The integral group ring $\mathbb{Z} G$ of a group $G$ has only trivial central units, if the only central units of $\mathbb{Z} G$ are $\pm z$ for $z$ in the center of $G$. We show that the order of a finite solvable group $G$ with this property, can only have $2$, $3$, $5$ and $7$ as prime divisors, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.Comment: [v4]: 13 pages. Final version. To appear in "Forum Mathematicum" (already available online

Abstract. We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus Conjecture for the group PSL(2, 19). We also prove the Zassenhaus Conjecture for PSL (2, 23). In a second application we show that there are no normalized units of order 6 in the integral group rings of M 10 and PGL(2, 9).This completes the proof of a theorem of W. Kimmerle and A. Konovalov that the Prime Graph Question has an affirmative answer for all groups having an order divisible by at most three different primes.Througout this paper let G be a finite group, ZG the integral group ring of G and V(ZG) the group of augmentation one units in ZG, aka. normalized units. The most famous open conjecture regarding torsion units in ZG is:The Zassenhaus Conjecture (ZC): Let u ∈ V(ZG) be a torsion unit. Then there exist a unit x ∈ QG and g ∈ G such that x −1 ux = g.If for a unit u such x and g exist we say that u is rationally conjugate to g.

When considering the unit group of O F G (O F the ring of integers of an abelian number field F and a finite group G) certain components in the Wedderburn decomposition of F G cause problems for known generic constructions of units; these components are called exceptional. Exceptional components are divided into two types: type 1 are division rings, type 2 are 2 × 2-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (w.r.t. quotients) having those division rings in their Wedderburn decomposition over F . We also classify the exceptional components of type 2 appearing in group algebras of a finite group over number fields F by describing all 58 finite groups G having a faithful exceptional Wedderburn component of this type in F G.

Fixed point properties and the abelianization of arithmetic subgroups Γ of SLn(D) and its elementary subgroup En(D) are well understood except in the degenerate case of lower rank, i.e. n = 2 and Γ = SL2(O) with O an order in a division algebra D with a finite number of units. In this setting we determine Serre's property FA for E2(O) and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its Z-rank. Thenceforth, we investigate applications in integral representation theory of finite groups. We obtain a characterization of when the unit group U(ZG) of the integral group ring ZG satisfies Kazhdan's property (T), both in terms of the finite group G and in terms of the simple components of the semisimple algebra QG. Furthermore, it is shown that for U(ZG) this property is equivalent to a hereditary version of property FA, denoted HFA, and even the significantly weaker property FAb (i.e. every subgroup of finite index has finite abelianization). A crucial step for this is a reduction to arithmetic groups SLn(O) and finite groups G which have the so-called cut property. For such groups G we describe the simple epimorphic images of QG. Contents 1 2 A. B ÄCHLE, G. JANSSENS, E. JESPERS, A. KIEFER, AND D. TEMMERMAN 5. Property FR and HFR for E 2 (O) 29 5.1. Property FR for the groups G R,K with applications to FR for E 2 (O) 30 5.2. Property FR for the Borel with a view on GE 2 (O) 34 Chapter III. Applications to U (ZG) 36 6. Exceptional components and cut groups 36 6.1. FA and cut groups 36 6.2. Higher rank and exceptional components 38 6.3. Exceptional components of cut groups 40 7. Property HFA 43 8. Property FA 47 Appendix A. Groups with faithful exceptional 2 × 2 components 51 Appendix B. Some Group Presentations 53 References 54

In this article the study of the Prime Graph Question for the integral group ring of almost simple groups which have an order divisible by exactly 4 different primes is continued. We provide more details on the recently developed "lattice method" which involves the calculation of Littlewood-Richardson coefficients. We apply the method obtaining results complementary to those previously obtained using the HeLP-method. In particular the "lattice method" is applied to infinite series of groups for the first time. We also prove the Zassenhaus Conjecture for four more simple groups. Furthermore we show that the Prime Graph Question has a positive answer around the vertex 3 provided the Sylow 3-subgroup is of order 3.Prime Graph Question (PQ). If V(ZG) contains a unit of order pq, does G have an element of order pq, where p and q are different primes? This is equivalent to say that the prime graphs of G and V(ZG) coincide. The Prime Graph Question seems especially approachable since W. Kimmerle and A. Konovalov proved a strong reduction: The Prime Graph Question has a positive answer for some group G if and only if it has a positive answer for all almost simple images of G [KK17, Theorem 2.1]. A group is called almost simple if it is sandwiched between a non-abelian simple group S and the automorphism group of S. In this case S is called the socle of G.