Strongly dense free subgroups of semisimple algebraic groups: existence inside an arbitrary dense subgroup and dominance of double word maps (Problems 1 and 2 of Breuillard–Green–Guralnick–Tao)

Breuillard, Green, Guralnick, and Tao (2012) proved that a semisimple algebraic group over an uncountable algebraically closed field contains a free subgroup all of whose nonabelian subgroups are Zariski-dense ('strongly dense'), and posed two problems left open by their method: (1) whether every finitely generated Zariski-dense subgroup of such a group contains a strongly dense free subgroup — the strong-density analogue of the free-subgroup conclusion of the Tits alternative — and (2) to characterize the pairs of words $(w_1,w_2)$ in the free group $F_2$ for which the double word map $(a,b)\mapsto(w_1(a,b),w_2(a,b))$ is dominant. The characteristic-3, type-$C_2$ exception in the original existence theorem was resolved in the authors' 2015 sequel (Appendix D); partial positive results for Problem 1 cover the affine group of the plane, Hitchin representation components, and (most recently) hyperbolic 3-manifold groups; and Bou-Rabee–Larsen (2017) obtained a partial answer to Problem 2 for multivariable surface relators. Neither general question has been settled.

Background

A semisimple algebraic group is a connected linear algebraic group with no nontrivial connected solvable normal subgroup (for example $\mathrm{SL}_n$, $\mathrm{Sp}_{2n}$, or a special orthogonal group); its group of $k$-rational points is denoted $\mathbf{G}(k)$. Throughout, 'dense' means dense in the Zariski topology, the topology on $\mathbf{G}(k)$ whose closed sets are zero loci of polynomial equations in the matrix entries, so a subgroup is dense precisely when it lies in no proper algebraic subgroup.

Two classical facts frame the problems. First, the Tits alternative (Tits 1972) implies that every finitely generated dense subgroup $\Gamma$ of $\mathbf{G}(k)$ contains a nonabelian free subgroup that is still dense in $\mathbf{G}(k)$; a free group is a group with a generating set obeying no relations, and $F_2$ denotes the free group on two generators. Second, Borel's theorem on word maps (Borel 1983; see also Larsen 2004) states that for any nontrivial word $w\in F_2$ the associated word map $w:\mathbf{G}\times\mathbf{G}\to\mathbf{G}$, $(a,b)\mapsto w(a,b)$, is dominant, i.e. its image is Zariski-dense in $\mathbf{G}$ (equivalently, contains a nonempty Zariski-open subset).

Breuillard, Green, Guralnick, and Tao (2012) introduced a strengthening of density: a subgroup $\Gamma\leq\mathbf{G}(k)$ is strongly dense if every pair of noncommuting elements of $\Gamma$ generates a dense subgroup of $\mathbf{G}(k)$, so that every subgroup of $\Gamma$ is either abelian or dense. Their main theorem proves that if $k$ is an uncountable algebraically closed field, then a generic set of pairs $(a,b)\in\mathbf{G}(k)\times\mathbf{G}(k)$ — one containing the complement of a countable union of proper closed subvarieties — generates a free strongly dense subgroup, with one technical exception: groups involving a factor of type $C_2$ (the rank-2 symplectic group $\mathrm{Sp}_4$) over a field of characteristic $3$, where a key proposition of their method fails. This exception was removed in Appendix D of the authors' 2015 sequel, which proves the same generic statement for $\mathrm{Sp}_4(k)$ in every characteristic other than $2$. Strong density has since become a standard tool: it underpins simultaneous paradoxical decompositions of compact Lie groups, results that word values generate finite simple groups of Lie type, and uniform expansion bounds for finite quotients.

The 2012 paper ends by posing two problems that its method — which produces generic pairs rather than pairs inside a prescribed subgroup — cannot reach, and which together ask how robust the strong-density phenomenon is. The first seeks the strong-density analogue of the free-subgroup conclusion of the Tits alternative inside an arbitrary dense subgroup. The second concerns the algebraic-geometric engine of the technique: since Borel's theorem makes a single word map dominant, one asks when two word maps are simultaneously dominant, i.e. when the double word map $(a,b)\mapsto(w_1(a,b),w_2(a,b))$ has dense image in $\mathbf{G}\times\mathbf{G}$. Two cyclic subgroups $\langle w_1\rangle,\langle w_2\rangle\leq F_2$ are commensurate if their intersection has finite index in both (equivalently, $w_1$ and $w_2$ are powers of a common word); noncommensurality is necessary for dominance, and further obstructions come from conjugate words and from pairs lying in a subgroup of $F_2$ generated by two conjugates, where trace identities such as $\mathrm{tr}(A[A,B])=\mathrm{tr}(A)$ force algebraic constraints on the image (Horowitz 1972 exhibited the relevant trace-identity phenomena for $\mathrm{SL}_2$).

Problem Statement

The two problems below are the problems posed at the end of the introduction of Breuillard–Green–Guralnick–Tao (2012). They share one scientific objective — determining how generic and how robust the strong-density phenomenon is — and the second is the word-map dominance question that powers the technique behind the first. Throughout, $\mathbf{G}$ is a semisimple algebraic group (a connected linear algebraic group with no nontrivial connected solvable normal subgroup, such as $\mathrm{SL}_n$), $k$ is a field, $\mathbf{G}(k)$ denotes its group of $k$-rational points, 'dense' means Zariski-dense (dense in the topology whose closed sets are zero loci of polynomial equations in the matrix entries, so a subgroup is dense precisely when it lies in no proper algebraic subgroup), and $F_2$ is the free group on two generators.

Problem 1 (strong density inside an arbitrary dense subgroup). Let $k$ be an uncountable algebraically closed field (the standing setting of the source paper, and the hypothesis of its main theorem), and let $\Gamma\leq\mathbf{G}(k)$ be a finitely generated dense subgroup. The Tits alternative guarantees that $\Gamma$ contains a dense free subgroup: as the source puts it, 'J. Tits showed that every finitely generated dense subgroup $\Gamma$ of $\mathbf{G}(k)$ contains a dense free subgroup. This is the beef of the Tits alternative. It remains a challenging problem to determine whether or not $\Gamma$ always contains a strongly dense free subgroup.' Must $\Gamma$ contain a nonabelian free subgroup that is strongly dense in $\mathbf{G}$, i.e. such that every pair of noncommuting elements of that free subgroup generates a dense subgroup of $\mathbf{G}(k)$ (equivalently, in the terminology of later literature, a hereditarily Zariski-dense free subgroup)? Kapovich's 2019 BIRS problem list poses a related variant for real semisimple Lie groups as Question 7.3 (dropping the finite-generation hypothesis and adding a quantitative sub-question).

Problem 2 (dominance of double word maps). For a semisimple algebraic group $\mathbf{G}$, characterize the set of pairs of words $(w_1,w_2)\in F_2\times F_2$ such that the double word map

$\mathbf{G}\times\mathbf{G}\to\mathbf{G}\times\mathbf{G},\qquad (a,b)\mapsto(w_1(a,b),\,w_2(a,b))$

is dominant, i.e. has Zariski-dense image in $\mathbf{G}\times\mathbf{G}$ (equivalently, an image containing a nonempty Zariski-open subset). Known necessary conditions and obstructions: the cyclic subgroups $\langle w_1\rangle$ and $\langle w_2\rangle$ must not be commensurate (commensurate means their intersection has finite index in both, equivalently $w_1$ and $w_2$ are powers of a common word); if $w_1$ and $w_2$ are conjugate in $F_2$, the image is contained in the variety of conjugate pairs and has dimension at most $2\dim\mathbf{G}-\mathrm{rk}(\mathbf{G})$; and the same failure occurs whenever both words lie in a subgroup of $F_2$ generated by two conjugates (for example $w_2=[w_1,w_3]$, where the matrix identity $\mathrm{tr}(A[A,B])=\mathrm{tr}(A)$ forces a nontrivial algebraic constraint on the image). No sufficient condition is known.

References

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