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
Abstract. Let S be a finite semigroup and let A be a finite dimensional S-graded algebra. We investigate the exponential rate of growth of the sequence of graded codimensions c S n (A) of A, i.e lim n→∞ n c S n (A). For group gradings this is always an integer. Recently in [20] the first example of an algebra with a non-integer growth rate was found. We present a large class of algebras for which we prove that their growth rate can be equal to arbitrarily large non-integers. An explicit formula is given. Surprisingly, this class consists of an infinite family of algebras simple as an S-graded algebra. This is in strong contrast to the group graded case for which the growth rate of such algebras always equals dim(A). In light of the previous, we also handle the problem of classification of all S-graded simple algebras, which is of independent interest. We achieve this goal for an important class of semigroups that is crucial for a solution of the general problem.
We prove a theorem which gives a bijection between the support τ-tilting modules over a given finite-dimensional algebra A and the support τ-tilting 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 well-behaved. 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 τ-tilting-finite wild blocks with more than one simple module. We then go on to classify all support τtilting 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 two-term tilting complexes, and it turns out that a tame block has at most 32 different basic two-term 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 τ-rigid modules over (not necessarily symmetric) string algebras.
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