\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Aut}}({\mathbb {F}}_5)$$\end{document}Aut(F5) has property (T)

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

[1]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[2]  H. Nagao,et al.  Representations of Finite Groups , 1989, Group Theory for Physicists.

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

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

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

[6]  D. M. Ocampo-Giraldo A time-expanded network for the biomedical sample transportation problem , 2020 .

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

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

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

[10]  Angelos Koutsianas,et al.  Computing All Elliptic Curves Over an Arbitrary Number Field with Prescribed Primes of Bad Reduction , 2015, Exp. Math..

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

[12]  Martin R Bridson,et al.  Automorphism groups of free groups, surface groups and free abelian groups , 2005, math/0507612.

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

[14]  Alexander Schrijver,et al.  Invariant Semidefinite Programs , 2010, 1007.2905.

[15]  Alain Valette,et al.  Kazhdan's Property (T): List of symbols , 2008 .

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

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

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

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

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

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

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

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

[24]  Juan Luis Varona,et al.  Complex networks and decentralized search algorithms , 2006 .

[25]  O. Kharlampovich,et al.  Combinatorial and Geometric Group Theory , 2010 .

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

[27]  Alain Valette,et al.  Kazhdan's Property (T): KAZHDAN'S PROPERTY (T) , 2008 .

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

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

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

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

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

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