Alastair Litterick, Benjamin Martin
We prove a generalization of a conjecture of C. Marion on generation properties of finite groups of Lie type, by considering geometric properties of an appropriate representation variety and associated tangent spaces.
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We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method based on character varieties that can be used to distinguish between the profinite completions of certain groups.
For a simple algebraic group G in characteristic p, a triple (a,b,c) of positive integers is said to be rigid for G if the dimensions of the subvarieties of G of elements of order dividing a,b,c sum to 2dim G. In this paper we complete the proof of a conjecture of the third author, that for a rigid triple (a,b,c) for G with p>0, the triangle group T_{a,b,c} has only finitely many simple images of the form G(p^r). We also obtain further results on the more general form of the conjecture, where the images G(p^r) can be arbitrary quasisimple groups of type G.
Generalised triangle groups of type (2,3,2) with no cyclic essential representations
本研究针对Roseberger关于广义三角群的猜想，在两种例外情形下验证了无非循环本质表示的(2,3,2)型广义三角群满足该猜想。
A triangle group is denoted by $Δ(p,q,r)$ and has finite presentation $$ Δ(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $Δ(p,q,r)$ that was used in Everitt's proof of Higman's 1968 conjecture that every Fuchsian group has amongst its homomorphic images all but finitely many alternating groups. We see that some of these compositions must give imprimitive representations, and in particular situations, where the representations being composed are all equivalent copies of an alternating group in the same de…
Deformation theory and finite simple quotients of triangle groups II
本文是我们的第一篇文章[10]的继续，在这篇文章中，我们展示了如何利用表示簇的变形理论来研究三角形群的有限单直子。而在第一部分，我们主要使用变形的主同态从${\rm SO}(3,\R)$，在这一部分中，我们使用${\rm PGL}_2(\R)$以及变形的表示，这是非常不同的主同态。
There is a compactification of the space of representations of a finitely generated group into the groups of isometries of all spaces with $Δ$-thin triangles. The ideal points are actions on $\mathbb R$-trees. It is a geometric reformulation and extension of the Culler-Morgan-Shalen theory concerning limits of representations into $\operatorname{SL}(2,{\mathbb C})$ and more generally $\operatorname{O}(n, 1)$. This paper was written and circulated in the early 90's, but never published.
Algebraic Representations of Ergodic Actions and Super-Rigidity
我们重新回顾Margulis-Zimmer超刚性，并提供一些推广。特别地，我们针对任意域上具有绝对值的代数群，得到了高秩群或群的乘积中格的超刚性结果。我们还得到了一类广泛的群体的混合行动的上循环超刚性的结果。我们的方法是基于遍历行动的代数表示的系统研究。
Automorphism groups of simplicial complexes and rigidity for uniformly bounded representations
考虑自反巴拿赫空间的L^p-上同调，给出了希尔伯特空间上系数为一致有界表示的1-上同调为零的一个谱条件.
Anosov triangle reflection groups in SL(3,R)
本文研究了SL(3,R)中紧致双曲三角反射群的Anosov表示，证明其存在性条件及非凸边界映射特性。
Characters of the Grothendieck-Teichmueller group through rigidity of the Burau representation
我们提出的例子，字符的绝对伽罗瓦群的数域，可以通过他们的行动恢复自同构的profinite完成的辫群，使用“刚性”的方法。我们用来恢复它们的方法是通过辫群的经典表示，特别是通过Burau表示。这使得一个扩展这些字符的Grothendieck-Teichmueller群。
Long, Reid, and Thistlewaite have shown that some groups generated by representations of the $Δ334$ triangle group in $SL_3({\mathbb Z})$ are thin, while the status of others is unknown. In this paper we take a new approach: for each group we introduce a new graph that captures information about representations of $Δ334$ in the group. We provide examples of our graph for a variety of groups, and we use information about the graph for $SL_3({\mathbb Z}/2{\mathbb Z})$ to show that the chromatic number of the graph for $SL_3({\mathbb Z})$ is at most eight. By generating a portion of the graph for…
Quasiisometric embeddings between right-angled Artin groups: rigidity
通过引入定义图上的分支条件，研究了右抵消Artin群之间准等距嵌入的刚性问题，包括对准等距嵌入到树的积中的障碍、定义图的刚性特征分类以及高维RAAG作为准等距嵌入接收器的不可能性。
Categorical forms of the rigid direct-system representation for finite local-unit-aligned totally ordered monoids
本文研究有限局部单位对齐全序幺半群的刚性直接系统表示的范畴性，证明其对严格块态射和固有$\ au$-相容同态均为自函子，并给出相应等价性。
Entropy rigidity for cusped Hitchin representations
本文针对几何有限Fuchsian群的带尖点Hitchin表示建立熵刚性定理，引入(1,1,2)-超横截群类与射影可见群横截表示理论，推广了多个相关已有结论。
For any finite group $G$, a natural question to ask is the order of the smallest possible automorphism group for a Cayley graph on $G$. A particular Cayley graph whose automorphism group has this order is referred to as an MRR (Most Rigid Representation), and its Cayley index is a numerical indicator of this value. Study of GRRs showed that with the exception of two infinite families and seven individual groups, every group admits a Cayley graph whose MRR is a GRR, so that the Cayley index is 1. The full answer to the question of finding the smallest possible Cayley index for a Cayley graph on…
Fields of definition for triangle groups as Fuchsian groups
本文研究紧双曲三角形群在迹域上的定义域问题，证明恰好有十一个这样的群共轭于PSL₂(K)的子群，且对任意全实域L不存在额外情形，从而回答了Waterman–Machlachlan的问题并证实了McMullen的五个相关猜想。
Critical Exponent Rigidity for $Θ-$positive Representations
本文研究了Θ-正表示的临界指数刚性，证明了在几何有限情况下，晶格的临界指数等于一。
Representation gaps of rigid planar diagram monoids
本文定义了三类平面图幺半群的非枢轴版本，通过计算表示间隙与间隙比评估其密码学适用性，发现非枢轴幺半群通常更不适合密码学用途。
Existential Inclusions of Bi-exact Groups are Conjugacy Representation Rigid
本文证明非顺从双精确群的存在性嵌入中，非平凡共轭交是顺从的，进而结合Bekka和Kalantar的工作，得出拟正则表示的弱等价类在自共轭子群中决定共轭类。
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