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Register a new userA radius 1 irreducibility criterion for lattices in products of trees
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Authors
Pierre-Emmanuel Caprace
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Let $T_1, T_2$ be regular trees of degrees $d_1, d_2 geq 3$. Let also $Gamma leq mathrm{Aut}(T_1) times mathrm{Aut}(T_2)$ be a group acting freely and transitively on $VT_1 times VT_2$. For $i=1$ and $2$, assume that the local action of $Gamma$ on $T_i$ is $2$-transitive; if moreover $d_i geq 7$, assume that the local action contains $mathrm{Alt}(d_i)$. We show that $Gamma$ is irreducible, unless $(d_1, d_2)$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $Gamma$ that can be checked purely in terms of its local action on a ball of radius~$1$ in $T_1$ and $T_2$. Under the same hypotheses, we show moreover that if $Gamma$ is irreducible, then it is hereditarily just-infinite, provided the local action on $T_i$ is not the affine group $mathbf F_5 rtimes mathbf F_5^*$. The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.
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We study lattices in a product $G = G_1 times dots times G_n$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_i$ is non-compact and every closed normal subgroup of $G_i$ is discrete or cocompact (e.g. $G_i$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $Gamma$ with dense projections is finite. The same result holds if $Gamma$ is non-uniform, provided $G$ has Kazhdans property (T). We show that for any compact subset $K subset G$, the collection of discrete subgroups $Gamma leq G$ with $G = Gamma K$ and dense projections is uniformly discrete, hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $Aut(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.
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