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We show under weak hypotheses that
, the Roller boundary of a finite-dimensional CAT(0) cube complex
is the Furstenberg–Poisson boundary of a sufficiently nice random walk on an acting group
. In particular, we show that if
admits a non-elementary proper action on
is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a
-stationary measure on
making it the Furstenberg–Poisson boundary for the
-random walk on
. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.
In this paper, we investigate the abstract homomorphisms of the special linear group SLn(
) over complete discrete valuation rings with finite residue field into the general linear group GLm(
) over the field of real numbers. We show that for m < 2n, every such homomorphism factors through a finite index subgroup of SLn(
with positive characteristic, this result holds for all m ∈
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