Overview
This research investigates a new family of generalized Euler characteristics defined for discrete groups, which can also be viewed in the context of orbifolds. These characteristics are indexed by natural numbers $n$. For the specific case where $n=0$, this generalization corresponds to the classical orbifold Euler characteristic, a concept previously explored by Wall and Serre. When $n \geq 1$ and considering finite groups, the generalized characteristic aligns with the chromatic cardinality, a notion examined by Ben-Moshe, Carmeli, Schlank, and Yanovski. The study suggests a natural interpretation of these generalized Euler characteristics, for any $n$, in terms of Morava $E$-theories.
Research Context
The classical orbifold Euler characteristic, typically associated with $n=0$ in this generalized framework, has been a subject of study by Wall and Serre. For finite groups and $n \geq 1$, the concept of chromatic cardinality provides a precedent for the generalized characteristic discussed. This work extends these earlier concepts to a broader framework applicable to infinite groups and more general values of $n$. The underlying context involves understanding the structure and properties of discrete groups and orbifolds through numerical invariants.
Approach
The core of the approach involves establishing a theoretical framework for the generalized Euler characteristics. A significant component of this framework is the development of a duality theory for the generalized cohomology of infinite groups $G$. This duality is specifically applied to groups possessing a finite universal space for proper actions, denoted as $\underline{E}G$. The duality is articulated through a new duality functor operating on the category of proper $G$-equivariant spectra. As a consequence of this theoretical development, the research demonstrates the vanishing of Klein's generalized Farrell-Tate cohomology when computed with $T(n)$-local coefficients for such groups.
The research also includes computational components, applying the developed framework to calculate the generalized orbifold Euler characteristics in various specific instances. These computations encompass:
- Many mapping class groups, for which the classical calculation was previously obtained by Harer and Zagier.
- Numerous arithmetic groups, whose classical orbifold Euler characteristics were computed by Harder.
Findings
- A family of generalized Euler characteristics for discrete groups and orbifolds, indexed by natural numbers $n$, has been introduced.
- The $n=0$ case of this family corresponds to the classical orbifold Euler characteristic.
- The $n \geq 1$ case for finite groups corresponds to the chromatic cardinality.
- These generalized Euler characteristics admit an interpretation in terms of Morava $E$-theories for any $n$.
- A duality theory for the generalized cohomology of infinite groups $G$ with a finite universal space for proper actions $\underline{E}G$ has been developed.
- This duality is expressed via a new duality functor on the category of proper $G$-equivariant spectra.
- The vanishing of Klein's generalized Farrell-Tate cohomology with $T(n)$-local coefficients was proven for infinite groups with finite universal space for proper actions.
- Generalized orbifold Euler characteristics were computed for many mapping class groups.
- Generalized orbifold Euler characteristics were computed for many arithmetic groups.
Why This Matters
The generalized Euler characteristics offer new invariants for studying discrete groups and orbifolds, extending classical concepts. The established duality theory for generalized cohomology provides a new tool for understanding the structure of infinite groups with specific topological properties. The demonstrated vanishing of certain generalized cohomology groups contributes to the theoretical understanding of these algebraic structures.