$\operatorname{Aut}(\mathbb{F}_5)$ has property $(T)$

We give a constructive, computer-assisted proof that $\operatorname{Aut}(\mathbb{F}_5)$, the automorphism group of the free group on $5$ generators, has Kazhdan's property $(T)$.

[1]  Tim Netzer,et al.  Kazhdan’s Property (T) via Semidefinite Optimization , 2015, Exp. Math..

[2]  S. Gersten.  A presentation for the special automorphism group of a free group , 1984 .

[3]  Stephen P. Boyd,et al.  Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding , 2013, Journal of Optimization Theory and Applications.

[4]  S. Popa.  Some rigidity results for non-commutative Bernoulli shifts☆ , 2006 .

[5]  Jean-Pierre Serre,et al.  Linear representations of finite groups , 1977, Graduate texts in mathematics.

[6]  J. Ellenberg.  Superstrong approximation for monodromy groups , 2012, 1210.3757.

[7]  O. Bogopolski,et al.  Subgroups of Small Index in Aut(Fn) and Kazhdan's Property (T) , 2010 .

[8]  Robert H. Gilman.  Finite quotients of the automorphism group of a free group , 1977 .

[9]  Linear Representations of the Automorphism Group of a Free Group , 2006, math/0606182.

[10]  Konrad Schmuedgen.  Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas , 2009 .

[11]  Iain Dunning,et al.  JuMP: A Modeling Language for Mathematical Optimization , 2015, SIAM Rev..

[12]  W. Magnus,et al.  Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations , 1966 .

[13]  James McCool,et al.  A faithful polynomial representation of Out F3 , 1989, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  Marek Kaluba,et al.  Certifying numerical estimates of spectral gaps , 2017, Groups Complex. Cryptol..

[15]  G. Jameson.  Ordered Linear Spaces , 1970 .

[16]  Yoshihiro Kanno,et al.  A numerical algorithm for block-diagonal decomposition of matrix $${*}$$-algebras with application to semidefinite programming , 2010 .

[17]  N. Ozawa.  NONCOMMUTATIVE REAL ALGEBRAIC GEOMETRY OF KAZHDAN’S PROPERTY (T) , 2013, Journal of the Institute of Mathematics of Jussieu.

[18]  Fredrik Johansson,et al.  Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language , 2017, ISSAC.

[19]  E. Breuillard.  EXPANDER GRAPHS, PROPERTY (τ) AND APPROXIMATE GROUPS , 2013 .

[20]  Koji Fujiwara,et al.  Computing Kazhdan Constants by Semidefinite Programming , 2019, Exp. Math..

[21]  Yehuda Shalom,et al.  The algebraization of Kazhdans property (T) , 2006 .

[22]  Martin Kassabov,et al.  Symmetric groups and expander graphs , 2005 .

[23]  Igor Pak,et al.  The product replacement algorithm and Kazhdan’s property (T) , 2000 .

[24]  Alan Edelman,et al.  Julia: A Fresh Approach to Numerical Computing , 2014, SIAM Rev..

[25]  A. Neumaier,et al.  Computer-assisted proofs , 2006, 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006).

[26]  Etienne de Klerk,et al.  Numerical block diagonalization of matrix *-algebras with application to semidefinite programming , 2011, Math. Program..

[27]  Mikhail Ershov,et al.  Property (T) for noncommutative universal lattices , 2008, 0809.4095.