I'm trying to give an example of Dehn's algorithm on a 2-torus and to do so I want to find a word about 20 letters or so in length that is equal to 1 so that I can apply the algorithm. I'm having trouble doing this. I can't seem to find a proper hyperbolic tiling of octagons on the internet so that I could trace out a path on the covering space, and I can't seem to be able to do it just looking at a 2-torus (how would I know if it's null homotopic?). Any ideas on how to generate this? Thanks!
2026-03-25 17:39:00.1774460340
How can I find a medium-length word equal to 1 on a 2-torus?
35 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ALGEBRAIC-TOPOLOGY
- How to compute homology group of $S^1 \times S^n$
- the degree of a map from $S^2$ to $S^2$
- Show $f$ and $g$ are both homeomorphism mapping of $T^2$ but $f$ is not homotopy equivalent with $g.$
- Chain homotopy on linear chains: confusion from Hatcher's book
- Compute Thom and Euler class
- Are these cycles boundaries?
- a problem related with path lifting property
- Bott and Tu exercise 6.5 - Reducing the structure group of a vector bundle to $O(n)$
- Cohomology groups of a torus minus a finite number of disjoint open disks
- CW-structure on $S^n$ and orientations
Related Questions in HOMOTOPY-THEORY
- how to prove this homotopic problem
- Are $[0,1]$ and $(0,1)$ homotopy equivalent?
- two maps are not homotopic equivalent
- the quotien space of $ S^1\times S^1$
- Can $X=SO(n)\setminus\{I_n\}$be homeomorphic to or homotopic equivalent to product of spheres?
- Why do $S^1 \wedge - $ and $Maps(S^1,-)$ form a Quillen adjunction?
- Is $S^{n-1}$ a deformation retract of $S^{n}$ \ {$k$ points}?
- Connection between Mayer-Vietoris and higher dimensional Seifert-Van Kampen Theorems
- Why is the number of exotic spheres equivalent to $S^7,S^{11},S^{15},S^{27}$ equal to perfect numbers?
- Are the maps homotopic?
Related Questions in HYPERBOLIC-GEOMETRY
- Sharing endpoint at infinity
- CAT(0) references request
- Do the loops "Snakes" by M.C. Escher correspond to a regular tilling of the hyperbolic plane?
- How to find the Fuschian group associated with a region of the complex plane
- Hyperbolic circles in the hyperbolic model
- Area of an hyperbolic triangle made by two geodesic and an horocycle
- Concavity of distance to the boundary in Riemannian manifolds
- Differential Equation of Circles orthogonal to a fixed Circle
- Is there a volume formula for hyperbolic tetrahedron
- Can you generalize the Triangle group to other polygons?
Related Questions in FUNDAMENTAL-GROUPS
- Help resolving this contradiction in descriptions of the fundamental groups of the figure eight and n-torus
- $f$ has square root if and only if image of $f_*$ contained in $2 \mathbb Z$
- Homotopic maps between pointed sets induce same group homomorphism
- If $H \le \pi_1(X,x)$ is conjugate to $P_*(\pi_1(Y, y))$, then $H \cong P_*(\pi_1(Y, y'))$ for some $y' \in P^{-1}(x)$
- Calculating the fundamental group of $S^1$ with SvK
- Monodromy representation.
- A set of generators of $\pi_1(X,x_0)$ where $X=U\cup V$ and $U\cap V$ path connected
- Is the fundamental group of the image a subgroup of the fundamental group of the domain?
- Showing that $\pi_1(X/G) \cong G$.
- Fundamental group of a mapping cone
Related Questions in LOW-DIMENSIONAL-TOPOLOGY
- Getting a self-homeomorphism of the cylinder from a self-homeomorphism of the circle
- Does $S^2\times[-1,1]$ decompose as $B^3\#B^3$
- Homologically zero circles in smooth manifolds
- Can we really move disks around a compact surface like this?
- Why is this not a valid proof of the Poincare Conjecture?
- Regarding Surgery and Orientation
- Can a one-dimensional shape have volume?
- The inside of a closed compact surface $\sum_g$
- How do you prove that this set is open?
- Understanding cobordisms constructed from a Heegaard triple
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
The universal covering space is the hyperbolic plane, which is contractible, so every closed path is null homotopic.
So, "all" you need to do is draw a closed path in the 1-skeleton of the tiling on the hyperbolic plane.
One reason this is hard to visualize is that area grows so swiftly in the hyperbolic plane, in fact it grows exponentially as a function of radius. So you quickly run out of room to visualize, because the count of octagons grows exponentially as you move away from a central octagon. If you start with one central octagon in the tiling, then the number of octagons that touch it is $48$, which is a lot to draw. And then the number of additional octagons that touch those is $272$ (I think, it's hard to count), which is impossible to draw. The number of octagons in each additional layer continues to grow with an exponential base between 5 and 6.
This document has a picture of the 1st layer of 48 octagons. And, as I said, the 2nd layer is impossible to visualize.
If you want an example to play with, you might pick something which doesn't eat up volume so quickly. I would recommend the $(2,3,7)$ tiling of the hyperbolic plane, which is invariant under the Coxeter group $$\langle a,b,c \mid a^2=b^2=c^2=(ab)^2=(bc)^3=(ca)^7=1\rangle $$ Its fundamental domain is the triangle with angles $\pi/2$, $\pi/3$, $\pi/7$, with group generators $a,b,c$ being the reflections in the three sides. This tiling has only $14$ tiles touching the central one.
I'll add one more thing, though. Drawing, or attempting to draw, any one of these tilings should lead to an excellent intuition for the proof of Dehn's algorithm.