Searcharxiv⌕ Search

arXiv subjects

Kai-Uwe Bux

Publications and source records attributed to Kai-Uwe Bux.

34 records · Page 2Linked to original sources
arXiv2011

The congruence subgroup property for $Aut F_2$: A group-theoretic proof of Asada's theorem

The goal of this paper is to give a group-theoretic proof of the congruence subgroup property for $Aut(F_2)$, the group of automorphisms of a free group on two generators. This result was first proved by Asada using techniques from anabelian geometry, and our proof is, to a large extent, a translation of Asada's proof into group-theoretic language. This translation enables us to simplify many parts of Asada's original argument and prove a quantitative version of the congruence subgroup property for $Aut(F_2)$.

math.GR↗
arXiv2011

Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem

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.

math.GR↗
arXiv2010

Finiteness Properties of Chevalley Groups over a Polynomial Ring over a Finite Field

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.

math.GR↗
arXiv2008

Connectivity Properties of Horospheres in Euclidean Buildings and Applications to Finiteness Properties of Discrete Groups

Let G(O_S) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O_S) established in an earlier paper is sharp in this case. The geometric analysis underlying our result determines the conectivity properties of horospheres in thick Euclidean buildings.

math.GR↗
arXiv2008

Some Remarks on the Braided Thompson Group BV

Matthew Brin and Patrick Dehornoy independently discovered a braided version BV of Thompson's group V. In this paper, we discuss some properties of BV that might make the group interesting for group based cryptography. In particular, we show that BV does not admit a non-trivial linear representation.

math.GR↗
arXiv2007

The Dimension of the Torelli group

We prove that the cohomological dimension of the Torelli group for a closed connected orientable surface of genus g at least 2 is equal to 3g-5. This answers a question of Mess, who proved the lower bound and settled the case of g=2. We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be 2g-3. For g at least 2, we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the "complex of cycles", on which the Torelli group acts.

math.GT↗
arXiv2006

Dimension of the Torelli group for Out(F_n)

Let T_n be the kernel of the natural map from Out(F_n) to GL(n,Z). We use combinatorial Morse theory to prove that T_n has an Eilenberg-MacLane space which is (2n-4)-dimensional and that H_{2n-4}(T_n,Z) is not finitely generated (n at least 3). In particular, this recovers the result of Krstic-McCool that T_3 is not finitely presented. We also give a new proof of the fact, due to Magnus, that T_n is finitely generated.

math.GT↗
arXiv2004

On the growth of iterated monodromy groups

Nekrashevych conjectured that the iterated monodromy groups of quadratic polynomials with preperiodic critical orbit have intermediate growth. We illustrate some of the difficulties that arise in attacking this conjecture and prove subexponential growth for the iterated monodromy group of $z^2+i$. This is the first non-trivial example supporting the conjecture.

math.GR↗
arXiv2004

Finiteness properties of soluble arithmetic groups over global function fields

Let G be a Chevalley group scheme and B<=G a Borel subgroup scheme, both defined over Z. Let K be a global function field, S be a finite non-empty set of places over K, and O_S be the corresponding S-arithmetic ring. Then, the S-arithmetic group B(O_S) is of type F_{|S|-1} but not of type FP_{|S|}. Moreover one can derive lower and upper bounds for the geometric invariants Σ^m(B(O_S)). These are sharp if G has rank 1. For higher ranks, the estimates imply that normal subgroups of B(O_S) with abelian quotients, generically, satisfy strong finiteness conditions.

math.GR↗
arXiv2003

Tangling and Braiding the Chessboard Complex

We describe a series of complexes that relate to the braid groups as the matching complexes relate to the symmetric groups. A modified construction applies as well to other complexes based on edge sets in graphs. We show that our constructions will yield Cohen-Macauley complexes provided the underlying complexes are Cohen-Macauley. Finally, we discuss a related series of complexes to provide some positive evidence that the braided Houghton groups, introduced by F. Degenhardt, are a series of groups with linearly increasing finiteness length as are the (unbraided) Houghton groups.

math.GT↗
arXiv2003

Thompson's Group F is not Minimally Almost Convex

We prove that Richard Thompson's group F is not minimally almost convex with respect to the two standard generators. This improves upon a recent result of S. Cleary and J. Taback. We make use of the forest diagrams for elements of F introduced by J. Belk and K. Brown. These diagrams seem particularly well-suited for understanding the Cayley graph for the two standard generators.

math.GR↗
arXiv2002

Finiteness Properties of S-Arithmetic Groups - a Survey

We give an overview about finiteness properties of soluble S-arithmetic groups. Both, the number field case and the function field case are covered. The main result is: If B is a Borel subgroup in a Chevalley group and R is an S-arithmetic ring, then the group B(R) has finiteness length |S|-1 in the function field case, and infinite finiteness length in the number field case.

math.GR↗