2025/09/26 by Mayukh Mukherjee, Mukherjee, Mayukh
Mathematics · #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR) #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2509.22260
openalex publication_date 2025/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a BV framework on Cayley graphs, which yields a sharp discrete Wulff isoperimetric inequality with the best constant tied to the generating set/stencil S, a quantitative Γ-convergence of discrete to continuum anisotropic perimeter, and a gauge construction in Heisenberg group that neutralizes shear, yielding an BV+shear identity and scale-sharp compactness. As an application we revisit a question of Gromov (2008) on the ratio between the isoperimetric profile and its greatest nondecreasing minorant. We show that this is uniformly bounded on non-amenable and two-ended groups, and our Wulff inequality and Γ-convergence give a constant-tracked proof for virtually nilpotent groups. We isolate a Tempered Følner criterion (TF) (exhaustion principle with controlled increment), which forces bounded ratio in general. We verify (TF) in two families: finite-lamp wreath products over (TF) bases and lamplighters over amenable bases. For semidirect products \mathbb Zd\rtimesA\mathbb Z we construct layer-nested sets of logarithmic height that are A-covariantly nested, Følner, and satisfy (TF)(ii) with a constant independent of A\inGL(d,\mathbb Z); and when A is hyperbolic (no eigenvalue on the unit circle) we have full (TF). We also formulate ``gap conjectures'' that would settle the question for all amenable Cayley graphs. The BV viewpoint has spectral and analytic consequences: we derive constant-tracked Cheeger-type, Faber-Krahn, Nash inequalities, etc. The (TF) control further yields a tempered Property A, leading to explicit coarse embeddings into Hilbert space with compression ρ(t) \gtrsim t1/2/log t. Finally, we show that (TF) is a robust reflection of bi-equivariant geometry, and in virtually nilpotent classes this doubles the sharp Wulff constant asymptotically-refining and answering another question of Gromov.