I have recently studying Tits' alternative. The theorem statement goes like the following:
Tits' alternative: Let $G$ be any finitely generated linear group over a field. Then one of the following is true,
$(1)$ $G$ contains a solvable normal subgroup of finite index,
$(2)$ $G$ contains a non-abelian free subgroup (of rank at least $2$).
I am in search of applications of this wonderful theorem in number theory. Any help, resources or reference will be appreciated. Thanks in advance.
The same question was asked by me in Mathoverflow. But due to lack of proper response, I posted it here.
I am not sure how broadly you interpret Number Theory (NT), but the following papers have NT as their primary Math Subject classification and quote Tits' Alternative:
Lubotzky, Alexander; Venkataramana, Tyakal Nanjundiah, The congruence topology, Grothendieck duality and thin groups, Algebra Number Theory 13 (2019), no. 6, 1281–1298, ZBL07103974.
Gao, Ziyang; Habegger, Philipp, Heights in families of abelian varieties and the geometric Bogomolov conjecture, Ann. Math. (2) 189, No. 2, 527-604 (2019). ZBL1432.11060.
Björklund, Michael, Small product sets in compact groups, Fundam. Math. 238, No. 1, 1-27 (2017). ZBL1427.11101.
Bourgain, Jean; Varjú, Péter P., Expansion in $SL_d(\mathbb Z/q\mathbb Z)$, $q$ arbitrary., Invent. Math. 188, No. 1, 151-173 (2012). ZBL1247.20052.
Vu, Van H.; Wood, Melanie Matchett; Wood, Philip Matchett, Mapping incidences, J. Lond. Math. Soc., II. Ser. 84, No. 2, 433-445 (2011). ZBL1234.05049.
Breuillard, Emmanuel, A height gap theorem for finite subsets of $GL_d(\overline{\mathbb{Q}})$ and nonamenable subgroups, Ann. Math. (2) 174, No. 2, 1057-1110 (2011). ZBL1243.11071.
Bourgain, Jean; Gamburd, Alex; Sarnak, Peter, Affine linear sieve, expanders, and sum-product, Invent. Math. 179, No. 3, 559-644 (2010). ZBL1239.11103.
Chang, Mei-Chu, Convolution of discrete measures on linear groups, J. Funct. Anal. 247, No. 2, 417-437 (2007). ZBL1130.22003.
Hajir, Farshid, On the growth of (p)-class groups in (p)-class field towers, J. Algebra 188, No. 1, 256-271 (1997). ZBL0879.11069.