Gemara

59a (Mishnah): A Cheresh gestures or reacts to gestures (to make transactions);

Ben Beseirah says, he is Kofetz or responds to Kefitzah for Metaltelim.

Version #1 (Rav Nachman): They argue only about Metaltelim, but all agree that gestures suffice for Gitin.

Even though the Mishnah explicitly says that they argue about Metaltelim, one might have thought that they argue even about Metaltelim. Rav Nachman teaches that this is not so.

Version #2 (Rav Nachman): Just like they argue about Metaltelim, they also argue about Gitin.

71a (Rav Kahana): If a Cheresh (deaf person) can communicate by writing, we write and give a Get to his wife if he requests this.

Question (Rav Yosef): We already know this from our Mishnah!

(Mishnah): If a man became unable to speak and others asked him if we should write a Get for his wife and he nodded his head, we test him three times. If he answers properly no and yes, we write and give the Get.

Answer (R. Zeira): The Mishnah discusses a mute. He is not like a (total) Cheresh:

(Beraisa): One who can hear but not speak is a mute. He is like a Pike'ach (sound-minded person) in every respect.

Question (against Rav Kahana - Beraisa): Regarding Metaltelim we rely on gestures, Kefitzah (lip movements) and writing of a Cheresh, but not for Gitin.

Answer: Tana'im argue about whether or not he can divorce:

(Seifa - R. Shimon ben Gamliel): Someone deaf from birth cannot divorce, but if a hearing man became deaf, he writes (a Get) and witnesses sign it.

72a (Beraisa): If a scribe wrote a Get and witnesses signed it, it is Batel unless the scribe and witnesses heard the husband's voice command them.

This teaches unlike Rav Kahana (71a, who said that if a Cheresh can communicate by writing, we write and give a Get to his wife).

Yevamos 112b (Mishnah): If a Cheresh married a Pikachas, or a Pike'ach married a Chereshes, he can choose to divorce her or stay married. Just like he married her through gesturing, he may divorce by gesturing.

Rishonim

The Rif and Rosh (Gitin 27b and 5:18) bring the Mishnah (Gitin 59a) and both versions of Rav Nachman.

Rosh (ibid.): The Rif brings Rav Nachman's teaching. This implies that he rules like Ben Beseira; he is lenient like the second version. I disagree. Throughout Shas the Amora'im explain Tana'im argue in order to understand the Mishnah, even if the Halachah does not follow the Tana they explain. One cannot be lenient without a proof.

Beis Yosef (EH 121 DH u'Mah she'Chosav Rabeinu): The Rosh says that the Rif holds that Kefitzah is weaker than gesturing, and the Halachah follows Ben Beseira. He must say that the Mishnah (Yevamos 112b) discusses marriage and divorce through gesturing because all agree to this, but really even Kefitzah suffices. The Rambam mentions only gesturing. Presumably, he learns from that Stam Mishnah. Perhaps the Rif agrees; he brought Rav Nachman's words only to show that we require gesturing also for Gitin. We are not lenient to allow Kefitzah to prevent Igun.

Rif and Rosh (Gitin 34a and 6:3): Regarding Metaltelim we rely on gestures and Kefitzah of a Cheresh and what he writes, but not for Gitin. This applies only to a Pike'ach who became deaf after marriage, but if a Cheresh married, we check him. If he wants, we divorce his wife. A Mishnah teaches that if a Cheresh married a Pikachas or vice-versa, he can choose to divorce or keep her. Just like he married her through gesturing, he may divorce her by gesturing. If a Pike'ach married a Pikachas and she went deaf, he may divorce her. If she went insane, he may not divorce her. If he went deaf or insane, he may never divorce her. Another Mishnah teaches that a Cheresh gestures or reacts to gestures. Ben Beseirah says, he is Kofetz or responds to Kefitzah for Metaltelim. Just like they argue about Metaltelim, they also argue about Gitin.

Ran (DH Romez): Gesturing is with the hand and head. Kefitzah is with the lips; it is less clear than gesturing.

Aruch (Erech Remez): Remizah (gesturing) is with the eye or hand. It is the Targum of "Koretz Ayin" and "Sholach Etzba".

Rif and Rosh (Gitin 33a and 7:19): If a scribe wrote a Get... it is valid only if the scribe and witnesses heard the husband's voice command them. This teaches unlike Rav Kahana, who said that if a Cheresh can communicate by writing, we write and give a Get to his wife.

Gra (EH 120:15): Rav Kahana holds like R. Shimon ben Gamliel. The Beraisa unlike him shows that Chachamim argue with R. Shimon ben Gamliel, like R. Yochanan taught.

Rosh (ibid.): R. Tam says that even if a Pike'ach wrote that we should write and give a Get to his wife, we do not do so. The Mishnah allows writing a Get for one who nodded his head, for nodding is better than writing. Alternatively, it is a leniency for one who became unable to speak, to enable his wife to remarry.

Rambam (17): If a Pike'ach married and became a Cheresh, he cannot divorce until he gets healthy. But if a Cheresh married, he can divorce by gesturing. His Kidushin is not mid'Oraisa. Just like he married through gesturing, he may divorce by gesturing.

Poskim

Shulchan Aruch (EH 120:5): A scribe and witnesses may not write and sign a Get unless they hear from the husband, whether he is Pike'ach, became mute, cannot hear, and whether or not he can speak.

Gra (ibid.): The Poskim rule like Answer #1. The Sugya (Rav Yosef and R. Zeira) hold like Answer #2, but the Beraisa (unlike Rav Kahana) refutes them.

Beis Yosef (DH Kasuv b'Kundrisin): The Ri'az (Gitin 7:1:6) says that a man may write a command to write and sign a Get and give it to the witnesses. If he sends it to the witnesses through another party, it is a Safek. It seems that he holds like the Rambam, who allows a written command, but does not allow making a Shali'ach to make a Shali'ach.

Shulchan Aruch (121:6): We rely on gestures to divorce only for one who became mute, or for a mute who can hear. A total Cheresh cannot divorce through gestures if he married when he was Pike'ach, or if a Yevamah fell to him from a Pike'ach brother. If a Cheresh married, he can divorce through gestures even if she is a Pikachas.

Beis Yosef (DH u'Mah): The Yerushalmi says that 'just like he married her through gesturing, he may divorce by gesturing' applies to a Cheresh who was Mekadesh with money, but not to Kidushei Bi'ah. The Nimukei Yosef (Yevamos 41a DH Cheresh) brings this, but the Poskim do not.

Beis Shmu'el (9) and Taz (10): The Bach cites Tosfos to say that a written command is better than gesturing. This is wrong. Tosfos said so only at the beginning of the Sugya. All agree that the Sugya concludes that writing is worse, except for the Rambam who says that it is equal. If he married through Kefitzah he may divorce similarly, even though we normally require gesturing.

Note: Perhaps one who was Mekadesh through Kefitzah may divorce similarly, but we do not allow this after Nisu'in, for perhaps relations shows desire for Kidushin more than Kefitzah shows desire to divorce.

Beis Shmu'el (9): However, if he married through writing, he may divorce similarly.

Rebuttal (R. Akiva Eiger 4): A written command is invalid. This is not because it does not testify to sanity. Rather, the scribe must be commanded orally or through bodily motion!