geometric-group-theory

An inequality from the “Interlacing-1” paper

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This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132) For the argument to work one seems to need that for the symmetric ±1 signing adjacency matrix of the graph, As, it holds that, max−root(∑s∈{0,1}mdet(xI−As))≤max−root(Es∈{0,1}m[det(xI−As)]) But why should this inequality be true? and the argument works because the polynomial on the … Read more

finite generation and indeterminates

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It’s a soft, maybe naive question. Let $G\le \text{SL}(n,\mathbb{Z}[x])$ be a subgroup. Can we prove the finite generation of $G$ by evaluating $x$ at some (possibly infinite) class of complex numbers? For example, for every $m\in\mathbb{Z}$, define the evaluating maps at $x=m$, $\phi_m:\text{SL}(n,\mathbb{Z}[x])\to\text{SL}(n,\mathbb{Z})$. If for each $m\in\mathbb{Z}$, $\phi_m(G)$ is finitely generated, then can we conclude … Read more

Visual boundary vs. ideal boundary of hyperbolic manifolds?

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I apologize in advance if this is elementary; I have very little experience with hyperbolic manifolds, but I’m using hyperbolic manifolds for part of a current project. Given a discrete torsion-free subgroup of isometries acting on hyperbolic space Γ↪Isom(Hn), then Γ also acts on the boundary sphere Sn−1 of hyperbolic space. As I understand it, … Read more

Words Growth in Finite Groups

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Let G be a finite group with a set of generators and let Γ be its Cayley Graph. Let bk be the number of elements in the ball of radius k. I am interested in what is known about the sequence (bk). (Of course its length is the diameter of the graph.) This is not … Read more

Subgroup membership problem for Noetherian groups

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I am interested in the status of the subgroup membership problem (MP) for finitely presented Noetherian groups. That is, given a finite presentation ⟨X,R⟩ for a Noetherian group, Input:u,v1,…,vn words in X±Decide:u∈⟨v1,…,vn⟩ Of course, relevant to this is the possibility that every finitely presented Noetherian group is virtually polycyclic (and so has solvable MP). Is this still … Read more

Hilbert space compression of lamplighter over lamplighter groups

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C2≀Z is the lamplighter group but I’m currently looking at the lamplighter group with this group as a base space. Question: Consider the group C2≀(C2≀Z), what is its compression exponent? [EDIT: I just noticed this is mentioned as an open question by Naor & Peres in Lp Compression, traveling salesmen and stable walks. I also … Read more

Amenability of the group of outer automorphisms of a connected compact Lie group

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Forgive my ignorance on the Lie theory. I have the following questions in my current work concerning a certain property of compact connected Lie groups. First, allow me to fix some notations. Let $G$ be a connected compact Lie group, $\mathfrak{g}$ its Lie algebra. Then $\operatorname{Aut}(G)$, the group of (smooth) automorphisms of the $G$, is … Read more

Branched 2-fold covering over edge of a 3-orbifold

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I am reading the paper “On some generalized triangle groups and three-dimensional orbifolds” by Vinberg, Mennike and Khelling (Tran. Moscow Math. Soc. 1995 (56)). Let k,l,m>0, at most one of them equal to 2 and (k,l,m)≠(2,3,3). Let ˜Γ=⟨X,Y,T | Xk=Yl=(XYYX)m=T2=1,XT=X−1,YT=Y−1⟩. The paper claims the following complex is a fundamental domain for the action of Γ on hyperbolic … Read more

Do positive-density subgroups intersect nontrivially?

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Let G be an infinite finitely generated group and S a generating set. Define density with respect to the sequence of balls Sn. If H1,H2≤G have positive density, must H1∩H2 be (a) nontrivial? (b) positive-density? Obviously if H1 and H2 have finite index then so does H1∩H2, so such subgroups are not relevant. It is … Read more

Amalgamated subgroup of an HNN extension finitely generated

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Baumslag proved that if G=A∗CB is an amalgamated free product where A and B are finitely presented, G is finitely presented if and only if C is finitely generated. Similarly, by using Britton’s lemma, it can be shown that if H is an HNN extension ⟨D,t∣t−1bt=b,∀b∈L⟩, where L is a subgroup in D and D … Read more