Tucker},title={Generating All Graph Coverings by Permutation Voltage Assignments},
 journal={Discrete Mathematics},volume={18},number={3},year={1977},pages={273--283},
 doi={10.1016/0012-365X(77)90131-5},note={\href{https://doi.org/10.1016/0012-365X(77)90131-5}{\nolinkurl{doi:10.1016/0012-365X(77)90131-5}}}
}
@article{MoserTardos2010,
 author={Robin A. Moser and G{\'a}bor Tardos},title={A Constructive Proof of the General {Lov{\'a}sz Local Lemma}},
 journal={Journal of the ACM},volume={57},number={2},year={2010},pages={11:1--11:15},
 doi={10.1145/1667053.1667060},note={\href{https://doi.org/10.1145/1667053.1667060}{\nolinkurl{doi:10.1145/1667053.1667060}}}
}
@article{Dodziuk1984,
 author={J{\'o}zef Dodziuk},title={Difference Equations, Isoperimetric Inequality and Transience of Certain Random Walks},
 journal={Transactions of the American Mathematical Society},volume={284},number={2},year={1984},pages={787--794},
 doi={10.1090/S0002-9947-1984-0743744-X},note={\href{https://doi.org/10.1090/S0002-9947-1984-0743744-X}{\nolinkurl{doi:10.1090/S0002-9947-1984-0743744-X}}}
}
@article{AM1985,
 author={Noga Alon and Vitali D. Milman},title={{$\lambda_1$}, Isoperimetric Inequalities for Graphs, and Superconcentrators},
 journal={Journal of Combinatorial Theory, Series B},volume={38},year={1985},pages={73--88},
 doi={10.1016/0095-8956(85)90092-9},note={\href{https://doi.org/10.1016/0095-8956(85)90092-9}{\nolinkurl{doi:10.1016/0095-8956(85)90092-9}}}
}
@article{WY2012,
 author={Rufus Willett and Guoliang Yu},title={Higher Index Theory for Certain Expanders and {Gromov} Monster Groups, {I}},
 journal={Advances in Mathematics},volume={229},number={3},year={2012},pages={1380--1416},
 doi={10.1016/j.aim.2011.10.024},note={\href{https://doi.org/10.1016/j.aim.2011.10.024}{\nolinkurl{doi:10.1016/j.aim.2011.10.024}}}
}

@article{Magnus1935,
 author={Wilhelm Magnus},
 title={Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring},
 journal={Mathematische Annalen},volume={111},year={1935},pages={259--280},
 doi={10.1007/BF01472217},note={\href{https://doi.org/10.1007/BF01472217}{\nolinkurl{doi:10.1007/BF01472217}}}
}

@article{PimsnerVoiculescu1982,
 author={Mihai Pimsner and Dan Voiculescu},
 title={{$K$}-Groups of Reduced Crossed Products by Free Groups},
 journal={Journal of Operator Theory},
 volume={8}, number={1}, year={1982}, pages={131--156},
 note={\href{https://www.theta.ro/jot/archive/1982-008-001/1982-008-001-006.pdf}{Publisher full text}}
}

@article{Rieffel1988,
 author={Marc A. Rieffel},
 title={Projective Modules over Higher-Dimensional Non-Commutative Tori},
 journal={Canadian Journal of Mathematics},
 volume={40}, number={2}, year={1988}, pages={257--338},
 doi={10.4153/CJM-1988-012-9},
 note={\href{https://doi.org/10.4153/CJM-1988-012-9}{\nolinkurl{doi:10.4153/CJM-1988-012-9}}}
}

@misc{EchterhoffWilliams2012,
 author={Siegfried Echterhoff and Dana P. Williams},
 title={Structure of Crossed Products by Strictly Proper Actions on Continuous-Trace Algebras},
 year={2012},
 note={arXiv:1208.4488v1, \href{https://arxiv.org/abs/1208.4488v1}{August 22, 2012}}
}

@article{Schwartz1980,
 author={Jacob T. Schwartz},
 title={Fast Probabilistic Algorithms for Verification of Polynomial Identities},
 journal={Journal of the ACM},
 volume={27}, number={4}, year={1980}, pages={701--717},
 doi={10.1145/322217.322225},
 note={\href{https://doi.org/10.1145/322217.322225}{\nolinkurl{doi:10.1145/322217.322225}}}
}

@article{Gersten1996,
 author={S. M. Gersten},
 title={Subgroups of Word Hyperbolic Groups in Dimension 2},
 journal={Journal of the London Mathematical Society},
 series={2}, volume={54}, number={2}, year={1996}, pages={261--283},
 doi={10.1112/jlms/54.2.261},
 note={\href{https://doi.org/10.1112/jlms/54.2.261}{\nolinkurl{doi:10.1112/jlms/54.2.261}}}
}

@incollection{Brown2010,
 author={Kenneth S. Brown},
 title={Lectures on the Cohomology of Groups},
 booktitle={Cohomology of Groups and Algebraic {$K$}-Theory},
 series={Advanced Lectures in Mathematics}, volume={12},
 publisher={International Press}, address={Somerville, MA},
 year={2010}, pages={131--166},
 note={\href{https://pi.math.cornell.edu/~kbrown/papers/cohomology_hangzhou.pdf}{Author version}}
}

@article{Hoeffding1963,
 author={Wassily Hoeffding},
 title={Probability Inequalities for Sums of Bounded Random Variables},
 journal={Journal of the American Statistical Association},
 volume={58}, number={301}, year={1963}, pages={13--30},
 doi={10.1080/01621459.1963.10500830},
 note={\href{https://doi.org/10.1080/01621459.1963.10500830}{\nolinkurl{doi:10.1080/01621459.1963.10500830}}}
}

@misc{OpenAIBass2026,
 author={{OpenAI}},
 title={{The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture}},
 howpublished={OpenAI Math Release preprint
               \href{https://github.com/openai/math/blob/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf}{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026}},
 year={2026}
}

@inproceedings{Zippel1979,
  author = {Zippel, Richard},
  title = {Probabilistic algorithms for sparse polynomials},
  editor = {Ng, Edward W.},
  booktitle = {Symbolic and Algebraic Computation},
  series = {Lecture Notes in Computer Science},
  volume = {72},
  publisher = {Springer},
  year = {1979},
  pages = {216--226},
  doi = {10.1007/3-540-09519-5_73}
}

@article{DeMilloLipton1978,
  author = {DeMillo, Richard A. and Lipton, Richard J.},
  title = {A probabilistic remark on algebraic program testing},
  journal = {Information Processing Letters},
  volume = {7},
  number = {4},
  year = {1978},
  pages = {193--195},
  doi = {10.1016/0020-0190(78)90067-4}
}