Sh. Mozes showed that the commensurator of the lattice ${\rm PSL}_2 \bigl({\bf F}_p[t{}^{-1}] \bigr)$ is dense in the full automorphism group of the Bruhat-Tits tree of valency $p+1$, the latter group being much bigger than ${\rm PSL}_2 \bigl({\bf F}_p((t)) \bigr)$. By G.A. Margulis' criterion, this density is a generalized arithmeticity result. We show that the density of the commensurator holds for many tree-lattices among those called of Nagao type by H. Bass and A. Lubotzky. The result covers many lattices obtained via Moufang twin trees.

2004-02-18math.GR

Integrability of Moufang Foundations - A Contribution to the Classification of Twin Buildings

Buildings have been introduced by J. Tits in order to study semi-simple algebraic groups from a geometrical point of view. One of the most important results in the theory of buildings is the classification of irreducible spherical buildings of rank at least 3. About 25 years ago, M. Ronan and J. Tits defined the class of twin buildings, which generalize spherical buildings in a natural way. The motivation of their definition is provided by the theory of Kac-Moody groups. A 2-spherical twin building is uniquely determined by its local structure in almost all cases: The so-called foundation is t…

2014-10-20math.GR

Generators of nonassociative simple Moufang loops over finite prime fields

We present an elementary proof that the nonassociative simple Moufang loops over finite prime fields are generated by three elements. In the last section, we conclude that integral Cayley numbers of unit norm are generated multiplicatively by three elements.

2007-01-24math.GR

Abelian-by-cyclic Moufang loops

We use groups with triality to construct a series of nonassociative Moufang loops. Certain members of this series contain an abelian normal subloop with the corresponding quotient being a cyclic group. In particular, we give a new series of examples of finite abelian-by-cyclic Moufang loops. The previously known [A. Rajah, J. Alg., 235 (2001), 66-93] loops of this type of odd order 3q^3, with prime q congruent to 1 mod 3, are particular cases of our series. Some of the examples are shown to be embeddable into a Cayley algebra.

2012-02-15math.GR

Trees, contraction groups, and Moufang sets

We study closed subgroups $G$ of the automorphism group of a locally finite tree $T$ acting doubly transitively on the boundary. We show that if the stabiliser of some end is metabelian, then there is a local field $k$ such that $\mathrm{PSL}_2(k) \leq G \leq \mathrm{PGL}_2(k)$. We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if $G$ is (virtually) a rank one simple $p$-adic analytic group for some prime $p$. A key point is that if some contraction group is closed, then $G$ is boundary-Moufang, meaning that the boundary $\partial T$ is a…

2012-12-03math.GR

On topological twin buildings and topological split Kac-Moody groups

We prove that a two-spherical split Kac-Moody group over a local field naturally provides a topological twin building in the sense of Kramer. This existence result and the local-to-global principle for twin building topologies combined with the theory of Moufang foundations as introduced and studied by Mühlherr, Ronan, and Tits allows one to immediately obtain a classification of two-spherical split Moufang topological twin buildings whose underlying Coxeter diagram contains no loop and no isolated vertices.

2013-09-29math.COmath.GNmath.GR

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纯孪生群与rondle的换位子

Commutators of pure twin groups and rondles

Shinshu University（信州大学）

本文建立了关于三点修改的有限阶rondle不变量与纯孪生群子群的下中心级数之间的关系，并构造了无穷多个无穷序列的素平面曲线，使得每个序列中的曲线在任意固定阶数下共享相同的不变量。

2026-05-24math.GRmath.GT