Abstract
It is well known that surface groups admit free and proper actions on finite products of infinite valence trees. In this note, we address the question of whether there can be a free and proper action on a finite product of bounded valence trees. We provide both some obstructions and an arithmetic criterion for existence. The bulk of the paper is devoted to an approach to verifying the arithmetic criterion which involves studying the character variety of certain surface groups over a field of positive characteristic. These methods may be useful for attempting to determine when groups admit good linear representations in other contexts.
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DF was partially supported by NSF Grant DMS-1308291. ML was partially supported by NSF Grant DMS-1401419. MS was supported by NSF Grant DMS- 1361000, and also acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric Structures And Representation Varieties” (the GEAR Network). RS was partially supported by NSF Grant DMS-1307164. This project was begun at MSRI during the special semester on Dynamics on Moduli Spaces of Geometric Structures, the authors thank the Institute for its support and hospitality.
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Fisher, D., Larsen, M., Spatzier, R. et al. Character varieties and actions on products of trees. Isr. J. Math. 225, 889–907 (2018). https://doi.org/10.1007/s11856-018-1683-3
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DOI: https://doi.org/10.1007/s11856-018-1683-3