Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

Industry Linkages and Audit Firms' Industry Portfolio Choice: Evidence from Product Language

Audit firms are organized along industry lines and industry specialization is a prominent feature of the audit market. Yet, we know little about how audit firms make their industry portfolio decisions, i.e., how audit firms decide which set of industries to specialize in. In this study, I examine how the linkages between industries in the product space affect audit firms’ industry portfolio choice. Using text-based product space measures to capture these industry linkages, I find that both Big 4 and small audit firms tend to specialize in industry-pairs that 1) are close to each other in the product space (i.e., have more similar product language) and 2) have a greater number of “between-industries” in the product space (i.e., have a greater number of industries with product language that is similar to both industries in the pair). Consistent with the basic tradeoff between specialization and coordination, these results suggest that specializing in industries that have more similar product language and more linkages to other industries in the product space allow audit firms greater flexibility to transfer industry-specific expertise across industries as well as greater mobility in the product space, hence enhancing its competitive advantage. Additional analysis using the collapse of Arthur Andersen as an exogenous supply shock in the audit market finds consistent results. Taken together, the findings suggest that industry linkages in the product space play an important role in shaping the audit market structure.