We study the dual action of Lie elements on faces of the adjoint braid arrangement, interpreted as the discrete differentiation of functions on faces across hyperplanes. We encode flags of faces with layered binary trees, allowing for the representation of Lie elements by antisymmetrized layered binary forests. This induces an action of layered binary forests on functions by discrete differentiation, which we call the forest derivative. The forest derivative has antisymmetry and satisfies the Jacobi identity. We show that the restriction of the forest derivative to functions which satisfy the Steinmann relations is additionally delayered, and thus forms a left comodule of the Lie cooperad. Dually, this endows the adjoint braid arrangement modulo the Steinmann relations with the structure of a Lie algebra internal to the category of linear species. Contents 1. Introduction 1 2. The Adjoint Braid Arrangement 6 3. The Forest Derivative 11 4. The Action of Lie Elements on Faces 17 5. Semisimple Differentiability and the Steinmann Relations 19 6. A Lie Algebra in Species 24 References 26 Date: March 4, 2019.

We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finiteindex embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and construct a finite-index embedding of the fundamental group into the Coxeter group whose Davis complex is the universal cover of the polygonal complex. All of the groups which we embed have minimal index among torsion-free subgroups, and therefore are maximal among torsion-free subgroups.

The commutative Hopf monoid of set compositions is a fundamental Hopf monoid internal to vector species, having Bosonic Fock image the Hopf algebra of quasisymmetric functions. We construct a geometric realization of this Hopf monoid over the adjoint braid arrangement, which identifies the monomial basis with signed characteristic functions of open permutohedral tangent cones. We show that the indecomposable quotient is obtained by identifying functions which differ only on hyperplanes, so that the resulting Lie coalgebra consists of functions on chambers of the adjoint braid arrangement. These functions are characterized by the Steinmann relations of axiomatic QFT, demonstrating an equivalence between the Steinmann relations, tangent cones to (generalized) permutohedra, and having algebraic structure in species. Our results give the pure mathematical interpretation of a classical construction in axiomatic QFT. We show that generalized time-ordered functions correspond to the cocommutative Hopf monoid of set compositions, and generalized retarded functions correspond to its primitive part.

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