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Register a new userQuotients by finite equivalence relations
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Authors
Janos Kollar
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This note studies the existence of quotients by finite set theoretic equivalence relations. May 18: Substantial revisions with a new appendix by C. Raicu
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Let $E$ be an elliptic curve. When the symmetric group $Sigma_{g+1}$ of order $(g+1)!$ acts on $E^{g+1}$ in the natural way, the subgroup $E_0^{g+1}$, consisting of those $(g+1)$-tuples whose coordinates sum to zero, is stable under the action of $Sigma_{g+1}$. It is isomorphic to $E^g$. This paper concerns the structure of the quotient variety $E^g/Sigma$ when $Sigma$ is a subgroup of $Sigma_{g+1}$ generated by simple transpositions. In an earlier paper we observed that $E^g/Sigma$ is a bundle over a suitable power, $E^N$, with fibers that are products of projective spaces. This paper shows that $E^g/Sigma$ has an etale cover by a product of copies of $E$ and projective spaces with an abelian Galois group.
Let $C(Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $Gamma$. We investigate the extent to which $C(Gamma)$ determines $Gamma$ when $Gamma$ is a group of geometric interest. If $Gamma_1$ is a lattice in ${rm{PSL}}(2,R)$ and $Gamma_2$ is a lattice in any connected Lie group, then $C(Gamma_1) = C(Gamma_2)$ implies that $Gamma_1$ is isomorphic to $Gamma_2$. If $F$ is a free group and $Gamma$ is a right-angled Artin group or a residually free group (with one extra condition), then $C(F)=C(Gamma)$ implies that $FcongGamma$. If $Gamma_1<{rm{PSL}}(2,Bbb C)$ and $Gamma_2< G$ are non-uniform arithmetic lattices, where $G$ is a semi-simple Lie group with trivial centre and no compact factors, then $C(Gamma_1)= C(Gamma_2)$ implies that $G cong {rm{PSL}}(2,Bbb C)$ and that $Gamma_2$ belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.
In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in the spirit of Mumfords geometric invariant theory (GIT). The article surveys some recent work on geometric invariant theory and quotients of varieties by linear algebraic group actions, as well as background material on linear algebraic groups, Mumfords GIT and some of the challenges that the non-reductive setting presents. The earlier work of two of the authors in the setting of unipotent group actions is extended to deal with actions of any linear algebraic group. Given the data of a linearisation for an action of a linear algebraic group H on an irreducible variety $X$, an open subset of stable points $X^s$ is defined which admits a geometric quotient variety $X^s/H$. We construct projective completions of the quotient $X^s/H$ by considering a suitable extension of the group action to an action of a reductive group on a reductive envelope and using Mumfords GIT. In good cases one can also compute the stable locus $X^s$ in terms of stability (in the sense of Mumford for reductive groups) for the reductive envelope.
In this paper we prove that the etale sheafification of the functor arising from the quotient of an algebraic supergroup by a closed subsupergroup is representable by a smooth superscheme.
We introduce notions of continuous orbit equivalence and strong (respective, weak) continuous orbit equivalence for automorphism systems of {e}tale equivalence relations, and characterize them in terms of the semi-direct product groupoids, as well as their reduced groupoid $C^*$-algebras with canonical Cartan subalgebras. In particular, we study topological rigidity of expansive automorphism actions on compact (connected) metrizable groups.
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