We exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank $2$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{SL}_2(\mathbb{R})$. This algorithm, together with methods for checking whether a two-generator subgroup of $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{SL}_2(\mathbb{R})$ is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields.

Supplementary materials are available with this article.

We give a computationally effective criterion for determining whether a finite-index subgroup of $\mathrm{SL}_2(\mathbf{Z})$ is a congruence subgroup, extending earlier work of Hsu for subgroups of $\mathrm{PSL}_2(\mathbf{Z})$.

We describe algorithms that allow the computation of fundamental domains in the Bruhat–Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.

We show that the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on its natural module is isomorphic to the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on the union of the right cosets of ${ \mathrm{O} }_{2m}^{+ } ({2}^{f} )$ and ${ \mathrm{O} }_{2m}^{- } ({2}^{f} )$.

We deal with aspects of direct and inverse problems in parameterized Picard–Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) $G$ is a PPV Galois group over these fields if and only if $G$ contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs $G$, including unipotent groups, $G$ is such a group if and only if it has differential type $0$. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a parameterized linear differential equation if its Galois group has differential type $0$.

We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature $(3, n, \infty )$, with $n\geq 4$. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.

We describe an effective algorithm to compute a set of representatives for the conjugacy classes of Hall subgroups of a finite permutation or matrix group. Our algorithm uses the general approach of the so-called ‘trivial Fitting model’.

Let G⊂SU(2,1) be nonelementary and S be its minimal generating system. In this paper, we show that if S satisfies some conditions, then S can be replaced by a minimal generating system S1consisting only of loxodromic elements.

The normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth ndim (G).

In this note, we prove that the Gauss–Picard modular group PU(2,1;Θ1) has Property (FA). Our result gives a positive answer to a question by Stover [‘Property (FA) and lattices in SU(2,1)’, Internat. J. Algebra Comput.17 (2007), 1335–1347] for the group PU(2,1;Θ1).

We describe a practical algorithm for primitivity testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For an imprimitive group, a system of imprimitivity can be constructed. An implementation of the algorithm in Magma is publicly available.

It is well known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of SL(2,ℝ), then its translates by the group form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such a hyperbolically convex fundamental domain for any discrete subgroup can be obtained by using Dirichlet’s and Ford’s polygon constructions. However, these two results are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction. A third, and most important and practical, method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. If Γ2 is a subgroup of Γ1 such that Γ1=Γ2⋅{L1,L2,…,Lm} and 𝔽 is the closure of a fundamental domain of the bigger group Γ1, then the set is a fundamental domain of Γ2. One can ask at this juncture, is it possible to choose the right coset suitably so that the set ℛ is a convex hyperbolic polygon? We will answer this question affirmatively for Hecke modular groups.

In this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups of PSL(2,ℤ). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integer f, which determines a maximal discrete subgroup Γ0(f)+ of SL(2,ℝ); a decision procedure to determine whether a given element of Γ0(f)+ is in a subgroup G; and the index of G in Γ0(f)+. The output consists of: a fundamental domain for G, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators of G. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup.

J. W. Anderson (1996) asked whether two finitely generated Kleinian groups with the same set of axes are commensurable. We give some partial solutions.

Let R be a ring with 1 and En (R) be the subgroup of GLn(R) generated by the matrices I + reij, r ∈ R, i ≠ j. We prove that the subgroup of consisting of the matrices of shape , where and , is (2, 3, 7)-generated whenever R is finitely generated and n, are large enough.

Two projective nonsingular complex algebraic curves X and Y defined over the field R of real numbers can be isomorphic while their sets X(R) and Y(R) of R-rational points could be even non homeomorphic. This leads to the count of the number of real forms of a complex algebraic curve X, that is, those nonisomorphic real algebraic curves whose complexifications are isomorphic to X. In this paper we compute, as a function of genus, the maximum number of such real forms that a complex algebraic curve admits.

We study commutators in pseudo-orthogonal groups O2nR (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO2nR. We estimate the number of commutators, c(O2nR) and c(GO2nR), needed to represent every element in the commutator subgroup. We show that c(O2nR) ≤ 4 if R satisfies the ∧-stable condition and either n ≥ 3 or n = 2 and 1 is the sum of two units in R, and that c(GO2nR) ≤ 3 when the involution is trivial and ∧ = R∈. We also show that c(O2nR) ≤ 3 and c(GO2nR) ≤ 2 for the ordinary orthogonal group O2nR over a commutative ring R of absolute stable rank 1 where either n ≥ 3 or n = 2 and 1 is the sum of two units in R.

We provide a number of explicit examples of small volume hyperbolic 3-manifolds and 3-orbifolds with various geometric properties. These include a sequence of orbifolds with torsion of order q interpolating between the smallest volume cusped orbifold (q = 6) and the smallest volume limit orbifold (q → ∞), hyperbolic 3-manifolds with automorphism groups with large orders in relation to volume and in arithmetic progression, and the smallest volume hyperbolic manifolds with totally geodesic surfaces. In each case we provide a presentation for the associated Kleinian group and exhibit a fundamental domain and an integral formula for the co-volume. We discuss other interesting properties of these groups.

Every invertible n-by-n matrix over a ring R satisfying the first Bass stable range condition is the product of n simple automorphisms, and there are invertible matrices which cannot be written as the products of a smaller number of simple automorphisms. This generalizes results of Ellers on division rings and local rings.

In this note, for any given simple group obtained from an orthogonal or unitary group of non-zero index, by a procedure similar to the construction of Chevalley groups and twisted groups, we construct a simple group which is identified with the given simple classical group. The simple groups constructed in this note can be interpreted as generalized simple groups of Lie type. Thus all simple groups of Lie type of types An, Bn, Cn and Dn and all generalized simple groups of Lie type constructed in this note exhaust all simple classical groups with non-zero indices.