# Scl in graphs of groups

## TL;DR

This paper proves that stable commutator length (scl) in groups acting on trees with cyclic stabilizers is rational and explores its predictable variation and convergence in certain families, drawing an analogy to hyperbolic Dehn surgery.

## Contribution

It establishes the rationality of scl in these groups and analyzes its behavior in surgery families, providing a homological perspective on geometric convergence.

## Key findings

- scl is rational in groups acting on trees with cyclic stabilizers
- scl varies predictably in surgery families
- scl converges to rational limits in these families

## Abstract

Let G be a group acting on a tree with cyclic edge and vertex stabilizers. Then stable commutator length (scl) is rational in G. Furthermore, scl varies predictably and converges to rational limits in so-called "surgery" families. This is a homological analog of the phenomenon of geometric convergence in hyperbolic Dehn surgery.