Geometric group action

In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.

Definition

In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions:

  1. Each element of G acts as an isometry of X.
  2. The action is cocompact, i.e. the quotient space X/G is a compact space.
  3. The action is properly discontinuous, with each point having a finite stabilizer.

Uniqueness

If a group G acts geometrically upon two geometries X and Y, then X and Y are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which G acts geometrically is quasi-isometric to the Cayley graph of G.

Examples

Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

gollark: The basic raw-value-reading bit, which I could easily implement myself in python, works fine, but some DMP code I had to copy from the arduino library (which is probably why it breaks horribly?) causes æææ.
gollark: ```MPU6050 3-axis acceleromter example programASAN:DEADLYSIGNAL===================================================================2032==ERROR: AddressSanitizer: SEGV on unknown address 0x00000013 (pc 0x0014255c bp 0x7ea10018 sp 0x7ea10000 T0) #0 0x14255b (/home/pi/mputest/a.out+0x14255b) #1 0xcdec3 (/home/pi/mputest/a.out+0xcdec3) #2 0x76c256bf (/lib/arm-linux-gnueabihf/libc.so.6+0x2c6bf)AddressSanitizer can not provide additional info.SUMMARY: AddressSanitizer: SEGV (/home/pi/mputest/a.out+0x14255b) ==2032==ABORTING```This is very unhelpful.
gollark: Now, *technically* I could implement all the filtering and sensor fusion algorithms and calibration myself in python, however no.
gollark: So, I want to read some values from an I2C device. Now, you might think "foolish gollark that's something like 50 lines of python at absolute most", and it is except to get anything but raw values I need to use some on-chip "digital motion processor" which is extremely poorly documented.
gollark: Yes, I was replying to ubq.

References

  • Cannon, James W. (2002). "Geometric Group Theory". Handbook of geometric topology. North-Holland. pp. 261–305. ISBN 0-444-82432-4.


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