Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms

Abstract

We prove that every mapping torus of any free group endomorphism is residually finite. We show how to use a not yet published result of E. Hrushovski to extend our result to arbitrary linear groups. The proof uses algebraic self-maps of affine spaces over finite fields. In particular, we prove that when such a map is dominant, the set of its fixed closed scheme points is Zariski dense in the affine space.

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

“…In the latter paper this machinery has been further developed which led naturally to some new concepts in arithmetical dynamics. Another impressive application of arithmetical dynamics to a hard group-theoretic problem has been demonstrated by Borisov and Sapir [18,19], see subsection 4.2.…”

“…Another instance is related to the work of Borisov and Sapir [18,19], see subsection 4.2, where somewhat similar philosophy led to an answer to another long-standing group-theoretic question (this time, from the theory of infinite groups).…”

Algorithmically complex residually finite groups

“…For example, for the free group F 2 , ρ F 2 (n) n 3 by [9]. A lower bound for the depth function for a free group is equivalent to n 2 virtually residually nilpotent (proved by Borisov and the third author [6,7]) but the only upper bound one can deduce from the proof is exponential. Although many of these groups have small cancelation presentations and so are covered by the results from [1], there are some groups of this kind for which the depth function is not known.…”

Largeness of LERF and 1-relator groups

“…There is an algorithm, due to Moldavanskii [56] in the caseˇ1.G/ D 1 and Brown [20] forˇ1.G/ D 2, as to whether a 2-generator 1-relator group is of this form. One could try looking for H Ä f G with H an ascending HNN extension of a finitely generated free group (which is shown to be residually finite in [16]), but the problem is of course that although H will have a deficiency 1 presentation, it will not in general be 2-generated (indeed as soon as d.H=H 0 / 3 it cannot be).…”

“…No progress in the 2-generator 1-relator case, but [5] shows that cyclically pinched 1-relator groups are often virtually free-by-cyclic (though the free subgroup will be infinitely generated for more than 2 generators). [16]). Is the property of being residually finite generic amongst 2-generator 1-relator presentations?…”