Chromatic Euler characteristics and duality for infinite groups
Abstract
We study a family of generalizations of the notion of Euler characteristic of discrete groups (or of orbifolds, depending on one's perspective) indexed on the natural numbers. For n=0, this is the classical orbifold Euler characteristic as studied by Wall and Serre, whereas for n ≥ 1 and finite groups, this is the chromatic cardinality as studied by Ben-Moshe--Carmeli--Schlank--Yanovski. For general n, we show that our generalized Euler characteristic admits a natural interpretation in terms of the Morava E-theories. Our work involves showing that the generalized cohomology of infinite groups G with finite universal space for proper actions underlineEG has a good theory of duality, as expressed by a new duality functor on the category of proper G-equivariant spectra. In particular, for such groups we prove the vanishing of Klein's generalized Farrell--Tate cohomology with T(n)-local coefficients. We compute our generalized orbifold Euler characteristics in a large number of examples. This includes many mapping class groups, where the classical calculation is a result of Harer--Zagier, and many arithmetic groups, whose classical orbifold Euler characteristics were computed by Harder.
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BibTeX
@misc{heuts2026chromatic,
title = {{Chromatic Euler characteristics and duality for infinite groups}},
author = {Gijs Heuts and Irakli Patchkoria},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.28107},
eprint = {2606.28107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.28107},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Chromatic Euler characteristics and duality for infinite groups
AU - Gijs Heuts
AU - Irakli Patchkoria
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2606.28107
UR - https://arxiv.org/abs/2606.28107
AB - We study a family of generalizations of the notion of Euler characteristic of discrete groups (or of orbifolds, depending on one's perspective) indexed on the natural numbers. For n=0, this is the classical orbifold Euler characteristic as studied by Wall and Serre, whereas for n ≥ 1 and finite groups, this is the chromatic cardinality as studied by Ben-Moshe--Carmeli--Schlank--Yanovski. For general n, we show that our generalized Euler characteristic admits a natural interpretation in terms of the Morava E-theories. Our work involves showing that the generalized cohomology of infinite groups G with finite universal space for proper actions underlineEG has a good theory of duality, as expressed by a new duality functor on the category of proper G-equivariant spectra. In particular, for such groups we prove the vanishing of Klein's generalized Farrell--Tate cohomology with T(n)-local coefficients. We compute our generalized orbifold Euler characteristics in a large number of examples. This includes many mapping class groups, where the classical calculation is a result of Harer--Zagier, and many arithmetic groups, whose classical orbifold Euler characteristics were computed by Harder.
ER -