We prove that every class-preserving endomorphism of the adjoint Chevalley group and of its elementary subgroup over a commutative ring is inner for the types A1, A2, and B2 when 2 is invertible, and for type G2 when 2 and 3 are invertible. Consequently, all these groups are Sha-rigid.
We prove that every locally inner (class-preserving) endomorphism of adjoint Chevalley groups and their elementary subgroups over commutative rings is inner for the root systems A1, A2, B2 (assuming 2 is invertible in the ring), and for G2 (assuming 2 and 3 are invertible). As a consequence, these groups are Sha-rigid. The proofs are direct and do not rely on classification of automorphisms or structural results about injective endomorphisms.
. In 1977, Makanin established the decidability of equations in free monoids. A key ingredient in his proof is the exponent of periodicity: for a word w, it is the largest exponent e such that w contains a nonempty factor of the form pe. Makanin showed the following for a system of equations in free monoids: if the system has a solution with a sufficiently large exponent of periodicity, then it has infinitely many solutions. However, the converse - whether the existence of infinitely many solutions implies the existence of solutions with arbitrarily large exponent of periodicity - remains open. In this paper, we investigate the analogous problem for quadratic equations in finitely generated groups. We use normal forms to define the exponent of periodicity. We then identify structural conditions on groups and their normal forms that guarantee that infinite solution sets of quadratic systems have an unbounded exponent of periodicity. We prove that these conditions are preserved under graph products and, in particular, hold for all finitely generated right-angled Artin groups. In addition, we show that they also hold for finitely generated (graph products of) torsion-free nilpotent and hyperbolic groups, and we characterize the Baumslag-Solitar groups satisfying them.
Myasnikov, Ushakov, and Won introduced power circuits in 2012 to construct a polynomial-time algorithm for the word problem in the Baumslag group, which has a non-elementary Dehn function. Power circuits are computational structures that support addition and the operation (x, y) 7 -> x & centerdot; 2(y) on integers. They also posed the question of decidability of the Diophantine problem over the structure < N>0; +, x & centerdot; 2(y), <=, 1 >, which is closely related to power circuits. In this paper, we prove that the Diophantine problem over this structure is undecidable.
Given a group G = H1 (& lowast;)(A) H2 which is the free product of two finitely generated groups H1 and H2 with amalgamation over a cyclic subgroup A which is malnormal in G, we study relations between the structure of its subgroups and the structure of the group G itself. Firstly, we show that if H-1 and H-2 are 3-free products of cyclics of rank > 3 then G is also a 3-free product of cyclics. Secondly, we prove that if H-1 and H-2 are 4-free products of cyclics of rank >= 4 then every 4-generated subgroup of G is a free product of <= 4 cyclics or a 1-relator quotient of a free product of four cyclic groups. Here a group is called an n-free product of cyclics if every n-generated subgroup is a free product of <= n cyclic groups. These results are based on ubiquitous applications of the Nielsen method for amalgamated free products which we recall carefully. Lastly, given an infinite, finitely presented group which is not free, but all of its infinite index subgroups are free, a well-known conjecture says that it is isomorphic to a surface group. We revisit and elaborate on predominantly group theoretic proofs of this conjecture for cyclically amalgamated products as above, as well as for certain HNN extensions.
In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words.
. Let ohm be a finite set of finitary operation symbols. An ohm-expanded group is a group (written additively and called the additive group of the ohm-expanded group) with an ohm-algebra structure. We use the black-box model of computation in ohm-expanded groups. In this model, elements of a finite ohm-expanded group H are represented (not necessarily uniquely) by bit strings of the same length, say, n. Given representations of elements of H, equality testing and the fundamental operations of H are performed by an oracle. Assume that H is distributive, i.e., all its fundamental operations associated with nonnullary operation symbols in ohm are distributive over addition. Suppose s = (s1, ... , sm) is a generating system of H. In this paper, we present probabilistic polynomial-time black-box ohm-expanded group algorithms for the following problems: (i) given (1n, s), construct a generating system of the additive group of H, (ii) given (1n, s, (t1, ... , tk)) with t1, ... , tk is an element of H, find a generating system of the additive group of the ideal in H generated by {t1, ... , tk}, and (iii) given (1n, s), decide whether H is an element of 3, where 3 is an arbitrary finitely based variety of distributive ohm-expanded groups with nilpotent additive groups. The error probability of these algorithms is exponentially small in n. In particular, this can be applied to groups, rings, R-modules, and R-algebras, where R is a fixed finitely generated commutative associative ring with 1. Rings and R-algebras may be here with or without 1, where 1 is considered as a nullary fundamental operation.
Motivated by a classic theorem of Birman and Series about the set of complete simple geodesics on a hyperbolic surface, we study the Hausdorff dimension of the set of endpoints in ∂ F_r of some abstract algebraic laminations associated with free group automorphisms. For an exponentially growing outer automorphism ϕ∈ Out(F_r) we show that the set of endpoints ℰ_L⊆∂ F_r of any of the attracting laminations L of ϕ has Hausdorff dimension 0 for any tree T∈ cv_r and any visual metric on the boundary ∂ T=∂ F_r. If ϕ∈ Out(F_r) is an atoroidal and fully irreducible, we deduce the same conclusion for the set of endpoints of the ending lamination Λ_ϕ of ϕ that gets collapsed by the Cannon-Thurston map ∂ F_r→∂ G_ϕ for the associated free-by-cyclic group G_ϕ=F_r⋊_ϕℤ.
Regardless of the choice of parameters, knowledge of a single signed message, i.e., a pair message/signature, produced by Kahrobaei-Koupparis digital signature scheme, proposed in [D. Kahrobaei and C. Koupparis, 2012], is sufficient to forge a valid signature for any other message.
Let p and n be positive integers. Assume additionally that p not equal 3 is a prime and that n > 2. Let R be a field of characteristic p. A very special consequence of a result of Bunina and Kunyavskii (2023, arXiv:2308.10076) is that SLn(R) is co-Hopfian as a group if and only if R is co-Hopfian as a ring. In this paper, we prove that if k is the algebraic closure of the 2 element field, then SL2(k) is a co-Hopfian group. Since this k is trivially seen to be co-Hopfian as a ring our result somewhat extends that of Bunina and Kunyavskii. We apply our result to prove that the class of groups satisfying Turner's Retract Theorem (called Turner groups here) is not closed under elementary equivalence thereby answering a question posed by the authors in (2017, Comm. Algebra).
In this paper we provide an alternative solution to a result by Juhász that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation $G(m) = \langle a,b \; | \; _{m}(a,b) = {}_{m}(b,a) \rangle$, where $m\geq 3$ is odd, and $_{m}(a,b)$ is the word $abab \dots$ of length $m$, is solvable. Our solution provides an implementable linear time algorithm, by considering an alternative group presentation to that of a torus knot group, and working with geodesic normal forms. An application of this result is that the conjugacy problem is solvable in extensions of odd dihedral Artin groups.
We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in $H_3(\mathbb Z)$ can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform $3$-step nilpotent group has decidable Submonoid Membership.
We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius n in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually abelian groups, we prove that this function is always asymptotically equivalent to a polynomial. Furthermore, we show that in any affine Coxeter group, the degree of polynomial growth of a conjugacy class is equivalent to the reflection length of any element of that class.
For any group G and integer k >= 2 the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group AC(k)(G), on the subset N-k(G) subset of G(k) of all k-tuples that generate G as a normal subgroup (provided N-k(G) is non-empty). The famous Andrews-Curtis Conjecture is that if G is free of rank k, then AC(k)(G) acts transitively on N-k(G). The set N-k(G) may have a rather complex structure, so it is easier to study the full Andrews-Curtis group FAC(G) generated by AC-transformations on a much simpler set G(k). Our goal here is to investigate the natural epimorphism lambda: FAC(k)(G)-* AC(k)(G). We show that if G is non-elementary torsion-free hyperbolic, then FAC(k)(G) acts faithfully on every nontrivial orbit of G(k), hence lambda: FAC(k)(G)-* AC(k()G) is an isomorphism.
In this paper, we prove a criterion for a predicate structure to be equationally Noetherian.
We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with countable underlying sets: totally ordered sets, fields, and groups. This answers an old question by B. Plotkin for groups.
The isomorphism problem for infinite finitely presented groups is probably the hardest among standard algorithmic problems in group theory. Classes of groups where it has been completely solved are nilpotent groups, hyperbolic groups, and limit groups. In this short paper, we address the problem of isomorphism to particular groups, including free groups. We also address the algorithmic problem of embedding a finitely presented group in a given limit group.
For a commutative finite Z- algebra, i.e., for a commutative ring R whose additive group is finitely generated, it is known that the group of units of R is finitely generated, as well. Our main results are algorithms to compute generators and the structure of this group. This is achieved by reducing the task first to the case of reduced rings, then to torsion-free reduced rings, and finally to an order in a reduced ring. The simplified cases are treated via a calculation of exponent lattices and various algorithms to compute the minimal primes, primitive idempotents, and other basic objects. All algorithms have been implemented and are available as a SageMath package. Whenever possible, the time complexity of the described methods is tracked carefully.
In this paper we analyze computational properties of the Diophantine problem (and its search variant) for spherical equations Pi(m)(i =1) z(i)(-1) c(i)z(i) = 1 (and their variants) over the class of finite metabelian groups G(p,n) = Z(p)(n) (sic) Z(p*), where n is an element of N and p is prime. We prove that the problem of finding solutions for certain constrained spherical equations is computationally hard on average (assuming that some lattice approximation problem is hard in the worst case).
