## Olivetti Club

Unfortunately, groups were introduced to me as sets equipped with a binary operation that satisfies a few properties.

Nevertheless, I persisted in my studies and learned about symmetry groups of geometric objects, and moreover that

every group is the group of symmetries of some object, namely its *Cayley graph*.

Equivalence is an important theme in math, and the geometric group theoretic flavor of equivalence comes from *quasi-isometries*.

$X$ and $Y$, two metric spaces, are *quasi-isometric* provided that there is a map $f:(X, d_X)\to(Y, d_Y)$ and constants $A\geq 1$, $B,C\geq 0$ such that

$$\forall\, p, q\in X : \frac{1}{A}d_X(p,q)-B \leq d_Y(f(p),f(q)) \leq A d_X(p,q)+B,$$

$$\forall \, y\in Y : \exists\, x\in X : d_Y(y, f(x))\leq C.$$

Perhaps you are wondering why metric spaces were mentioned all of a sudden.

Excellent question... in this talk we will break down the barriers between groups and graphs and spaces.

Consequentially, we will be able to use group theory to study geometry and vice versa via the *Milnor-Schwarz lemma*.

The goal will be to motivate, explicate, and manipulate the italicized phrases.

Everyone is welcome.

Don't write your abstract as an acrostic, kids.