# All Questions

Tagged with gt.geometric-topology gr.group-theory

236 questions

**5**

votes

**1**answer

155 views

### “Dimension” of discrete subgroups of infinite covolume in Lie groups

Let $G$ be a semisimple Lie group with finite center, $K$ a maximal compact
subgroup, and $d=\dim(G/K)$. Let $\Gamma$ be a non-cocompact discrete subgroup of $G$. [Edit: assume that $\Gamma$ is ...

**3**

votes

**0**answers

103 views

### Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$

Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...

**2**

votes

**1**answer

76 views

### A Backtrack as a Single Word in a Group Presentation yields a Complex that isn't of the Same Homotopy Type?

By "backtrack" I mean a subword of a relator in a group presentation of the form $x x^{-1}$.
Let $X = \langle a \rangle$ as a presentation complex.
Let $Y = \langle a$ | $aa^{-1} \rangle$ as a ...

**4**

votes

**0**answers

66 views

### Hilbert space compression of lamplighter over lamplighter groups

$C_2 \wr \mathbb{Z}$ is the lamplighter group but I'm currently looking at the lamplighter group with this group as a base space.
Question: Consider the group $C_2 \wr (C_2 \wr \mathbb{Z})$, what is ...

**14**

votes

**0**answers

209 views

### What is the centralizer of a Coxeter element?

Let $(W,S)$ be a Coxeter system (of finite rank) and $c \in W$ a Coxeter element.
If $W$ is finite, then the centralizer $C_W(c)$ is the cyclic group generated by $c$ (e.g. see the book "Reflection ...

**4**

votes

**1**answer

119 views

### non-proper parabolic isometries of hyperbolic spaces

In his seminal paper on hyperbolic groups (see Section 8.1) Gromov defines an isometry $f$ of a hyperbolic space $X$ to be parabolic if the orbit of any point $x\in X$ under the action of $\langle f\...

**4**

votes

**2**answers

205 views

### Is the *unreduced* Burau representation unitary?

In a 1984 paper, Squier wrote down an explicit Hermitian form $J$ relative to which the reduced Burau representation is unitary. The form is valued in the ring $\mathbb{Z}[s^{\pm 1}]$, where $s^2 = t$ ...

**7**

votes

**0**answers

175 views

### Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...

**8**

votes

**2**answers

325 views

### Hyperbolic $3$-manifold groups that embed in compact Lie groups

Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple ...

**6**

votes

**2**answers

176 views

### Generalization of Bieberbach's second theorem

Let $F_0$ and $F_1$ be compact flat manifolds of dimensions $k$ and $m$, respectively, where $k \geq m$. Suppose $f : \pi_1(F_0) \to \pi_1(F_1)$ is a surjective homomorphism. Consider the covering ...

**31**

votes

**2**answers

608 views

### Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...

**6**

votes

**0**answers

115 views

### Automorphism groups of cocompact Fuchsian groups as mapping class groups

Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...

**12**

votes

**1**answer

227 views

### Equivalence of surjections from a surface group to a free group

Let $g \geq 2$. Let $S = \langle a_1,b_2,...,a_g,b_g | [a_1,b_1] \cdots [a_g,b_g] \rangle$ be the fundamental group of a genus $g$ surface and let $F_g$ be a free group with $g$ generators. Given ...

**4**

votes

**0**answers

161 views

### Uniqueness of the boundary of a hierarchically hyperbolic group

Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...

**5**

votes

**0**answers

130 views

### The preimage of bounded real intervals under homomorphisms on hyperbolic groups

Let $G$ be a hyperbolic group with a fixed (finite, symmetric) generating set and suppose that $\varphi : G \to \mathbb{R}$ is a group homomorphism. Write $W_n = \{ g \in G: |g|=n\}$, where $|g|$ ...