arXiv subjects

Daniel Studenmund

Publications and source records attributed to Daniel Studenmund.

15 recordsLinked to original sources

arXiv2026

Piecewise isometry groups of Euclidean tessellations

Given a tessellation of Euclidean or hyperbolic space, the piecewise isometry group is the group whose elements are given by cutting space into finitely many tessellated convex subsets and gluing them back together. Groups of piecewise isometries of tessellations generalize Houghton's groups and Thompson's group $V$, and for cubical tessellations were studied by Bieri and Sach. We prove structure results about groups of piecewise isometries of sufficiently nice tessellations of Euclidean space, such as tessellations associated to crystallographic root systems, in particular proving that they are elementary amenable. Future work in progress will prove finite generation and higher finiteness properties.

math.GR↗

arXiv2022

Counting flat cycles in the homology of locally symmetric spaces

Locally symmetric spaces like $SL(n,\mathbb Z)\backslash SL_n(\mathbb R)/SO(n)$ contain immersed compact flat manifolds of dimension equal to the real rank. We give a lower bound for the contribution of these cycles to the homology of congruence covers. Similar results are proved for other families of locally symmetric spaces.

math.NT↗

arXiv2021

Hall's universal group is a subgroup of the abstract commensurator of a free group

P. Hall constructed a universal countable locally finite group U, determined up to isomorphism by two properties: every finite group C is a subgroup of U, and every embedding of C into U is conjugate in U. Every countable locally finite group is a subgroup of U. We prove that U is a subgroup of the abstract commensurator of a finite-rank nonabelian free group.

math.GR↗

arXiv2021

Topological Models of Abstract Commensurators

The full solenoid over a topological space $X$ is the inverse limit of all finite covers. When $X$ is a compact Hausdorff space admitting a locally path connected universal cover, we relate the pointed homotopy equivalences of the full solenoid to the abstract commensurator of the fundamental group $π_1(X)$. The relationship is an isomorphism when $X$ is an aspherical CW complex. If $X$ is additionally a geodesic metric space and $π_1(X)$ is residually finite, we show that this topological model is compatible with the realization of the abstract commensurator as a subgroup of the quasi-isometry group of $π_1(X)$. This is a general topological analogue of work of Biswas, Nag, Odden, Sullivan, and others on the universal hyperbolic solenoid, the full solenoid over a closed surface of genus at least two.

math.GR↗

arXiv2021

The dualizing module and top-dimensional cohomology group of $\text{GL}_n(\mathcal{O})$

For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group. In contrast to $\text{SL}_n(\mathcal{O})$, we prove that the dualizing module of $\text{GL}_n(\mathcal{O})$ is sometimes the Steinberg module, but sometimes instead is a variant that takes into account a sort of orientation. Using this, we obtain vanishing and nonvanishing theorems for the cohomology of $\text{GL}_n(\mathcal{O})$ in its virtual cohomological dimension.

math.NT↗

arXiv2019

On abstract commensurators of surface groups

Let $Γ$ be the fundamental group of a surface of finite type and Comm$(Γ)$ be its abstract commensurator. Then Comm$(Γ)$ contains the solvable Baumslag--Solitar groups $\langle a ,b : a b a^{-1} = b^n \rangle$ for any $n > 1$. Moreover, the Baumslag--Solitar group $\langle a ,b : a b^2 a^{-1} = b^3 \rangle$ has an image in Comm$(Γ)$ that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm$(Γ)$ are concrete objects amenable to computational methods. For example, we give a proof that $\langle a ,b : a b^2 a^{-1} = b^3 \rangle$ is not residually finite without the use of normal forms of HNN extensions.

math.GR↗

arXiv2018

Commensurability growths of algebraic groups

Fixing a subgroup $Γ$ in a group $G$, the full commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ of $G$ with $[Γ: Γ\cap Δ][Δ: Γ\cap Δ] \leq n$. For pairs $Γ\leq G$, where $G$ is a Chevalley group scheme defined over $\mathbb{Z}$ and $Γ$ is an arithmetic lattice in $G$, we give precise estimates for the full commensurability growth, relating it to subgroup growth and a computable invariant that depends only on $G$.

math.GR↗

arXiv2018

Commensurability growth of branch groups

Fixing a subgroup $Γ$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ of $G$ with $[Γ: Γ\cap Δ][Δ: Γ\cap Δ] = n$. For pairs $Γ\leq A$, where $A$ is the automorphism group of a $p$-regular tree and $Γ$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $Γ$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this function is precisely $\aleph_0$ for any $n = p^k$.

math.GR↗

arXiv2018

Arithmetic lattices in unipotent algebraic groups

Fixing an arithmetic lattice $Γ$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ with $[Γ: Γ\cap Δ] [Δ: Γ\cap Δ] = n$. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smith's "Subgroups of finite index in nilpotent groups" apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic $\mathbb{Z}$-group with arithmetic lattice $Γ$, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in $p^{-s}$, where the degrees of the numerator and denominator are independent of $p$. This gives regularity results for the set of arithmetic lattices in $G$.

math.GR↗

arXiv2017

Semidualities from products of trees

Let $K$ be a global function field of characteristic $p$, and let $Γ$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $Γ$ is a $p$-torsion element. We define semiduality groups, and we show that $Γ$ is a $\mathbb{Z}[1/p]$-semiduality group if $Γ$ acts as a lattice on a product of trees. We also give other examples of semiduality groups, including lamplighter groups, Diestel-Leader groups, and countable sums of finite groups.

math.GR↗

arXiv2015

The topology of local commensurability graphs

We initiate the study of the $p$-local commensurability graph of a group, where $p$ is a prime. This graph has vertices consisting of all finite-index subgroups of a group, where an edge is drawn between $A$ and $B$ if $[A : A\cap B]$ and $[B: A\cap B]$ are both powers of $p$. We show that any component of the $p$-local commensurability graph of a group with all nilpotent finite quotients is complete. Further, this topological criterion characterizes such groups. In contrast to this result, we show that for any prime $p$ the $p$-local commensurability graph of any large group (e.g. a nonabelian free group or a surface group of genus two or more or, more generally, any virtually special group) has geodesics of arbitrarily long length.

math.GR↗

arXiv2015

Abstract commensurators of lattices in Lie groups

Let Gamma be a lattice in a simply-connected solvable Lie group. We construct a Q-defined algebraic group A such that the abstract commensurator of Gamma is isomorphic to A(Q) and Aut(Gamma) is commensurable with A(Z). Our proof uses the algebraic hull construction, due to Mostow, to define an algebraic group H so that commensurations of Gamma extend to Q-defined automorphisms of H. We prove an analogous result for lattices in connected linear Lie groups whose semisimple quotient satisfies superrigidity.

math.GR↗

arXiv2015

Full residual finiteness growths of nilpotent groups

Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of $G$ has finite index in the center of $G$ we show that the growth is precisely $n^b$, where $b$ is the product of the nilpotency class and dimension of $G$. In the general case, we give a method for finding an upper bound of the form $n^b$ where $b$ is a natural number determined by what we call a terraced filtration of $G$. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.

math.GR↗

arXiv2014

Nonarchimedean superrigidity of solvable S-arithmetic groups

Let Gamma be an S-arithmetic subgroup of a solvable algebraic group G over an algebraic number field F, such that the finite set S contains at least one place that is nonarchimedean. We construct a certain group H, such that if L is any local field and alpha is any homomorphism from Gamma to GL(n,L), then alpha virtually extends (modulo a bounded error) to a continuous homomorphism defined on some finite-index subgroup of H. In the special case where F is the field of rational numbers, the real-rank of G is 0, and Gamma is Zariski-dense in G, we may let H = G_S. We also point out a generalization that does not require G to be solvable.

math.GR↗

arXiv2013

Commensurators of solvable S-arithmetic groups

We show that the abstract commensurator of an S-arithmetic subgroup of a solvable algebraic group over Q is isomorphic to the Q-points of an algebraic group, and compare this with examples of nonlinear abstract commensurators of S-arithmetic groups in positive characteristic. In particular, we include a description of the abstract commensurator of the lamplighter group.

math.GR↗