Tag - Linear groups

Umberto Zannier: Bounded generation in linear groups and exponential parametrizations

31st December, 2022

In fairly recent joint work with Corvaja, Rapinchuk, Ren, we applied results from Diophantine S-unit theory to problems of 'bounded generation' in linear groups: this property is a strong form of finite generation and is useful for several issues in the setting. Focusing on 'anisotropic groups' (i.e. containing only semi-simple elements), we could give a simple essentially complete description of those with the property. More recently, in further joint work also with Demeio, we proved the natural expectation that sets boundedly generated by semi-simple elements (in linear groups over number fields) are 'sparse'. Actually, this holds for all sets obtained by exponential parametrizations. As a special consequence, this gives back the previous results with a different approach and additional precision and generality.

Volker Diekert: Decidability of membership problems for 2×2 matrices over ℚ

7th October, 2022

We consider membership problems in matrix semigroups. Using symbolic algorithms on words and finite automata, we prove various new decidability results for 2×2 matrices over ℚ. For that, we introduce the concept of flat rational sets: if M is a monoid and N is a submonoid, then flat rational sets of M over N are finite unions of the form L0g1L1⋯gtLt where all Li are rational subsets of N and gi ∈ M. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL2(ℚ) over GL2(ℤ) is decidable (in singly exponential time). It is possible that such a strong decidability result cannot be pushed any further for groups sitting between GL2(ℤ) and GL2(ℚ).

We also show a dichotomy for non-trivial group extension of GL2(ℤ) in GL2(ℚ): if G is a f.g. group such that GL2(ℤ) < G ≤ GL2(ℚ), then either G ≅ GL2(ℤ) × ℤk, for some k ≥ 1, or G contains an extension of the Baumslag-Solitar group BS(1,q), with q ≥ 2, of infinite index. In the first case of the dichotomy the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable. In the second case, decidability of the membership problem for rational subsets in G is open.

Our result improves various natural decidability results for 2×2 matrices with rational entries, and it also supports them with concrete complexity bounds for the first time.

Claudio Bravo: Quotients of the Bruhat-Tits tree function field analogues of the Hecke congruence subgroups

1st July, 2022

Let C be a smooth, projective, and geometrically connected curve defined over a finite field F. For each closed point P∞ of C, let R be the ring of functions that are regular outside P∞, and let K be the completion path P∞ of the function field of C. In order to study groups of the form GL2(R), Serre describes the quotient graph GL2(R)∖T, where T is the Bruhat-Tits tree defined from SL2(K). In particular, Serre shows that GL2(R)∖T is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. It is not hard to see that finite index subgroups inherit this property. In this exposition we describe the quotient graph H∖T defined from the action on T of the group H consisting of matrices that are upper triangular modulo I, where I is an ideal of R. More specifically, we give an explicit formula for the cusp number H∖T. Then By, using Bass-Serre theory, we describe the combinatorial structure of H. These groups play, in the function field context, the same role as the Hecke Congruence subgroups of SL2(ℤ). Moreover, not that the groups studied by Serre correspond to the case where the ideal I coincides with the ring R.

Eugene Plotkin: On logical rigidity of groups

4th December, 2020

We will survey a series of recent developments in the area of first-order descriptions of linear groups. The goal is to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups and Kac-Moody groups. We also dwell on the principal problem of isotipicity of finitely generated groups.