
Higher modular groups as amalgamated products and a dichotomy for integral group rings, J. Algebra 604 (2022), 185–223
joint with A. Bächle, E. Jespers, A. Kiefer and D. Temmerman
arXiv 
▸ Abstract

▸ We show that $\mathcal{U}(\mathbb{Z}G)$, the unit group of the integral group ring $\mathbb{Z} G$, either
satisfies Kazhdan's property (T) or is, up to commensurability, a nontrivial amalgamated product, in case $G$ is a finite group
satisfying some mild conditions. Crucial in the proof is the construction of amalgamated decompositions of the elementary group $\operatorname{E}_2(\mathcal{O})$,
where $\mathcal{O}$ is an order in a rational division algebra. A major step is to introduce subgroups $\operatorname{E}_2(\Gamma_n(\mathbb{Z}))$ inside the
socalled higher modular groups $\operatorname{SL}_+(\Gamma_n(\mathbb{Z}))$, which are discrete subgroups of certain $2 \times 2$ matrix groups with
entries in a Clifford algebra. The groups $\operatorname{E}_2(\Gamma_n(\mathbb{Z}))$ mimic the elementary groups in linear groups over rings. We prove that
$\operatorname{E}_2(\Gamma_n(\mathbb{Z}))$ has in general a nontrivial decomposition as a free product with amalgamated subgroup $\operatorname{E}_2(\Gamma_{n1}(\mathbb{Z}))$.
From this we obtain that also the higher modular groups do have a very clearly structured amalgam decompositions in low dimensions.

Codimension Growth of Lie algebras with a generalized action, 2021,
Proc. Amer. Math. Soc. , 150 (2022), no. 7, 2741–2754
arXiv 
▸ Abstract

▸ Let $L$ be a finite dimensional Lie $F$algebra endowed with a generalized action by an associative algebra $H$.
We investigate the exponential growth rate of the sequence of Hgraded codimensions $C_n^H(L)$ of $L$ which is a measure for the number
of nonpolynomial $H$identities of $L$. More precisely, we construct the first example of an $S$graded Lie algebra having a noninteger,
even irrational, exponential growth rate $\lim_{n \rightarrow \infty} \sqrt[n]{c_n^S(L)}$. Hereby $S$ is a semigroup and an exact value is given. On the other hand,
returning to general $H$, if $L$ is semisimple and also semisimple for the $H$action we prove the analog of Amitsur's conjecture (i.e. $\lim_{n \rightarrow \infty} \sqrt[n]{c_n^H(L)} \in \mathbb{Z}$).
Moreover if $H=FS$ is a semigroup algebra the semisimplicity on $L$ can be dropped which is in strong contract to the associative setting.

Abelianization and fixed point properties of linear groups and units in integral group rings, Math. Nachr. , 296 (2023), no. 1, 8–56
joint with A. Bächle, E. Jespers, A. Kiefer and D. Temmerman
arXiv 
▸ Abstract

▸ Fixed point properties and the abelianization of arithmetic subgroups $\Gamma$ of $\operatorname{SL}_n(D)$ and its elementary
subgroup $\operatorname{E}_n(D)$ are well understood except in the degenerate case of lower rank, i.e. $n=2$ and $\Gamma = \operatorname{SL}_2(\mathcal{O})$
with $\mathcal{O}$ an order in a division algebra $D$ with a finite number of units. In this setting we determine Serre's property
$\operatorname{FA}$ for $\operatorname{E}_2(\mathcal{O})$ and its subgroups of finite index. We construct a generic and computable exact sequence
describing its abelianization, affording a closed formula for its $\mathbb{Z}$rank. Thenceforth, we investigate applications in
integral representation theory of finite groups. We obtain a characterization of when the unit group $\mathcal{U}(\mathbb{Z}G)$ of
the integral group ring $\mathbb{Z} G$ satisfies Kazhdan's property $(\operatorname{T})$, both in terms of the finite group $G$ and
in terms of the simple components of the semisimple algebra $\mathbb{Q}G$. Furthermore, it is shown that for $\mathcal{U}( \mathbb{Z} G)$
this property is equivalent to a hereditary version of property $\operatorname{FA}$, denoted $\operatorname{HFA}$, and even the significantly
weaker property $\operatorname{FAb}$ (i.e. every subgroup of finite index has finite abelianization). A crucial step for this is a reduction to
arithmetic groups $\operatorname{SL}_n(\mathcal{O})$ and finite groups $G$ which have the socalled cut property.
For such groups $G$ we describe the simple epimorphic images of $\mathbb{Q} G$.

A glimpse into the Asymptotics of polynomial identities, Nieuw Arch. Wiskd. (5) 18 (2017), no. 1, 6064
Journal 
▸ Abstract

▸ This a survey paper on the most significant results in the combinatorial theory of polynomial identities.

Free products in the unit group of the integral group ring of a finite group, Proc. Amer. Math. Soc. 145 (2017), no. 7, 27712783
joint with E. Jespers and D. Temmerman
arXiv 
Journal 
▸ Abstract

▸ Let $G$ be a finite group and let $p$ be a prime. We continue the search for generic constructions of free products and
free monoids in the unit group $\mathcal{U}(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z} G$. For a nilpotent group $G$ with a
noncentral element $g$ of order $p$, explicit generic constructions are given of two periodic units $b_1$ and $b_2$ in
$\mathcal{U}(\mathbb{Z}G)$ such that $\langle b_1 , b_2\rangle =\langle b_1\rangle \star \langle b_2 \rangle =\mathbb{Z}_p \star \mathbb{Z}_{p}$,
a free product of two cyclic groups of prime order. Moreover, if $G$ is nilpotent of class $2$ and $g$ has order $p^n$, then also concrete
generators for free products $\mathbb{Z}_{p^k} \star \mathbb{Z}_{p^m}$ are constructed (with $1\leq k,m\leq n $). As an application, for finite nilpotent
groups, we obtain earlier results of MarciniakSehgal and GoncalvesPassman. Further, for an arbitrary finite group $G$ we give
generic constructions of free monoids in $\mathcal{U}(\mathbb{Z}G)$ that generate an infinite solvable subgroup.

A reduction theorem for $\tau$rigid modules, Math. Z. (2018), 137
joint with F. Eisele and T. Raedschelders
arXiv 
Journal (open access) 
GAPcode 
▸ Abstract

▸ We prove a theorem which gives a bijection between the support $\tau$tilting modules over a given finitedimensional algebra
A and the support τtilting modules over $A/I$, where $I$ is the ideal generated by the intersection of the center of $A$ and the radical of $A$.
This bijection is both explicit and wellbehaved. We give various corollaries of this, with a particular focus on blocks of group rings of
finite groups. In particular we show that there are $\tau$tilting finite wild blocks with more than one simple module. We then go on to classify
all support $\tau$tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame
blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic twoterm tilting complexes,
and it turns out that a tame block has at most $32$ different basic twoterm tilting complexes. We do this by using the aforementioned
reduction theorem, which reduces the problem to ten different algebras only depending on the ground field $k$, all of which happen to be
string algebras. To deal with these ten algebras we give a combinatorial classification of all $\tau$rigid modules over
(not necessarily symmetric) string algebras

The Polynomial Part of the Codimension Growth of Affine PI algebras, Adv. Math. 309 (2017), 487511.
joint with E.Aljadeff and Y.Karasik
arXiv 
Journal 
▸ Abstract

▸ Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$algebra satisfying a polynomial identity (PI).
The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $\theta(cn^td^n)$, where $d$ is the well known (PI) exponent of $W$.
In this paper we establish an algebraic interpretation of the polynomial part (the constant $t$) by means of Kemer's theory. In particular,
we show that in case $W$ is a basic algebra, then $t=\frac{dq}{2}+s$, where $q$ is the number of simple component in $W/J(W)$ and $s+1$ is
the nilpotency degree of $J(W)$. Thus proving a conjecture of Giambruno.

Semigroup graded algebras and graded PIexponent, Israel J. Math. 220 (2017), no. 1, 387452.
joint with A.Gordienko and E. Jespers
arXiv 
Journal 
▸ Abstract

▸ Let $S$ be a semigroup. We study the structure of gradedsimple $S$graded algebras $A$ and the exponential rate
$\operatorname{PIexp}^{S}(A) := \lim\limits_{n \rightarrow \infty} \sqrt[n]{c_n^{S}(A)}$ of growth of
codimensions $c_n^{S}(A)$ of their graded polynomial identities. This is of great interest since such algebras
can have noninteger $\operatorname{PIexp}^{S}(A)$ despite being finite dimensional and associative. In addition, such algebras
can have a nontrivial Jacobson radical $J(A)$. All this is in strong contrast with the case when $S$ is a group since in the group case
$J(A)$ is trivial, $\operatorname{PIexp}^{S}(A)$ is always integer and, if the base field is algebraically closed, then
$\operatorname{PIexp}^{S}(A)$ equals $\operatorname{dim} A$. Without any restrictions on the base field $F$, we classify gradedsimple $S$graded
algebras $A$ for a class of semigroups $S$ which is complementary to the class of groups. We explicitly describe the structure of $J(A)$ showing that $J(A)$
is build up of pieces of a maximal $S$graded semisimple subalgebra of $A$ which turns out to be simple. When $F$ is algebraically closed,
we get an upper bound for $\mathop{\overline\lim}_{n\to\infty}\sqrt[n]{c_n^{S}}(A)$. If $A/J(A)\cong M_2(F)$
and $S$ is a right zero band, we show that this upper bound is sharp and $\operatorname{PIexp}^{S\text{}\mathrm{gr}}(A)$ indeed exists.
In particular, we present an infinite family of gradedsimple algebras $A$ with arbitrarily large noninteger
$\operatorname{PIexp}^{S}(A)$.

$\mathbb{Z}$S_{n}modules and polynomial identities with integer coefficients, Int. J. Algebra Comput. 23 (2013), no. 8, 19251943.
joint with A.Gordienko
arXiv 
Journal 
▸ Abstract

▸ We show that, like in the case of algebras over fields, the study of multilinear polynomial identities of unitary rings can be
reduced to the study of proper polynomial identities. In particular, the factors of series of $\mathbb{Z}S_n$submodules in the
$\mathbb{Z}S_n$modules of multilinear polynomial functions can be derived by the analog of Young's (or Pieri's) rule from the factors of
series in the corresponding $\mathbb{Z}S_n$modules of proper polynomial functions. As an application, we calculate the codimensions and a
basis of multilinear polynomial identities of unitary rings of upper triangular $2\times 2$ matrices and infinitely generated Grassmann
algebras over unitary rings. In addition, we calculate the factors of series of $\mathbb{Z}S_n$submodules for these algebras. Also we
establish relations between codimensions of rings and codimensions of algebras and show that the analog of Amitsur's conjecture holds
in all torsionfree rings, and all torsionfree rings with $1$ satisfy the analog of Regev's conjecture.

Primitive Central Idempotents of Rational Group Algebras, J. Algebra Appl. 12 (2013), no. 1, 125130.
arXiv 
Journal 
▸ Abstract

▸ We give a description of the primitive central idempotents of the rational group algebra $\mathbb{Q}G$ of a finite group $G$. Such a
description is already investigated by Jespers, Olteanu and del Río, but some unknown scalars are involved. Our description also gives
answers to their questions.