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We study a family of groups consisting of the simplest extensions of lamplighter groups. We use these groups to answer multiple open questions in combinatorial group theory, providing groups that exhibit various combinations of properties:
Decidable Subgroup Membership and undecidable Uniform Subgroup Membership Problem.
Rational volume growth series and undecidable Word Problem.
Recursive (even context-free) language of conjugacy geodesics, decidable Word Problem, and undecidable Conjugacy Problem.
We also consider the co-Word Problem, residual finiteness and the Isomorphism Problem within this class.
In this paper, we relate the Identity Problem with problems regarding ordered groups. For instance, in torsion-free nilpotent groups, it is equivalent to the problem asking if a given subset of the group extends to the positive cone of a left-order, and to the Word Problem in an associated lattice-ordered group. Our results are the following:
We give an independent proof that the Identity and Subgroup Problems are decidable in finitely generated nilpotent groups, solving by the same occassion the Word Problem for a family of lattice-ordered groups.
Reciprocally, the (known) decidability in some lattice-ordered groups allows us to solve the Normal Identity Problem in free nilpotent groups.
A related problem, the Fixed-Target Submonoid Membership is shown to be undecidable in some nilpotent groups.
We also explore the connection in a family of metabelian groups.
We provide the first example of virtually nilpotent group (together with a specific generating set) for which the Green series is not D-finite. The proof relies on a arithmetical miracle allowing us to count the number of solutions of a Diophantine equation, and the study of the subword complexity of multiplicative sequences.
We study the horofunction boundary of finitely generated nilpotent groups, and the natural group action on it. More specifically, we prove the followings:
For discrete Heisenberg groups, we classify the orbits of Busemann points. As a byproduct, we observe that the set of orbits is finite and the set of Busemann points is countable. Furthermore, by using the approximation with Lie groups, we observe that the entire horoboundary is uncountable.
For the discrete Cartan group, we exhibit an continuum of Busemann points, disproving a conjecture of Tointon and Yadin. As a byproduct, we prove that the group acts non-trivially on its reduced horoboundary, disproving a conjecture of Bader and Finkelshtein.
In this note, we provide examples of groups with transcendental spectral radius. We provide two constructions:
The first one provides finitely presented examples, and relies on links between decidability of the Word Problem and semi-computability of the spectral radius.
The second provides examples with decidable Word Problem, and relies on small-cancellation and the Rapid Decay property.
The Identity Problem in virtually solvable matrix groups over algebraic numbers, joint w/ Ruiwen Dong, arXiv link
The Identity Problem in virtually solvable matrix groups over algebraic numbers, joint w/ Ruiwen Dong, arXiv link
The Tits alternative states that a finitely generated matrix group either contains a nonabelian free subgroup , or it is virtually solvable. This paper considers two decision problems in virtually solvable matrix groups: the Identity Problem (does a given finitely generated subsemigroup contain the identity matrix?), and the Group Problem (is a given finitely generated subsemigroup a group?). We show that both problems are decidable in virtually solvable matrix groups over the field of algebraic numbers. Our proof extends the decidability result for nilpotent groups by Bodart, Ciobanu, Metcalfe and Shaffrir, and for metabelian groups by Dong (STOC'24). Since the Identity Problem and the Group Problem are known to be undecidable in matrix groups containing F2×F2, our result significantly reduces the decidability gap for both decision problems.
We study both the Submonoid Membership problem (SMP) and the Rational Subset Membership problem (RSMP) in finitely generated nilpotent groups. We give two reductions with important applications:
The SMP in any nilpotent group can be reduced to the RSMP in smaller groups. As a corollary, we prove the existence of a group with decidable SMP and undecidable RSMP, answering a question of Lohrey and Steinberg.
The Rational Subset Membership problem in H_3(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 give a criterion on pairs (G,S) - where G is a virtually s-step nilpotent group and S is a finite generating set - saying whether the geodesic growth is exponential or strictly sub-exponential. Whenever s=1,2, this goes further and we prove the geodesic growth is either exponential or polynomial. For s≥3 however, intermediate growth is possible. We provide an example of virtually 3-step nilpotent group with geodesic growth asymptotic to exp(n^{3/5} log(n)). This is the first known example of group with intermediate geodesic growth. Along the way, we prove results on the geometry of virtually nilpotent groups of independent interest, including asymptotics with error terms for their volume growth.
Dead ends and rationality of complete growth series, joint w/ Pierre Bagnoud, journal link, arXiv link
Dead ends and rationality of complete growth series, joint w/ Pierre Bagnoud, journal link, arXiv link
We are interested in the NG-rationality and NG-algebraicity of the complete growth series of finitely generated groups. It is shown that dead ends of arbitrarily large depths form an obstruction to NG-rationality. In the case of the 3-dimensional Heisenberg group H_3(Z), we prove that the complete series is not NG-algebraic for any generating set. Dead ends are also used to show that complete growth series of higher Heisenberg groups are not NG-rational for specific generating sets. Using a more general version of this obstruction, we prove that complete growth series of some lamplighter groups are not NG-rational either. This work provides the first examples of groups exhibiting a difference between rationality of standard growth series, and rationality of complete growth series.
N.B. Some additionnal results can be found in my thesis.
We provide new examples of groups without rational cross-sections (also called regular normal forms), using connections with bounded generation and rational orders on groups. Our examples contain a finitely presented HNN extension of the first Grigorchuk group. This last group is the first example of finitely presented group with solvable word problem and without rational cross-sections. It is also not autostackable, and has no left-regular complete rewriting system.
Algorithmic and geometric aspects of nilpotent groups (Doctoral Thesis)
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