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Register a new userOn Pro-$2$ Identities of $2times2$ Linear Groups
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Let $hat{F}$ be a free pro-$p$ non-abelian group, and let $Delta$ be a commutative Noetherian complete local ring with a maximal ideal $I$ such that $textrm{char}(Delta/I)=p>0$. In [Zu], Zubkov showed that when $p eq2$, the pro-$p$ congruence subgroup $$GL_{2}^{1}(Delta)=ker(GL_{2}(Delta)overset{DeltatoDelta/I}{longrightarrow}GL_{2}(Delta/I))$$ admits a pro-$p$ identity, i.e., there exists an element $1 eq winhat{F}$ that vanishes under any continuous homomorphism $hat{F}to GL_{2}^{1}(Delta)$. In this paper we investigate the case $p=2$. The main result is that when $textrm{char}(Delta)=2$, the pro-$2$ group $GL_{2}^{1}(Delta)$ admits a pro-$2$ identity. This result was obtained by the use of trace identities that originate in PI-theory.
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We initiate an investigation of lattices in a new class of locally compact groups, so called locally pro-$p$-complete Kac-Moody groups. We discover that in rank 2 their cocompact lattices are particularly well-behaved: under mild assumptions, a cocompact lattice in this completion contains no elements of order $p$. This statement is still an open question for the Caprace-Remy-Ronan completion. Using this, modulo results of Capdeboscq and Thomas, we classify edge-transitive cocompact lattices and describe a cocompact lattice of minimal covolume.
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