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Confined subgroups in groups with contracting elements

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arxiv 2405.09070 v1 pith:74UDVQ6L submitted 2024-05-15 math.GR math.DSmath.GT

classification math.GRmath.DSmath.GT
keywords growthsubgroupsconfinedgroupsdecompositionhopfactionsboundary
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In this paper, we study the growth of confined subgroups through boundary actions of groups with contracting elements. We establish that the growth rate of a confined subgroup is strictly greater than half of the ambient growth rate in groups with purely exponential growth. Along the way, several results are obtained on the Hopf decomposition for boundary actions of subgroups with respect to conformal measures. In particular, we prove that confined subgroups are conservative, and examples of subgroups with nontrivial Hopf decomposition are constructed. We show a connection between Hopf decomposition and quotient growth and provide a dichotomy on quotient growth of Schreier graphs for subgroups in hyperbolic groups.

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  1. Directional growth of coamenable normal subgroups: counterexamples and rigidity

    math.GR 2026-05 unverdicted novelty 8.0 of 10

    In higher rank, coamenable normal subgroups preserve the Riemannian critical exponent and growth indicators on the opposition-involution fixed locus but not the full limit cones, with explicit constructions showing th...

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