Mixed identities for simple locally finite groups

Abstract: A mixed identity of a group is a nontrivial word with constants that vanishes under every substitution of its variables. We derive lower bounds for the length of mixed identities in finite simple groups of Lie type, and characterise exactly those families of such groups of bounded rank which satisfy mixed identities of bounded length. We classify the infinite simple locally finite groups admitting a mixed identity: apart from an explicit list of alternating, finitary linear classical, and non-simply-laced groups of Lie type, no such group exists. For the groups in this list, we determine when mixed identities must be singular and obtain restrictions on their critical constants. Moreover, we prove that simple compact Lie groups do not admit mixed identities.

• The paper classifies infinite simple locally finite groups admitting mixed identities, identifying alternating, finitary classical, symplectic, and selected nonsimply laced Lie-type families.
• The paper proves that finite simple groups have field-size-independent mixed-identity lengths only in untwisted types B, C, F4, and G2, while other families satisfy quantitative lower bounds such as Ω(q^{1/a}) and Ω(log n/log log n).
• The paper combines algebraic-group word maps, polynomial estimates, Kegel covers, and ultraproducts to resolve major conjectural cases, while leaving open nonsimply laced infinite groups and certain even-dimensional orthogonal families.

This paper by Bradford, Ersoy, Schneider, and Thom undertakes a systematic study of mixed identities—nontrivial words with constants vanishing under every substitution of variables—for simple locally finite groups, both infinite and finite (2608.14537). The work combines three methodological strands: quantitative lower bounds on word length for finite simple groups of Lie type, the theory of word maps on simple algebraic groups in the tradition of Tomanov and Gordeev, and the structure theory of locally finite groups via Kegel covers and ultraproducts.

Background and terminology

A word $w \in C * \mathbf{F}_r$ with constants in a group $C$ is a mixed identity for $G \le C$ if its word map $w\colon G^r \to C$ has trivial image. The paper distinguishes several structural classes: a mixed identity is singular if its content (the image under the augmentation homomorphism $C * \mathbf{F}_r \to \mathbf{F}_r$) is trivial; it is strict if it has no critical constants, i.e. no constants $c_j$ sitting between consecutive occurrences of the same variable with opposite exponents (positions indexed by $J_-(w)$). Critical constants are the sole obstruction to strictness, and the paper tracks their size via a seminorm, introducing the critical length $\lvert w\rvert_{\mathrm{crit}}$.

The context is the classical theory of group laws: Jones's theorem that a fixed nontrivial law is satisfied by only finitely many nonabelian finite simple groups, Tomanov's characterization of linear groups admitting strict mixed identities as solvable-by-finite, and the recent program of the same authors bounding the length of shortest mixed identities for finite groups. Two guiding conjectures frame the paper: (i) every mixed identity of an infinite simple locally finite group is singular, and (ii) only finitely many isomorphism classes of nonabelian finite simple groups satisfy a nonsingular mixed identity of fixed length.

Classification for infinite simple locally finite groups

The first main theorem gives a complete classification. An infinite simple locally finite group admits a mixed identity if and only if it belongs to one of four families: the infinite alternating group $\Alt(\Omega)$; a finitary linear classical group over a finite field; a finitary symplectic group over an infinite locally finite field; or a group of Lie type $B_n$, $C$0, $C$1, or $C$2 over an infinite locally finite field. Moreover, in the first three cases every mixed identity is necessarily singular.

The dichotomy between simply laced and nonsimply laced types drives the result. For simply laced types $C$3, $C$4, $C$5, $C$6, $C$7—and for the twisted Steinberg groups $C$8, $C$9, $G \le C$0, $G \le C$1—no mixed identity exists with constants in the ambient algebraic group over $G \le C$2. The reason is that in simply laced type the G-small semisimple elements (semisimple elements acting trivially on all long roots of a maximal torus) are precisely central, and the center is trivial in the adjoint group. For nonsimply laced types $G \le C$3, $G \le C$4, $G \le C$5, $G \le C$6, Gordeev's construction yields singular mixed identities of bounded length, with critical constants that can be taken to be any G-small semisimple element together with long root elements (and short root elements only in the exceptional characteristic cases $G \le C$7 for $G \le C$8, or $G \le C$9 for $w\colon G^r \to C$0). For the generalized Suzuki and Ree groups $w\colon G^r \to C$1, $w\colon G^r \to C$2, $w\colon G^r \to C$3, the situation is subtle: mixed identities exist with constants in the ambient algebraic group (inherited from $w\colon G^r \to C$4, $w\colon G^r \to C$5, $w\colon G^r \to C$6 respectively), but none with constants in the twisted group itself, since the required G-small unipotent constants lie outside every finite subgroup—the twist swaps the classes of long and short root elements, so no fixed point can be G-small of either class.

A corollary of the simply laced analysis resolves a conjecture of Larsen and Shalev: a simple compact real Lie group admits no mixed identity with noncentral critical constants; in particular, a centreless simple compact Lie group has no mixed identity at all, since compact groups contain no almost unipotent elements and Zariski density transfers identities to the complex algebraic group.

Quantitative bounds for finite simple groups

The second main theorem establishes that for a fixed Lie type $w\colon G^r \to C$7, the groups $w\colon G^r \to C$8 admit mixed identities of length bounded independently of $w\colon G^r \to C$9 if and only if $C * \mathbf{F}_r \to \mathbf{F}_r$0 and $C * \mathbf{F}_r \to \mathbf{F}_r$1. For all remaining types the shortest mixed identity has length $C * \mathbf{F}_r \to \mathbf{F}_r$2, and for the Suzuki and Ree groups specifically the lower bound is $C * \mathbf{F}_r \to \mathbf{F}_r$3.

The proof of these lower bounds proceeds by a reduction from the finite group to the ambient algebraic group over $C * \mathbf{F}_r \to \mathbf{F}_r$4 via a Schwartz–Zippel estimate of Breuillard–Green–Guralnick–Tao: a polynomial of degree $C * \mathbf{F}_r \to \mathbf{F}_r$5 nonvanishing on $C * \mathbf{F}_r \to \mathbf{F}_r$6 cannot vanish on more than an $C * \mathbf{F}_r \to \mathbf{F}_r$7 proportion of $C * \mathbf{F}_r \to \mathbf{F}_r$8. Concretely, a short mixed identity for $C * \mathbf{F}_r \to \mathbf{F}_r$9 would yield a matrix polynomial (built from $c_j$0 and its diagonal differences) of controlled degree vanishing identically on $c_j$1, forcing it to be a mixed identity for $c_j$2—which is impossible for simply laced types by the absence of noncentral G-small constants. The paper also provides a self-contained alternative proof of the Schwartz–Zippel bound in both the untwisted and Steinberg cases, the latter requiring a careful analysis of twisted polynomial equations and the structure of locally finite fields admitting automorphisms of order $c_j$3.

For variable rank, the paper proves a rank-dependent lower bound: every nonsingular mixed identity of $c_j$4 or of the classical groups of types $c_j$5, $c_j$6, $c_j$7, $c_j$8, $c_j$9, $J_-(w)$0 has length $J_-(w)$1. The argument combines geometric propagation estimates—showing via Warning's second theorem that a word with small critical constants can move a $J_-(w)$2-dimensional totally isotropic subspace to an independent one when $J_-(w)$3—with an iterated reduction argument bounding the diameter of word images, yielding $J_-(w)$4 for a mixed identity on a group of natural module dimension $J_-(w)$5. In the unitary case the constant $J_-(w)$6 is absorbed using the independent bound $J_-(w)$7 for $J_-(w)$8.

These results resolve a Jones-type conjecture for families of finite simple groups satisfying mixed identities of bounded length, with one exception: the paper cannot exclude families of even-degree orthogonal groups $J_-(w)$9 in which both rank and field size tend to infinity. This is stated plainly as the remaining gap in the classification.

Nonlinear groups: Kegel covers and ultraproducts

The analysis of nonlinear infinite simple locally finite groups rests on Kegel's theorem that every simple locally finite group admits a Kegel cover, and on a reduction showing that a cofinal subfamily of factors is either alternating, classical of a fixed type, or of one of finitely many exceptional types. Two structural dichotomies are established: a group is linear (of bounded degree) if and only if its Kegel factors have bounded rank or belong to a fixed twisted exceptional type; and a group admitting a nonsingular mixed identity of length $\lvert w\rvert_{\mathrm{crit}}$0 forces its factors to satisfy mixed identities of the same length and content, which by the rank bounds is incompatible with unbounded rank or unbounded field size (except in the unitary case, where $\lvert w\rvert_{\mathrm{crit}}$1 again excludes unbounded $\lvert w\rvert_{\mathrm{crit}}$2).

When a singular mixed identity exists, a propagation lemma shows that all elements have uniformly bounded projective rank norm in cofinally many factors, and an ultraproduct construction—over $\lvert w\rvert_{\mathrm{crit}}$3 permutation representations in the alternating case, or over the ultraproduct field $\lvert w\rvert_{\mathrm{crit}}$4 in the classical case—produces a faithful finitary linear representation. Combined with Hall's classification of periodic simple groups of finitary linear transformations, this yields the classification of singular mixed identities: the group must be $\lvert w\rvert_{\mathrm{crit}}$5, a finitary classical group over a finite field, or a finitary symplectic group over an infinite locally finite field. In the orthogonal case over infinite fields, the argument shows that the required critical constants of the form $\lvert w\rvert_{\mathrm{crit}}$6 would have unbounded rank length along the direct system, giving a contradiction.

The paper also proves the logical implication that Conjecture (ii) (finiteness of finite simple groups with short nonsingular mixed identities) implies Conjecture (i) (all infinite simple locally finite mixed identities are singular), via Kegel covers; the converse is stated to be unclear. Hall's universal locally finite group serves as an instructive example: it is nonlinear, of type $\lvert w\rvert_{\mathrm{crit}}$7, and admits no mixed identity whatsoever, even singular ones—showing that the existence of a singular mixed identity is strictly stronger than the absence of nonsingular ones.

Almost simple groups

An appendix exhibits a phenomenon with no analogue in the simple case: for $\lvert w\rvert_{\mathrm{crit}}$8 odd, the almost simple group $\lvert w\rvert_{\mathrm{crit}}$9, with socle $\Alt(\Omega)$0 of index at most four, satisfies a mixed identity of length eight, built from an Eichler transformation and a reflection, even though the shortest mixed identity of its socle has length $\Alt(\Omega)$1 by the main theorem. This sharpens the previously known fact that shortest mixed identities need not be monotone under passage from socle to overgroup, and it explains why the even-degree orthogonal families are precisely the residual obstruction in the finite classification.

Limitations and open questions

The paper is explicit about what remains open. The classification of infinite simple locally finite groups with mixed identities is complete, but the conjecture that all such mixed identities are singular remains open for the nonsimply laced types $\Alt(\Omega)$2, $\Alt(\Omega)$3, $\Alt(\Omega)$4, $\Alt(\Omega)$5 over infinite locally finite fields. For finite groups, the bounded-length classification is unresolved for families $\Alt(\Omega)$6 with both $\Alt(\Omega)$7 and $\Alt(\Omega)$8; correspondingly, the rank bound for nonsingular mixed identities covers the classical types but the exceptional types of unbounded field size are handled only through the $\Alt(\Omega)$9 bounds, and the implication from Conjecture (ii) to Conjecture (i) is one-directional. The quantitative bounds in the finite case depend on the dimension of the chosen module, which is uniform only for the exceptional types.

Conclusion

The paper delivers a near-complete structural theory of mixed identities for simple locally finite groups: a full existence classification in the infinite case, exact determination of the finite simple families admitting mixed identities of length bounded in the field size (modulo the even-degree orthogonal families), rank-dependent lower bounds for nonsingular identities, and a proof of the Larsen–Shalev conjecture for compact Lie groups. The interplay between the algebraic-geometric analysis of critical constants and the combinatorial structure theory of locally finite groups is the technical core, and the remaining open cases—singularity for nonsimply laced infinite groups and bounded-rank-versus-bounded-field orthogonal families—are precisely identified.