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Abstract We considerG, a linear algebraic group defined over$$\Bbbk $$, an algebraically closed field (ACF). By considering$$\Bbbk $$as an embedded residue field of an algebraically closed valued fieldK, we can associate to it a compactG-space$$S^\mu _G(\Bbbk )$$consisting of$$\mu $$-types onG. We show that for each$$p_\mu \in S^\mu _G(\Bbbk )$$,$$\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$$is a solvable infinite algebraic group when$$p_\mu $$is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of$$\mathrm {Stab}\left (p_\mu \right )$$in terms of the dimension ofp.
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This content will become publicly available on July 17, 2026
Aspherical complex surfaces, the Singer conjecture, and Gromov–Lück inequality $\chi \ge |\sigma |$
Abstract We discuss the Singer conjecture and Gromov–Lück inequality$$\chi\geq |\sigma|$$for aspherical complex surfaces. We give a proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group that does not rely on Gromov’s Kähler groups theory. Without the residually finiteness assumption, we observe that this conjecture can be proven for all aspherical complex surfaces except possibly those in Class$$\mathrm{VII}_0^+$$(a positive answer to the global spherical shell conjecture would rule out the existence of aspherical surfaces in this class). We also sharpen the Gromov-Lück inequality for aspherical complex surfaces that are not in Class$$\mathrm{VII}_0^+$$. This is achieved by connecting the circle of ideas of the Singer conjecture with the study of Reid’s conjecture.
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- Award ID(s):
- 2104662
- PAR ID:
- 10627898
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Mathematical Proceedings of the Cambridge Philosophical Society
- ISSN:
- 0305-0041
- Page Range / eLocation ID:
- 1 to 15
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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