arXiv subjects
Publications and source records attributed to Stefan Witzel.
Heinrich Tietze has shown that for a closed connected subset of euclidean space being convex is a local property. We generalize this to CAT(0)-spaces and locally compact CAT(κ) spaces. As an application we give a construction of certain convex sets in euclidean buildings.
We prove that the Brin-Thompson groups sV, also called higher dimensional Thompson's groups, are of type F_\infty for all natural numbers s. This result was previously shown for s up to 3, by considering the action of sV on a naturally associated space. Our key step is to retract this space to a subspace sX which is easier to analyze.
We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it is related to the embedding into a right-angled Coxeter group described by Davis and Januszkiewicz. We compare the approaches and also adapt the argument that the action extends to finite index supergroup that is a graph product of finite groups.
It is known that the Burau representation of the 4-strand braid group is faithful if and only if certain matrices f and k generate a (non-abelian) free group. Regarding f and k as isometries of a euclidean building we show that f^3 and k^3 generate a free group. We give two proofs, one utilizing the metric geometry of the building, and the other using simplicial retractions.
We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m-1 and classical finiteness length n-1 for all 0 < m <= n. The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of Brown's criterion due to Martin Fluch and the author.
In these notes we determine the finiteness length of the groups G(O_S) where G is an F_q-isotropic, connected, noncommutative, almost simple F_q-group and O_S is one of F_q[t], F_q[t^{-1}], and F_q[t,t^{-1}]. That is, k = F_q(t) and S contains one or both of the places s_0 and s_\infty corresponding to the polynomial p(t) = t respectively to the point at infinity. The statement is that the finiteness length of G(O_S) is n-1 if S contains one of the two places and is 2n-1 if it contains both places, where n is the F_q-rank of G. For example, the group SL_3(F_q[t,t^{-1}]) is of type F_3 but not of type F_4, a fact that was previously unknown.
We translate Brown's criterion for homological finiteness properties to the setting of Bredon homology.
We adapt and refine methods developed by Abramenko and Devillers--Köhl--Mühlherr in order to establish the sphericity of the Phan geometries of type B_n and C_n, and their generalizations. As an application we determine the finiteness length of the unitary form of certain hyperbolic Kac--Moody groups. We also reproduce the finiteness length of the unitary form of the groups Sp_{2n}(GF(q^2)[t,t^{-1}]). Another application is the first published proof of the Phan-type theorem of type F_4. Within the revision of the classification of the finite simple groups this concludes the revision of Phan's theorems and their extension to the non-simply laced diagrams. We also reproduce the Phan-type theorems of types B_n and C_n.
We show that the finiteness length of an $S$-arithmetic subgroup $Γ$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places in $S$. This determines the finiteness properties for arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups. Our main tool is Behr-Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.
It is known from work by H. Abels and P. Abramenko that for a classical Fq-group G of rank n the arithemetic lattice G(Fq[t]) of Fq[t]-rational points is of type Fn-1 provided that q is large enough. We show that the statement is true without any assumption on q and for any isotropic, absolutely almost simple group G defined over Fq.
We show that if G is a Chevalley group of rank n and F_q[t,t^{-1}] is the ring of Laurent polynomials over a finite field, then G(F_q[t,t^{-1}]) is of type F_{2n-1}. This bound is optimal because it is known -- and we show again -- that the group is not of type F_{2n}.