اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe asymptotic geometry of right-angled Artin groups, I
188
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study atomic right-angled Artin groups -- those whose defining graph has no cycles of length less than five, and no separating vertices, separating edges, or separating vertex stars. We show that these groups are not quasi-isometrically rigid, but that an intermediate form of rigidity does hold. We deduce from this that two atomic groups are quasi-isometric iff they are isomorphic.
قيم البحث
اقرأ أيضاً
The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible s
pherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of locally reducible Artin groups, which includes all $2$-dimensional Artin groups, and for spherical Artin groups of any type other than $E_6$, $E_7$, $E_8$. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.
We show that if a right-angled Artin group $A(Gamma)$ has a non-trivial, minimal action on a tree $T$ which is not a line, then $Gamma$ contains a separating subgraph $Lambda$ such that $A(Lambda)$ stabilizes an edge in $T$.
We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of l
ength $n$) with $ngeq 5$. We construct another eight forbidden graphs and show that every graph $K$ on $le 8$ vertices either contains one of our examples, or contains a hole of length $ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a RAAG to contain no hyperbolic surface subgroups. We prove that for one of these forbidden subgraphs $P_2(6)$, the right angled Artin group $A(P_2(6))$ is a subgroup of a (right angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).
We characterize when (and how) a Right-Angled Artin group splits nontrivially over an abelian subgroup.
For each natural number $d$ we construct a $3$-generated group $H_d$, which is a subdirect product of free groups, such that the cohomological dimension of $H_d$ is $d$. Given a group $F$ and a normal subgroup $N lhd F$ we prove that any right angled
Artin group containing the special HNN-extension of $F$ with respect to $N$ must also contain $F/N$. We apply this to construct, for every $d in mathbb{N}$, a $4$-generated group $G_d$, embeddable into a right angled Artin group, such that the cohomological dimension of $G_d$ is $2$ but the cohomological dimension of any right angled Artin group, containing $G_d$, is at least $d$. These examples are used to show the non-existence of certain universal right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular we show that for every $nin mathbb{N}$ there exists $delta(n) in mathbb{N}$ such that any $n$-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the $delta(n)$-th direct power of the free group of rank $2$. As another corollary we derive that any $n$-generated finitely presented residually free group embeds into the direct product of at most $delta(n)$ limit groups.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
