Marion's finiteness conjecture for rigid triangle generation

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Let GG be a simple algebraic group over an algebraically closed field of characteristic p>0p>0. For a positive integer mm, let jm(G)j_m(G) be the dimension of the subvariety of elements of order dividing mm, and call a triple (a,b,c)(a,b,c) rigid for GG when

ja(G)+jb(G)+jc(G)=2dim⁡G.j_a(G)+j_b(G)+j_c(G)=2\dim G.

Call (a,b,c)(a,b,c) hyperbolic when 1/a+1/b+1/c<11/a+1/b+1/c<1. Marion's conjecture. Fix a prime pp. If (a,b,c)(a,b,c) is a rigid hyperbolic triple of primes for GG, then there are only finitely many positive integers rr such that the finite quasisimple group G(pr)G(p^r) is (a,b,c)(a,b,c)-generated.

The conjecture concerns the finiteness of field extensions yielding triangle generation in the rigid case, extending the known results for rank-two groups. Its resolution status is not specified in the supplied text.

References

Primary source

Timothy C. Burness, “Simple groups, generation and probabilistic methods”, arXiv:1710.10434 (2017).

Additional references

3 papers in this index state this conjecture (2017). The statement above is taken from the most recent of them; the others are arXiv:1706.07641, arXiv:1706.01653.

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