TITLE: How feasible is it to prove Kazhdan's property (T) by a computer?
QUESTION [90 upvotes]: Recently, I have proved that Kazhdan's property (T) is theoretically provable 
by computers (arXiv:1312.5431, 
explained below), but I'm quite lame with computers and have 
no idea what they actually can do. So, my question is how feasible is it to 
prove property (T) of a given group, say $\mathrm{Out}(F_{r>3})$ (a famous 
open problem), by solving the equation below by a computer? Even the case of 
$\mathrm{SL}_{r>2}({\mathbb Z})$ where property (T) is known is unclear.
A group $\Gamma$, generated by a finite subset $S$ and with its non-normalized Laplacian denoted by 
$$\Delta=\sum_{x\in S} (1-x)^*(1-x)=\sum_{x\in S} (2-x-x^{-1})\in{\mathbb Z}[\Gamma],$$
has property (T) iff the equation in ${\mathbb Z}[\Gamma]$,
$$ m \Delta^2 = n \Delta + \sum_{i=1}^k l_i \xi_i^*\xi_i $$
has a solution in $k,m,n,l_i\in{\mathbb Z}_{>0}$ and $\xi_i\in{\mathbb Z}[\Gamma]$.

REPLY [38 votes]: I think it is appropriate to let MO users know (the OP himself knows it well) that this question was recently solved: it is feasible to provide a computer based proof for property (T) using the Ozawa Equation and this method was applied successfully to the groups mentioned in the question. In particular, the famous open problem alluded to above is now solved.
The computation is done via semidefinite programming, as suggested by David Speyer in his answer.
It is now known that the group $\mathrm{Aut}(F_n)$ has property (T) iff $n\geq 4$.
For $n\geq 5$ this is shown in Kaluba-Kielak-Nowak.
The case $n=5$ was treated earlier in Kaluba-Nowak-Ozawa
and the case $n=4$ was recently settled in Nitsche.
Some earlier positive results were obtained in Netzer-Thom and Fujiwara-Kabaya.
$\mathrm{Aut}(F_2)$ and $\mathrm{Aut}(F_3)$ are known not to have (T).
This is all quite remarkable.