The virtually cyclic dimension of a group measures the complexity of building a classifying space for the family of virtually cyclic subgroups — the smallest dimension of a model for the universal space that the group acts on with virtually cyclic stabilizers. Computing this dimension is hard in general, but for groups with enough structure, bounds can be established.
Hernández Hernández and León Álvarez (arXiv:2603.13096) introduce the balanced property (Lück's Condition C) as a tool for bounding the virtually cyclic dimension of poly-surface and poly-free groups. A group is balanced if the commensurator of every infinite virtually cyclic subgroup is itself virtually cyclic. This prevents the commensurators from growing too large, which would force the classifying space to have higher dimension to accommodate them.
The paper proves that the balanced property is preserved under short exact sequences, direct limits, and acylindrical graph of groups decompositions. This stability means that once you establish the balanced property for basic building blocks (surface groups, free groups, hyperbolic groups), it automatically extends to any group built from these blocks by the preserved operations. The resulting class — normally poly-hyperbolic, normally poly-free, normally poly-surface groups — is large and includes pure braid groups of surfaces with boundary, Artin groups of FC-type, and right-angled Artin groups.
The bound is linear: for normally poly-surface groups of “length” n, the virtually cyclic dimension is at most a linear function of n. This is sharp enough to be useful for concrete calculations and general enough to cover the important examples in geometric group theory.